A Theorem on Bayesian Updating

and

Applications to Communication Gameslowast

Navin Kartikdagger Frances Xu LeeDagger Wing Suensect

August 8 2019

Abstract

We develop a result on Bayesian updating Roughly when two agents have different pri-

ors each believes that a (Blackwell) more informative experiment will on average bring

the otherrsquos posterior closer to his own prior We apply the result to two models of strate-

gic communication verifiable disclosure and costly falsification Under some conditions

sendersrsquo information revelation are strategic complements when concealing or falsifying

information is costly but strategic substitutes when disclosing verifiable information is

costly In the latter case (but not the others) a receiver can be worse off with additional

senders or better external information

lowastWe thank Nageeb Ali Ilan Guttman Keri Hu Alessandro Lizzeri Steve Matthews Ariel Pakes AndreaPrat Mike Riordan Satoru Takahashi Jianrong Tian Enrico Zanardo Jidong Zhou and various seminar andconference audiences for comments Sarba Raj Bartaula Dilip Ravindran and Teck Yong Tan provided excellentresearch assistance Kartik gratefully acknowledges financial support from the NSF (Grant SES-1459877)

daggerDepartment of Economics Columbia University E-mail nkartikcolumbiaeduDaggerQuinlan School of Business Loyola University Chicago E-mail francesxu312gmailcomsectFaculty of Business and Economics The University of Hong Kong E-mail wsueneconhkuhk

Contents

Page

1 Introduction 1

2 Information Validates the Prior 6

21 Binary states 7

22 Many states 9

3 Multi-Sender Disclosure Games 11

31 The model 12

32 A single-sender benchmark 14

33 Strategic substitutes and complements 17

34 Many senders 23

35 Sequential reporting 24

4 Costly Signaling 25

41 The model 25

42 Equilibrium and comparative statics 27

5 Conclusion 31

A Proofs 33

B Discussion of Theorem 2 41

References 43

C Supplementary Appendix (Not for Publication) 46

C1 Perfectly correlated signals in the disclosure application 46

C2 Non-linear utility functions in the disclosure application 47

C3 The role of linearity and MLRP in the signaling application 50

1 Introduction

Rational agents revise their beliefs upon receiving new information From an ex-ante point ofview however one cannot expect new information to systematically alter onersquos beliefs in anyparticular direction More precisely a fundamental property of Bayesian updating is that be-liefs are a martingale an agentrsquos expectation of his posterior belief is equal to his prior beliefBut what about an agentrsquos expectation of another agentrsquos posterior belief when their currentbeliefs are different Or intimately related should agents expect new information to system-atically affect any existing disagreement These questions are not only of intrinsic interestbut tackling them proves useful for analyzing common-prior environments with asymmetricinformation

This paper makes two contributions First we provide a result concerning the mutual ex-pectations of Bayesian agents who disagree on the distribution of a fundamental Second weuse this result to derive new insights about some canonical sender(s)-receiver communicationgames either with multiple senders or when a receiver obtains external information We iden-tify a unified mechanism that drives senders to reveal more information in some economicsettings and less in others in the latter cases a receiver can be harmed by having access tobetter external information or more senders

A theorem on Bayesian updating In Section 2 we develop the following result on Bayesianupdating (Throughout this introduction some technical details are suppressed) Let Ω =

0 1 be the possible states of the world Anne (A) and Bob (B) have mutually-known butpossibly-different priors over Ω βA isin (0 1) and βB isin (0 1) respectively where βi is theprobability individual i ascribes to state ω = 1 A signal s will be drawn from an informationstructure or experiment E given by a family of probability distributions pω(middot)ωisinΩ wherepω(s) is the probability of observing signal s when the true state is ω Anne and Bob agree onthe experiment Let βi(s) denote irsquos posterior (on ω = 1) after observing signal s derived byBayesian updating Let EE

i [βj(middot)] be irsquos ex-ante expectation of jrsquos posterior under experimentE where the expectation is taken over signals from irsquos point of view

Consider two experiments E and E that are comparable in the sense of Blackwell (19511953) specifically let E be a garbling of E or equivalently E be more informative than E 1 Weprove (Theorem 1) that for any i j isin AB

βi le (ge) βj =rArr EEi [βj(middot)] le (ge)EE

i [βj(middot)] (1)

1 Throughout binary comparisons should be understood in the weak sense unless indicated otherwise egldquogreater thanrdquo means ldquoat least as large asrdquo

1

In words if Anne holds a lower (resp higher) prior than Bob then Anne predicts that amore informative experiment will on average reduce (resp raise) Bobrsquos posterior by a largeramount than a less informative experiment Put differently Anne expects more informationto further validate her prior in the sense of bringing on average Bobrsquos posterior closer to herprior Of course Bob expects just the reverse We refer to this result as information validates theprior IVP hereafter

IVP subsumes the familiar martingale property of Bayesian updating To see this considerj = i in Equation 1 and take E to be an uninformative experimentmdashwhich leaves an agentrsquosposterior after any signal equal to his priormdashand take E to be any experiment IVP also impliesthat an agent expects another agentrsquos posterior to fall in between their two priors2

IVP implies an interesting point about disagreement When the disagreement between be-liefs βi and βj is quantified by the canonical metric |βi minus βj| a particular signal can lead tohigher posterior disagreement than the prior (By contrast as shown by Baliga Hanany andKlibanoff (2013) no signal can lead to polarization it cannot be that (βi minus βi)(βj minus βj) lt 0)Nevertheless IVP implies (Corollary 1) that both Anne and Bob predict that a more infor-mative experiment will on average reduce their posterior disagreement to a greater extentthan a less informative experiment in particular no matter the experiment they expect lowerposterior disagreement than their prior disagreement At the extreme both predict their pos-terior disagreement to be zero under a fully informative experimentmdasheven though they holddifferent expectations about what the otherrsquos posterior belief will be

Theorem 2 provides a generalization of IVP to non-binary state spaces using suitablestatistics of the agentsrsquo beliefs and likelihood-ratio ordering conditions on the agentsrsquo priorsand on the experiments

Applications to communication games IVP is a statistical result The second part of ourpaper shows why it is useful in strategic contexts The idea is that even in common-priorenvironments private information can lead to one agentmdashAnne an informed agentmdashholdinga different belief about a fundamental than anothermdashBob an uninformed agent Only Anneknows both their beliefs Annersquos strategic incentives may depend on how she expects newinformation to affect Bobrsquos belief This new information can be exogenous or endogenouseg owing to the strategic behavior of still other agents The belief difference between Anneand Bob can occur either on an equilibrium path because of Annersquos ldquopoolingrdquo behavior or

2 To see this note that (i) letting U denote an uninformative experiment EUi [βj(middot)] = βj and (ii) letting F

denote a fully informative experiment (one in which any signal reveals the state and hence leaves any twoagents with the same posterior no matter their priors) EF

i [βj(middot)] = βi The result follows using Equation 1 andthe fact that any experiment is more informative than U but less informative than F

2

off path because of a deviation by Anne as usual even off-path considerations will generallyaffect the on-path properties of an equilibrium

We develop these points in the context of two familiar classes of communication games

Voluntary disclosure Section 3 studies a family of voluntary disclosure games also referredto as persuasion games In these games biased sendersrsquo only instrument of influence onthe decision-maker (DM) is the certifiable or verifiable private information they are endowedwith see Milgrom (2008) and Dranove and Jin (2010) for surveys Senders cannot explicitlylie but can choose what information to disclose and what to withhold We consider a modelin which senders if informed draw signals that are independently distributed conditional onan underlying binary state that affects the DMrsquos payoff3 We allow for senders to either haveopposing or similar biases but assume that a senderrsquos payoff is state-independent and linear(increasing or decreasing) in the DMrsquos belief

Our focus is on how strategic disclosure interacts with message costs either disclosureor concealment of information can entail direct costs for each sender Disclosure costs arenatural when the process of certifying or publicizing information demands resources suchas time effort or hiring an outside expert there can also be other ldquoproprietary costsrdquo Asubset of the literature starting with Jovanovic (1982) and Verrecchia (1983) has modeledsuch a cost although primarily only in single-sender problems On the flip side there arecontexts in which it is the suppression of information that requires costly resources Besidesdirect costs there can also be a psychic disutility to concealing information or concealmentmay be discovered ex-post (by auditors whistleblowers or mere happenstance) and resultin negative consequences for the sender through explicit punishment or reputation loss Arecent example is the $70 million dollar fine imposed by the National Highway Traffic SafetyAdministration on Honda Motor in January 2015 because ldquoit did not report hundreds of deathand injury claims for the last 11 years nor did it report certain warranty and other claimsin the same periodrdquo (The New York Times 2015)

To determine his own disclosure it is essential for each sender to predict how othersendersrsquo messages will affect the DMrsquos posterior belief and how this depends on his ownmessage (disclosure or nondisclosure) Our methodological insight is to view sendersrsquo mes-sages as endogenously-determined experiments and bring the IVP theorem to bear This ap-proach allows us to provide a unified treatmentmdashregardless of whether senders have similar

3 We assume that senders are uninformed with some probability importantly an uninformed sender cannotcertify that he is uninformed This modeling device was introduced by Dye (1985) and Shin (1994ab) to preventldquounravelingrdquo (Grossman and Hart 1980 Milgrom 1981) and serves the same purpose here

3

or opposing biases and regardless of whether message costs are disclosure or concealmentcosts We thereby uncover new insights into the classic question of when competition pro-motes disclosure

Any senderrsquos equilibrium disclosure behavior follows a threshold rule of disclosing allsufficiently favorable signals We establish a simple but important benchmark without mes-sage costs any sender in the multi-sender disclosure game uses the same disclosure thresholdas he would in a single-sender game In other words there is a strategic irrelevance The intu-ition is that without message costs a senderrsquos objective is the same regardless of the presenceof other senders he simply wants to induce the most favorable ldquointerim beliefrdquo in the DMbased on his own message as this will lead to the most favorable posterior belief based on allsendersrsquo messages

How do message costs alter the irrelevance result Consider a concealment cost In asingle-sender setting a sender irsquos disclosure threshold will be such that the DMrsquos interpreta-tion of nondisclosure is more favorable than irsquos private belief at the thresholdmdashthis wedge isnecessary to compensate i for the concealment cost Nondisclosure thus generates an interimdisagreement between the DMrsquos belief and the threshold typersquos private belief Naturally dis-closure produces no such disagreement Now add a second sender j to the picture Our IVPtheorem implies that regardless of jrsquos behavior the threshold type of i predicts that jrsquos mes-sage will on average make the DMrsquos posterior less favorable to i as compared to the DMrsquosinterim belief following irsquos nondisclosure Consequently concealment is now less attractiveto sender i IVP further implies that jrsquos effect on i is stronger when j disclosure more iewhen j is more informative In sum sendersrsquo disclosures are strategic complements under aconcealment cost4

The logic reverses under a disclosure cost In a single-sender setting the DMrsquos interpre-tation of nondisclosure is now less favorable than irsquos private belief at the thresholdmdashthe gainfrom disclosing information must compensate i for the direct cost Reasoning analogously toabove IVP now implies that disclosure becomes less attractive when the other sender is moreinformative the threshold type of i expects the other senderrsquos message to make the DMrsquosposterior more favorable to i reducing the gains from disclosure Consequently sendersrsquodisclosures are strategic substitutes under a disclosure cost

These results have notable welfare implications In the case of concealment cost a DMalways benefits from an additional sender not only because of the information this sender

4 In a different model Bourjade and Jullien (2011) find an effect related to that we find under concealmentcost Loosely speaking ldquoreputation lossrdquo in their model plays a similar role to concealment cost in ours

4

provides but also because it improves disclosure from other senders In the case of disclosurecost however the strategic substitution result implies that while a DM gains some directbenefit from consulting an additional sender the indirect effect on other sendersrsquo behavioris deleterious to the DM In general the net effect is ambiguous it is not hard to constructexamples in which the DM is made strictly worse off by adding a sender even if this senderhas an opposite bias to that of an existing sender Thus competition between senders neednot increase information revelation nor benefit the DM5 The DM can even be made worseoff when disclosure costs become lower or a sender is more likely to be informed althougheither change would help the DM in a single-sender setting

Costly signaling games Our IVP theorem is also useful in games of asymmetric informa-tion even when the information asymmetry is eliminated in equilibrium We develop suchan application in Section 4 For concreteness we consider information transmission with ly-ing costs (Kartik 2009) but we also discuss how our themes apply to canonical signalingapplications like education signaling (Spence 1973)

Suppose a sender has imperfect information about a binary state and can falsify or ma-nipulate his information by incurring costs A receiver makes inferences about the state basedon both the senderrsquos signal and some other exogenous information6 The sender has linearpreferences over the receiverrsquos belief about the state Due to a bounded signal space thereis incomplete separation across sender types we focus on equilibria in which ldquolowrdquo typesseparate and ldquohighrdquo types pool What is the effect of better exogenous information (in theBlackwell sense) on the senderrsquos signaling

Even sender types who separate in equilibrium can induce an incorrect interim belief (iethe receiverrsquos belief based only on the senderrsquos signal before incorporating the exogenousinformation) by deviating We use IVP to establish that better exogenous information reducesthe senderrsquos expected benefit from falsifying his information to appear more favorable Intu-itively the sender expects any favorable but incorrect interim belief to get neutralized more

5 In quite different settings Milgrom and Roberts (1986) Dewatripont and Tirole (1999) Krishna and Morgan(2001) and Gentzkow and Kamenica (2017) offer formal analyses supporting the viewpoint that competitionbetween senders helpsmdashor at least cannot hurtmdasha DM Carlin Davies and Iannaccone (2012) present a resultin which increased competition leads to less voluntary disclosure Their model can be viewed as one in whichsenders bear a concealment cost that is assumed to decrease in the amount of disclosure by other senders ElliottGolub and Kirilenko (2014) show how a DM can be harmed by ldquoinformation improvementsrdquo in a cheap-talksetting but the essence of their mechanism is not the strategic interaction between senders

6 Weiss (1983) is an early paper that introduces exogenous information into a signaling environment his isnot however a model in which signaling has direct costs See also Feltovich Harbaugh and To (2002) Moresimilar to our setting are Daley and Green (2014) and Truyts (2015) we elaborate on the connections with thesepapers at the end of Section 4

5

by better exogenous information This means that better exogenous information makes costlydeviations less attractive to the sender which relaxes his incentive constraints Consequentlybetter exogenous information reduces wasteful signaling and enlarges the set of separatingtypesmdashevery type of the sender is better off and so is the receiver because there is more infor-mation revealed by the sender We explain how this result can be also viewed as a strategiccomplementary between multiple senders

Other applications While our paper applies IVP to two models of strategic communicationwe hope that the result will also be useful in other contexts Indeed the logic of IVP underliesand unifies the mechanisms in a few existing papers that study models with heterogeneouspriors under specific information structures In particular see the strategic ldquopersuasion mo-tiverdquo that generates bargaining delays in Yildiz (2004) motivational effects of difference ofopinion in Che and Kartik (2009) and Van den Steen (2010 Proposition 5) and a rationale fordeference in Hirsch (2016 Proposition 8) in a non-strategic setting see why minorities expectlower levels of intermediate bias in Sethi and Yildiz (2012 Proposition 5) We ourselves haveused IVPmdashspecifically invoking Theorem 1 belowmdashto study information acquisition prior todisclosure (Kartik Lee and Suen 2017) We briefly touch on some implications of IVP forBayesian persuasion with heterogeneous priors in the current paperrsquos conclusion Section 5

2 Information Validates the Prior

The backbone of our analysis is a pair of theorems below that relate the informativeness ofan experiment to the expectations of individuals with different beliefs Theorem 1 is a specialcase of Theorem 2 but for expositional clarity we introduce them separately

Throughout we use the following standard definitions concerning information structures(Blackwell 1953) Fix any finite state space Ω with generic element ω An experiment is E equiv(SS PωωisinΩ) where S is a measurable space of signals S is a σ-algebra on S and each Pω

is a probability measure over the signals in state ω An experiment E equiv (S S PωωisinΩ) is agarbling of experiment E if there is a Markov kernel from (SS) to (S S) denoted Q(middot|s)7 suchthat for each ω isin Ω and every set Σ isin S it holds that

Pω(Σ) =

983133

S

Q(Σ|s) dPω(s)

This definition captures the statistical notion that on a state-by-state basis the distribution

7 Ie (i) the map s 983041rarr Q(Σ|s) is S-measurable for every Σ isin S and (ii) the map Σ 983041rarr Q(Σ|s) is a probabilitymeasure on (S S) for every s isin S

6

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

1 Introduction

Rational agents revise their beliefs upon receiving new information From an ex-ante point ofview however one cannot expect new information to systematically alter onersquos beliefs in anyparticular direction More precisely a fundamental property of Bayesian updating is that be-liefs are a martingale an agentrsquos expectation of his posterior belief is equal to his prior beliefBut what about an agentrsquos expectation of another agentrsquos posterior belief when their currentbeliefs are different Or intimately related should agents expect new information to system-atically affect any existing disagreement These questions are not only of intrinsic interestbut tackling them proves useful for analyzing common-prior environments with asymmetricinformation

This paper makes two contributions First we provide a result concerning the mutual ex-pectations of Bayesian agents who disagree on the distribution of a fundamental Second weuse this result to derive new insights about some canonical sender(s)-receiver communicationgames either with multiple senders or when a receiver obtains external information We iden-tify a unified mechanism that drives senders to reveal more information in some economicsettings and less in others in the latter cases a receiver can be harmed by having access tobetter external information or more senders

A theorem on Bayesian updating In Section 2 we develop the following result on Bayesianupdating (Throughout this introduction some technical details are suppressed) Let Ω =

0 1 be the possible states of the world Anne (A) and Bob (B) have mutually-known butpossibly-different priors over Ω βA isin (0 1) and βB isin (0 1) respectively where βi is theprobability individual i ascribes to state ω = 1 A signal s will be drawn from an informationstructure or experiment E given by a family of probability distributions pω(middot)ωisinΩ wherepω(s) is the probability of observing signal s when the true state is ω Anne and Bob agree onthe experiment Let βi(s) denote irsquos posterior (on ω = 1) after observing signal s derived byBayesian updating Let EE

i [βj(middot)] be irsquos ex-ante expectation of jrsquos posterior under experimentE where the expectation is taken over signals from irsquos point of view

Consider two experiments E and E that are comparable in the sense of Blackwell (19511953) specifically let E be a garbling of E or equivalently E be more informative than E 1 Weprove (Theorem 1) that for any i j isin AB

βi le (ge) βj =rArr EEi [βj(middot)] le (ge)EE

i [βj(middot)] (1)

1 Throughout binary comparisons should be understood in the weak sense unless indicated otherwise egldquogreater thanrdquo means ldquoat least as large asrdquo

1

In words if Anne holds a lower (resp higher) prior than Bob then Anne predicts that amore informative experiment will on average reduce (resp raise) Bobrsquos posterior by a largeramount than a less informative experiment Put differently Anne expects more informationto further validate her prior in the sense of bringing on average Bobrsquos posterior closer to herprior Of course Bob expects just the reverse We refer to this result as information validates theprior IVP hereafter

IVP subsumes the familiar martingale property of Bayesian updating To see this considerj = i in Equation 1 and take E to be an uninformative experimentmdashwhich leaves an agentrsquosposterior after any signal equal to his priormdashand take E to be any experiment IVP also impliesthat an agent expects another agentrsquos posterior to fall in between their two priors2

IVP implies an interesting point about disagreement When the disagreement between be-liefs βi and βj is quantified by the canonical metric |βi minus βj| a particular signal can lead tohigher posterior disagreement than the prior (By contrast as shown by Baliga Hanany andKlibanoff (2013) no signal can lead to polarization it cannot be that (βi minus βi)(βj minus βj) lt 0)Nevertheless IVP implies (Corollary 1) that both Anne and Bob predict that a more infor-mative experiment will on average reduce their posterior disagreement to a greater extentthan a less informative experiment in particular no matter the experiment they expect lowerposterior disagreement than their prior disagreement At the extreme both predict their pos-terior disagreement to be zero under a fully informative experimentmdasheven though they holddifferent expectations about what the otherrsquos posterior belief will be

Theorem 2 provides a generalization of IVP to non-binary state spaces using suitablestatistics of the agentsrsquo beliefs and likelihood-ratio ordering conditions on the agentsrsquo priorsand on the experiments

Applications to communication games IVP is a statistical result The second part of ourpaper shows why it is useful in strategic contexts The idea is that even in common-priorenvironments private information can lead to one agentmdashAnne an informed agentmdashholdinga different belief about a fundamental than anothermdashBob an uninformed agent Only Anneknows both their beliefs Annersquos strategic incentives may depend on how she expects newinformation to affect Bobrsquos belief This new information can be exogenous or endogenouseg owing to the strategic behavior of still other agents The belief difference between Anneand Bob can occur either on an equilibrium path because of Annersquos ldquopoolingrdquo behavior or

2 To see this note that (i) letting U denote an uninformative experiment EUi [βj(middot)] = βj and (ii) letting F

denote a fully informative experiment (one in which any signal reveals the state and hence leaves any twoagents with the same posterior no matter their priors) EF

i [βj(middot)] = βi The result follows using Equation 1 andthe fact that any experiment is more informative than U but less informative than F

2

off path because of a deviation by Anne as usual even off-path considerations will generallyaffect the on-path properties of an equilibrium

We develop these points in the context of two familiar classes of communication games

Voluntary disclosure Section 3 studies a family of voluntary disclosure games also referredto as persuasion games In these games biased sendersrsquo only instrument of influence onthe decision-maker (DM) is the certifiable or verifiable private information they are endowedwith see Milgrom (2008) and Dranove and Jin (2010) for surveys Senders cannot explicitlylie but can choose what information to disclose and what to withhold We consider a modelin which senders if informed draw signals that are independently distributed conditional onan underlying binary state that affects the DMrsquos payoff3 We allow for senders to either haveopposing or similar biases but assume that a senderrsquos payoff is state-independent and linear(increasing or decreasing) in the DMrsquos belief

Our focus is on how strategic disclosure interacts with message costs either disclosureor concealment of information can entail direct costs for each sender Disclosure costs arenatural when the process of certifying or publicizing information demands resources suchas time effort or hiring an outside expert there can also be other ldquoproprietary costsrdquo Asubset of the literature starting with Jovanovic (1982) and Verrecchia (1983) has modeledsuch a cost although primarily only in single-sender problems On the flip side there arecontexts in which it is the suppression of information that requires costly resources Besidesdirect costs there can also be a psychic disutility to concealing information or concealmentmay be discovered ex-post (by auditors whistleblowers or mere happenstance) and resultin negative consequences for the sender through explicit punishment or reputation loss Arecent example is the $70 million dollar fine imposed by the National Highway Traffic SafetyAdministration on Honda Motor in January 2015 because ldquoit did not report hundreds of deathand injury claims for the last 11 years nor did it report certain warranty and other claimsin the same periodrdquo (The New York Times 2015)

To determine his own disclosure it is essential for each sender to predict how othersendersrsquo messages will affect the DMrsquos posterior belief and how this depends on his ownmessage (disclosure or nondisclosure) Our methodological insight is to view sendersrsquo mes-sages as endogenously-determined experiments and bring the IVP theorem to bear This ap-proach allows us to provide a unified treatmentmdashregardless of whether senders have similar

3 We assume that senders are uninformed with some probability importantly an uninformed sender cannotcertify that he is uninformed This modeling device was introduced by Dye (1985) and Shin (1994ab) to preventldquounravelingrdquo (Grossman and Hart 1980 Milgrom 1981) and serves the same purpose here

3

or opposing biases and regardless of whether message costs are disclosure or concealmentcosts We thereby uncover new insights into the classic question of when competition pro-motes disclosure

Any senderrsquos equilibrium disclosure behavior follows a threshold rule of disclosing allsufficiently favorable signals We establish a simple but important benchmark without mes-sage costs any sender in the multi-sender disclosure game uses the same disclosure thresholdas he would in a single-sender game In other words there is a strategic irrelevance The intu-ition is that without message costs a senderrsquos objective is the same regardless of the presenceof other senders he simply wants to induce the most favorable ldquointerim beliefrdquo in the DMbased on his own message as this will lead to the most favorable posterior belief based on allsendersrsquo messages

How do message costs alter the irrelevance result Consider a concealment cost In asingle-sender setting a sender irsquos disclosure threshold will be such that the DMrsquos interpreta-tion of nondisclosure is more favorable than irsquos private belief at the thresholdmdashthis wedge isnecessary to compensate i for the concealment cost Nondisclosure thus generates an interimdisagreement between the DMrsquos belief and the threshold typersquos private belief Naturally dis-closure produces no such disagreement Now add a second sender j to the picture Our IVPtheorem implies that regardless of jrsquos behavior the threshold type of i predicts that jrsquos mes-sage will on average make the DMrsquos posterior less favorable to i as compared to the DMrsquosinterim belief following irsquos nondisclosure Consequently concealment is now less attractiveto sender i IVP further implies that jrsquos effect on i is stronger when j disclosure more iewhen j is more informative In sum sendersrsquo disclosures are strategic complements under aconcealment cost4

The logic reverses under a disclosure cost In a single-sender setting the DMrsquos interpre-tation of nondisclosure is now less favorable than irsquos private belief at the thresholdmdashthe gainfrom disclosing information must compensate i for the direct cost Reasoning analogously toabove IVP now implies that disclosure becomes less attractive when the other sender is moreinformative the threshold type of i expects the other senderrsquos message to make the DMrsquosposterior more favorable to i reducing the gains from disclosure Consequently sendersrsquodisclosures are strategic substitutes under a disclosure cost

These results have notable welfare implications In the case of concealment cost a DMalways benefits from an additional sender not only because of the information this sender

4 In a different model Bourjade and Jullien (2011) find an effect related to that we find under concealmentcost Loosely speaking ldquoreputation lossrdquo in their model plays a similar role to concealment cost in ours

4

provides but also because it improves disclosure from other senders In the case of disclosurecost however the strategic substitution result implies that while a DM gains some directbenefit from consulting an additional sender the indirect effect on other sendersrsquo behavioris deleterious to the DM In general the net effect is ambiguous it is not hard to constructexamples in which the DM is made strictly worse off by adding a sender even if this senderhas an opposite bias to that of an existing sender Thus competition between senders neednot increase information revelation nor benefit the DM5 The DM can even be made worseoff when disclosure costs become lower or a sender is more likely to be informed althougheither change would help the DM in a single-sender setting

Costly signaling games Our IVP theorem is also useful in games of asymmetric informa-tion even when the information asymmetry is eliminated in equilibrium We develop suchan application in Section 4 For concreteness we consider information transmission with ly-ing costs (Kartik 2009) but we also discuss how our themes apply to canonical signalingapplications like education signaling (Spence 1973)

Suppose a sender has imperfect information about a binary state and can falsify or ma-nipulate his information by incurring costs A receiver makes inferences about the state basedon both the senderrsquos signal and some other exogenous information6 The sender has linearpreferences over the receiverrsquos belief about the state Due to a bounded signal space thereis incomplete separation across sender types we focus on equilibria in which ldquolowrdquo typesseparate and ldquohighrdquo types pool What is the effect of better exogenous information (in theBlackwell sense) on the senderrsquos signaling

Even sender types who separate in equilibrium can induce an incorrect interim belief (iethe receiverrsquos belief based only on the senderrsquos signal before incorporating the exogenousinformation) by deviating We use IVP to establish that better exogenous information reducesthe senderrsquos expected benefit from falsifying his information to appear more favorable Intu-itively the sender expects any favorable but incorrect interim belief to get neutralized more

5 In quite different settings Milgrom and Roberts (1986) Dewatripont and Tirole (1999) Krishna and Morgan(2001) and Gentzkow and Kamenica (2017) offer formal analyses supporting the viewpoint that competitionbetween senders helpsmdashor at least cannot hurtmdasha DM Carlin Davies and Iannaccone (2012) present a resultin which increased competition leads to less voluntary disclosure Their model can be viewed as one in whichsenders bear a concealment cost that is assumed to decrease in the amount of disclosure by other senders ElliottGolub and Kirilenko (2014) show how a DM can be harmed by ldquoinformation improvementsrdquo in a cheap-talksetting but the essence of their mechanism is not the strategic interaction between senders

6 Weiss (1983) is an early paper that introduces exogenous information into a signaling environment his isnot however a model in which signaling has direct costs See also Feltovich Harbaugh and To (2002) Moresimilar to our setting are Daley and Green (2014) and Truyts (2015) we elaborate on the connections with thesepapers at the end of Section 4

5

by better exogenous information This means that better exogenous information makes costlydeviations less attractive to the sender which relaxes his incentive constraints Consequentlybetter exogenous information reduces wasteful signaling and enlarges the set of separatingtypesmdashevery type of the sender is better off and so is the receiver because there is more infor-mation revealed by the sender We explain how this result can be also viewed as a strategiccomplementary between multiple senders

Other applications While our paper applies IVP to two models of strategic communicationwe hope that the result will also be useful in other contexts Indeed the logic of IVP underliesand unifies the mechanisms in a few existing papers that study models with heterogeneouspriors under specific information structures In particular see the strategic ldquopersuasion mo-tiverdquo that generates bargaining delays in Yildiz (2004) motivational effects of difference ofopinion in Che and Kartik (2009) and Van den Steen (2010 Proposition 5) and a rationale fordeference in Hirsch (2016 Proposition 8) in a non-strategic setting see why minorities expectlower levels of intermediate bias in Sethi and Yildiz (2012 Proposition 5) We ourselves haveused IVPmdashspecifically invoking Theorem 1 belowmdashto study information acquisition prior todisclosure (Kartik Lee and Suen 2017) We briefly touch on some implications of IVP forBayesian persuasion with heterogeneous priors in the current paperrsquos conclusion Section 5

2 Information Validates the Prior

The backbone of our analysis is a pair of theorems below that relate the informativeness ofan experiment to the expectations of individuals with different beliefs Theorem 1 is a specialcase of Theorem 2 but for expositional clarity we introduce them separately

Throughout we use the following standard definitions concerning information structures(Blackwell 1953) Fix any finite state space Ω with generic element ω An experiment is E equiv(SS PωωisinΩ) where S is a measurable space of signals S is a σ-algebra on S and each Pω

is a probability measure over the signals in state ω An experiment E equiv (S S PωωisinΩ) is agarbling of experiment E if there is a Markov kernel from (SS) to (S S) denoted Q(middot|s)7 suchthat for each ω isin Ω and every set Σ isin S it holds that

Pω(Σ) =

983133

S

Q(Σ|s) dPω(s)

This definition captures the statistical notion that on a state-by-state basis the distribution

7 Ie (i) the map s 983041rarr Q(Σ|s) is S-measurable for every Σ isin S and (ii) the map Σ 983041rarr Q(Σ|s) is a probabilitymeasure on (S S) for every s isin S

6

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

In words if Anne holds a lower (resp higher) prior than Bob then Anne predicts that amore informative experiment will on average reduce (resp raise) Bobrsquos posterior by a largeramount than a less informative experiment Put differently Anne expects more informationto further validate her prior in the sense of bringing on average Bobrsquos posterior closer to herprior Of course Bob expects just the reverse We refer to this result as information validates theprior IVP hereafter

IVP subsumes the familiar martingale property of Bayesian updating To see this considerj = i in Equation 1 and take E to be an uninformative experimentmdashwhich leaves an agentrsquosposterior after any signal equal to his priormdashand take E to be any experiment IVP also impliesthat an agent expects another agentrsquos posterior to fall in between their two priors2

IVP implies an interesting point about disagreement When the disagreement between be-liefs βi and βj is quantified by the canonical metric |βi minus βj| a particular signal can lead tohigher posterior disagreement than the prior (By contrast as shown by Baliga Hanany andKlibanoff (2013) no signal can lead to polarization it cannot be that (βi minus βi)(βj minus βj) lt 0)Nevertheless IVP implies (Corollary 1) that both Anne and Bob predict that a more infor-mative experiment will on average reduce their posterior disagreement to a greater extentthan a less informative experiment in particular no matter the experiment they expect lowerposterior disagreement than their prior disagreement At the extreme both predict their pos-terior disagreement to be zero under a fully informative experimentmdasheven though they holddifferent expectations about what the otherrsquos posterior belief will be

Theorem 2 provides a generalization of IVP to non-binary state spaces using suitablestatistics of the agentsrsquo beliefs and likelihood-ratio ordering conditions on the agentsrsquo priorsand on the experiments

Applications to communication games IVP is a statistical result The second part of ourpaper shows why it is useful in strategic contexts The idea is that even in common-priorenvironments private information can lead to one agentmdashAnne an informed agentmdashholdinga different belief about a fundamental than anothermdashBob an uninformed agent Only Anneknows both their beliefs Annersquos strategic incentives may depend on how she expects newinformation to affect Bobrsquos belief This new information can be exogenous or endogenouseg owing to the strategic behavior of still other agents The belief difference between Anneand Bob can occur either on an equilibrium path because of Annersquos ldquopoolingrdquo behavior or

2 To see this note that (i) letting U denote an uninformative experiment EUi [βj(middot)] = βj and (ii) letting F

denote a fully informative experiment (one in which any signal reveals the state and hence leaves any twoagents with the same posterior no matter their priors) EF

i [βj(middot)] = βi The result follows using Equation 1 andthe fact that any experiment is more informative than U but less informative than F

2

off path because of a deviation by Anne as usual even off-path considerations will generallyaffect the on-path properties of an equilibrium

We develop these points in the context of two familiar classes of communication games

Voluntary disclosure Section 3 studies a family of voluntary disclosure games also referredto as persuasion games In these games biased sendersrsquo only instrument of influence onthe decision-maker (DM) is the certifiable or verifiable private information they are endowedwith see Milgrom (2008) and Dranove and Jin (2010) for surveys Senders cannot explicitlylie but can choose what information to disclose and what to withhold We consider a modelin which senders if informed draw signals that are independently distributed conditional onan underlying binary state that affects the DMrsquos payoff3 We allow for senders to either haveopposing or similar biases but assume that a senderrsquos payoff is state-independent and linear(increasing or decreasing) in the DMrsquos belief

Our focus is on how strategic disclosure interacts with message costs either disclosureor concealment of information can entail direct costs for each sender Disclosure costs arenatural when the process of certifying or publicizing information demands resources suchas time effort or hiring an outside expert there can also be other ldquoproprietary costsrdquo Asubset of the literature starting with Jovanovic (1982) and Verrecchia (1983) has modeledsuch a cost although primarily only in single-sender problems On the flip side there arecontexts in which it is the suppression of information that requires costly resources Besidesdirect costs there can also be a psychic disutility to concealing information or concealmentmay be discovered ex-post (by auditors whistleblowers or mere happenstance) and resultin negative consequences for the sender through explicit punishment or reputation loss Arecent example is the $70 million dollar fine imposed by the National Highway Traffic SafetyAdministration on Honda Motor in January 2015 because ldquoit did not report hundreds of deathand injury claims for the last 11 years nor did it report certain warranty and other claimsin the same periodrdquo (The New York Times 2015)

To determine his own disclosure it is essential for each sender to predict how othersendersrsquo messages will affect the DMrsquos posterior belief and how this depends on his ownmessage (disclosure or nondisclosure) Our methodological insight is to view sendersrsquo mes-sages as endogenously-determined experiments and bring the IVP theorem to bear This ap-proach allows us to provide a unified treatmentmdashregardless of whether senders have similar

3 We assume that senders are uninformed with some probability importantly an uninformed sender cannotcertify that he is uninformed This modeling device was introduced by Dye (1985) and Shin (1994ab) to preventldquounravelingrdquo (Grossman and Hart 1980 Milgrom 1981) and serves the same purpose here

3

or opposing biases and regardless of whether message costs are disclosure or concealmentcosts We thereby uncover new insights into the classic question of when competition pro-motes disclosure

Any senderrsquos equilibrium disclosure behavior follows a threshold rule of disclosing allsufficiently favorable signals We establish a simple but important benchmark without mes-sage costs any sender in the multi-sender disclosure game uses the same disclosure thresholdas he would in a single-sender game In other words there is a strategic irrelevance The intu-ition is that without message costs a senderrsquos objective is the same regardless of the presenceof other senders he simply wants to induce the most favorable ldquointerim beliefrdquo in the DMbased on his own message as this will lead to the most favorable posterior belief based on allsendersrsquo messages

How do message costs alter the irrelevance result Consider a concealment cost In asingle-sender setting a sender irsquos disclosure threshold will be such that the DMrsquos interpreta-tion of nondisclosure is more favorable than irsquos private belief at the thresholdmdashthis wedge isnecessary to compensate i for the concealment cost Nondisclosure thus generates an interimdisagreement between the DMrsquos belief and the threshold typersquos private belief Naturally dis-closure produces no such disagreement Now add a second sender j to the picture Our IVPtheorem implies that regardless of jrsquos behavior the threshold type of i predicts that jrsquos mes-sage will on average make the DMrsquos posterior less favorable to i as compared to the DMrsquosinterim belief following irsquos nondisclosure Consequently concealment is now less attractiveto sender i IVP further implies that jrsquos effect on i is stronger when j disclosure more iewhen j is more informative In sum sendersrsquo disclosures are strategic complements under aconcealment cost4

The logic reverses under a disclosure cost In a single-sender setting the DMrsquos interpre-tation of nondisclosure is now less favorable than irsquos private belief at the thresholdmdashthe gainfrom disclosing information must compensate i for the direct cost Reasoning analogously toabove IVP now implies that disclosure becomes less attractive when the other sender is moreinformative the threshold type of i expects the other senderrsquos message to make the DMrsquosposterior more favorable to i reducing the gains from disclosure Consequently sendersrsquodisclosures are strategic substitutes under a disclosure cost

These results have notable welfare implications In the case of concealment cost a DMalways benefits from an additional sender not only because of the information this sender

4 In a different model Bourjade and Jullien (2011) find an effect related to that we find under concealmentcost Loosely speaking ldquoreputation lossrdquo in their model plays a similar role to concealment cost in ours

4

provides but also because it improves disclosure from other senders In the case of disclosurecost however the strategic substitution result implies that while a DM gains some directbenefit from consulting an additional sender the indirect effect on other sendersrsquo behavioris deleterious to the DM In general the net effect is ambiguous it is not hard to constructexamples in which the DM is made strictly worse off by adding a sender even if this senderhas an opposite bias to that of an existing sender Thus competition between senders neednot increase information revelation nor benefit the DM5 The DM can even be made worseoff when disclosure costs become lower or a sender is more likely to be informed althougheither change would help the DM in a single-sender setting

Costly signaling games Our IVP theorem is also useful in games of asymmetric informa-tion even when the information asymmetry is eliminated in equilibrium We develop suchan application in Section 4 For concreteness we consider information transmission with ly-ing costs (Kartik 2009) but we also discuss how our themes apply to canonical signalingapplications like education signaling (Spence 1973)

Suppose a sender has imperfect information about a binary state and can falsify or ma-nipulate his information by incurring costs A receiver makes inferences about the state basedon both the senderrsquos signal and some other exogenous information6 The sender has linearpreferences over the receiverrsquos belief about the state Due to a bounded signal space thereis incomplete separation across sender types we focus on equilibria in which ldquolowrdquo typesseparate and ldquohighrdquo types pool What is the effect of better exogenous information (in theBlackwell sense) on the senderrsquos signaling

Even sender types who separate in equilibrium can induce an incorrect interim belief (iethe receiverrsquos belief based only on the senderrsquos signal before incorporating the exogenousinformation) by deviating We use IVP to establish that better exogenous information reducesthe senderrsquos expected benefit from falsifying his information to appear more favorable Intu-itively the sender expects any favorable but incorrect interim belief to get neutralized more

5 In quite different settings Milgrom and Roberts (1986) Dewatripont and Tirole (1999) Krishna and Morgan(2001) and Gentzkow and Kamenica (2017) offer formal analyses supporting the viewpoint that competitionbetween senders helpsmdashor at least cannot hurtmdasha DM Carlin Davies and Iannaccone (2012) present a resultin which increased competition leads to less voluntary disclosure Their model can be viewed as one in whichsenders bear a concealment cost that is assumed to decrease in the amount of disclosure by other senders ElliottGolub and Kirilenko (2014) show how a DM can be harmed by ldquoinformation improvementsrdquo in a cheap-talksetting but the essence of their mechanism is not the strategic interaction between senders

6 Weiss (1983) is an early paper that introduces exogenous information into a signaling environment his isnot however a model in which signaling has direct costs See also Feltovich Harbaugh and To (2002) Moresimilar to our setting are Daley and Green (2014) and Truyts (2015) we elaborate on the connections with thesepapers at the end of Section 4

5

by better exogenous information This means that better exogenous information makes costlydeviations less attractive to the sender which relaxes his incentive constraints Consequentlybetter exogenous information reduces wasteful signaling and enlarges the set of separatingtypesmdashevery type of the sender is better off and so is the receiver because there is more infor-mation revealed by the sender We explain how this result can be also viewed as a strategiccomplementary between multiple senders

Other applications While our paper applies IVP to two models of strategic communicationwe hope that the result will also be useful in other contexts Indeed the logic of IVP underliesand unifies the mechanisms in a few existing papers that study models with heterogeneouspriors under specific information structures In particular see the strategic ldquopersuasion mo-tiverdquo that generates bargaining delays in Yildiz (2004) motivational effects of difference ofopinion in Che and Kartik (2009) and Van den Steen (2010 Proposition 5) and a rationale fordeference in Hirsch (2016 Proposition 8) in a non-strategic setting see why minorities expectlower levels of intermediate bias in Sethi and Yildiz (2012 Proposition 5) We ourselves haveused IVPmdashspecifically invoking Theorem 1 belowmdashto study information acquisition prior todisclosure (Kartik Lee and Suen 2017) We briefly touch on some implications of IVP forBayesian persuasion with heterogeneous priors in the current paperrsquos conclusion Section 5

2 Information Validates the Prior

The backbone of our analysis is a pair of theorems below that relate the informativeness ofan experiment to the expectations of individuals with different beliefs Theorem 1 is a specialcase of Theorem 2 but for expositional clarity we introduce them separately

Throughout we use the following standard definitions concerning information structures(Blackwell 1953) Fix any finite state space Ω with generic element ω An experiment is E equiv(SS PωωisinΩ) where S is a measurable space of signals S is a σ-algebra on S and each Pω

is a probability measure over the signals in state ω An experiment E equiv (S S PωωisinΩ) is agarbling of experiment E if there is a Markov kernel from (SS) to (S S) denoted Q(middot|s)7 suchthat for each ω isin Ω and every set Σ isin S it holds that

Pω(Σ) =

983133

S

Q(Σ|s) dPω(s)

This definition captures the statistical notion that on a state-by-state basis the distribution

7 Ie (i) the map s 983041rarr Q(Σ|s) is S-measurable for every Σ isin S and (ii) the map Σ 983041rarr Q(Σ|s) is a probabilitymeasure on (S S) for every s isin S

6

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

off path because of a deviation by Anne as usual even off-path considerations will generallyaffect the on-path properties of an equilibrium

We develop these points in the context of two familiar classes of communication games

Voluntary disclosure Section 3 studies a family of voluntary disclosure games also referredto as persuasion games In these games biased sendersrsquo only instrument of influence onthe decision-maker (DM) is the certifiable or verifiable private information they are endowedwith see Milgrom (2008) and Dranove and Jin (2010) for surveys Senders cannot explicitlylie but can choose what information to disclose and what to withhold We consider a modelin which senders if informed draw signals that are independently distributed conditional onan underlying binary state that affects the DMrsquos payoff3 We allow for senders to either haveopposing or similar biases but assume that a senderrsquos payoff is state-independent and linear(increasing or decreasing) in the DMrsquos belief

Our focus is on how strategic disclosure interacts with message costs either disclosureor concealment of information can entail direct costs for each sender Disclosure costs arenatural when the process of certifying or publicizing information demands resources suchas time effort or hiring an outside expert there can also be other ldquoproprietary costsrdquo Asubset of the literature starting with Jovanovic (1982) and Verrecchia (1983) has modeledsuch a cost although primarily only in single-sender problems On the flip side there arecontexts in which it is the suppression of information that requires costly resources Besidesdirect costs there can also be a psychic disutility to concealing information or concealmentmay be discovered ex-post (by auditors whistleblowers or mere happenstance) and resultin negative consequences for the sender through explicit punishment or reputation loss Arecent example is the $70 million dollar fine imposed by the National Highway Traffic SafetyAdministration on Honda Motor in January 2015 because ldquoit did not report hundreds of deathand injury claims for the last 11 years nor did it report certain warranty and other claimsin the same periodrdquo (The New York Times 2015)

To determine his own disclosure it is essential for each sender to predict how othersendersrsquo messages will affect the DMrsquos posterior belief and how this depends on his ownmessage (disclosure or nondisclosure) Our methodological insight is to view sendersrsquo mes-sages as endogenously-determined experiments and bring the IVP theorem to bear This ap-proach allows us to provide a unified treatmentmdashregardless of whether senders have similar

3 We assume that senders are uninformed with some probability importantly an uninformed sender cannotcertify that he is uninformed This modeling device was introduced by Dye (1985) and Shin (1994ab) to preventldquounravelingrdquo (Grossman and Hart 1980 Milgrom 1981) and serves the same purpose here

3

or opposing biases and regardless of whether message costs are disclosure or concealmentcosts We thereby uncover new insights into the classic question of when competition pro-motes disclosure

Any senderrsquos equilibrium disclosure behavior follows a threshold rule of disclosing allsufficiently favorable signals We establish a simple but important benchmark without mes-sage costs any sender in the multi-sender disclosure game uses the same disclosure thresholdas he would in a single-sender game In other words there is a strategic irrelevance The intu-ition is that without message costs a senderrsquos objective is the same regardless of the presenceof other senders he simply wants to induce the most favorable ldquointerim beliefrdquo in the DMbased on his own message as this will lead to the most favorable posterior belief based on allsendersrsquo messages

How do message costs alter the irrelevance result Consider a concealment cost In asingle-sender setting a sender irsquos disclosure threshold will be such that the DMrsquos interpreta-tion of nondisclosure is more favorable than irsquos private belief at the thresholdmdashthis wedge isnecessary to compensate i for the concealment cost Nondisclosure thus generates an interimdisagreement between the DMrsquos belief and the threshold typersquos private belief Naturally dis-closure produces no such disagreement Now add a second sender j to the picture Our IVPtheorem implies that regardless of jrsquos behavior the threshold type of i predicts that jrsquos mes-sage will on average make the DMrsquos posterior less favorable to i as compared to the DMrsquosinterim belief following irsquos nondisclosure Consequently concealment is now less attractiveto sender i IVP further implies that jrsquos effect on i is stronger when j disclosure more iewhen j is more informative In sum sendersrsquo disclosures are strategic complements under aconcealment cost4

The logic reverses under a disclosure cost In a single-sender setting the DMrsquos interpre-tation of nondisclosure is now less favorable than irsquos private belief at the thresholdmdashthe gainfrom disclosing information must compensate i for the direct cost Reasoning analogously toabove IVP now implies that disclosure becomes less attractive when the other sender is moreinformative the threshold type of i expects the other senderrsquos message to make the DMrsquosposterior more favorable to i reducing the gains from disclosure Consequently sendersrsquodisclosures are strategic substitutes under a disclosure cost

These results have notable welfare implications In the case of concealment cost a DMalways benefits from an additional sender not only because of the information this sender

4 In a different model Bourjade and Jullien (2011) find an effect related to that we find under concealmentcost Loosely speaking ldquoreputation lossrdquo in their model plays a similar role to concealment cost in ours

4

provides but also because it improves disclosure from other senders In the case of disclosurecost however the strategic substitution result implies that while a DM gains some directbenefit from consulting an additional sender the indirect effect on other sendersrsquo behavioris deleterious to the DM In general the net effect is ambiguous it is not hard to constructexamples in which the DM is made strictly worse off by adding a sender even if this senderhas an opposite bias to that of an existing sender Thus competition between senders neednot increase information revelation nor benefit the DM5 The DM can even be made worseoff when disclosure costs become lower or a sender is more likely to be informed althougheither change would help the DM in a single-sender setting

Costly signaling games Our IVP theorem is also useful in games of asymmetric informa-tion even when the information asymmetry is eliminated in equilibrium We develop suchan application in Section 4 For concreteness we consider information transmission with ly-ing costs (Kartik 2009) but we also discuss how our themes apply to canonical signalingapplications like education signaling (Spence 1973)

Suppose a sender has imperfect information about a binary state and can falsify or ma-nipulate his information by incurring costs A receiver makes inferences about the state basedon both the senderrsquos signal and some other exogenous information6 The sender has linearpreferences over the receiverrsquos belief about the state Due to a bounded signal space thereis incomplete separation across sender types we focus on equilibria in which ldquolowrdquo typesseparate and ldquohighrdquo types pool What is the effect of better exogenous information (in theBlackwell sense) on the senderrsquos signaling

Even sender types who separate in equilibrium can induce an incorrect interim belief (iethe receiverrsquos belief based only on the senderrsquos signal before incorporating the exogenousinformation) by deviating We use IVP to establish that better exogenous information reducesthe senderrsquos expected benefit from falsifying his information to appear more favorable Intu-itively the sender expects any favorable but incorrect interim belief to get neutralized more

5 In quite different settings Milgrom and Roberts (1986) Dewatripont and Tirole (1999) Krishna and Morgan(2001) and Gentzkow and Kamenica (2017) offer formal analyses supporting the viewpoint that competitionbetween senders helpsmdashor at least cannot hurtmdasha DM Carlin Davies and Iannaccone (2012) present a resultin which increased competition leads to less voluntary disclosure Their model can be viewed as one in whichsenders bear a concealment cost that is assumed to decrease in the amount of disclosure by other senders ElliottGolub and Kirilenko (2014) show how a DM can be harmed by ldquoinformation improvementsrdquo in a cheap-talksetting but the essence of their mechanism is not the strategic interaction between senders

6 Weiss (1983) is an early paper that introduces exogenous information into a signaling environment his isnot however a model in which signaling has direct costs See also Feltovich Harbaugh and To (2002) Moresimilar to our setting are Daley and Green (2014) and Truyts (2015) we elaborate on the connections with thesepapers at the end of Section 4

5

by better exogenous information This means that better exogenous information makes costlydeviations less attractive to the sender which relaxes his incentive constraints Consequentlybetter exogenous information reduces wasteful signaling and enlarges the set of separatingtypesmdashevery type of the sender is better off and so is the receiver because there is more infor-mation revealed by the sender We explain how this result can be also viewed as a strategiccomplementary between multiple senders

Other applications While our paper applies IVP to two models of strategic communicationwe hope that the result will also be useful in other contexts Indeed the logic of IVP underliesand unifies the mechanisms in a few existing papers that study models with heterogeneouspriors under specific information structures In particular see the strategic ldquopersuasion mo-tiverdquo that generates bargaining delays in Yildiz (2004) motivational effects of difference ofopinion in Che and Kartik (2009) and Van den Steen (2010 Proposition 5) and a rationale fordeference in Hirsch (2016 Proposition 8) in a non-strategic setting see why minorities expectlower levels of intermediate bias in Sethi and Yildiz (2012 Proposition 5) We ourselves haveused IVPmdashspecifically invoking Theorem 1 belowmdashto study information acquisition prior todisclosure (Kartik Lee and Suen 2017) We briefly touch on some implications of IVP forBayesian persuasion with heterogeneous priors in the current paperrsquos conclusion Section 5

2 Information Validates the Prior

The backbone of our analysis is a pair of theorems below that relate the informativeness ofan experiment to the expectations of individuals with different beliefs Theorem 1 is a specialcase of Theorem 2 but for expositional clarity we introduce them separately

Throughout we use the following standard definitions concerning information structures(Blackwell 1953) Fix any finite state space Ω with generic element ω An experiment is E equiv(SS PωωisinΩ) where S is a measurable space of signals S is a σ-algebra on S and each Pω

is a probability measure over the signals in state ω An experiment E equiv (S S PωωisinΩ) is agarbling of experiment E if there is a Markov kernel from (SS) to (S S) denoted Q(middot|s)7 suchthat for each ω isin Ω and every set Σ isin S it holds that

Pω(Σ) =

983133

S

Q(Σ|s) dPω(s)

This definition captures the statistical notion that on a state-by-state basis the distribution

7 Ie (i) the map s 983041rarr Q(Σ|s) is S-measurable for every Σ isin S and (ii) the map Σ 983041rarr Q(Σ|s) is a probabilitymeasure on (S S) for every s isin S

6

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

or opposing biases and regardless of whether message costs are disclosure or concealmentcosts We thereby uncover new insights into the classic question of when competition pro-motes disclosure

Any senderrsquos equilibrium disclosure behavior follows a threshold rule of disclosing allsufficiently favorable signals We establish a simple but important benchmark without mes-sage costs any sender in the multi-sender disclosure game uses the same disclosure thresholdas he would in a single-sender game In other words there is a strategic irrelevance The intu-ition is that without message costs a senderrsquos objective is the same regardless of the presenceof other senders he simply wants to induce the most favorable ldquointerim beliefrdquo in the DMbased on his own message as this will lead to the most favorable posterior belief based on allsendersrsquo messages

How do message costs alter the irrelevance result Consider a concealment cost In asingle-sender setting a sender irsquos disclosure threshold will be such that the DMrsquos interpreta-tion of nondisclosure is more favorable than irsquos private belief at the thresholdmdashthis wedge isnecessary to compensate i for the concealment cost Nondisclosure thus generates an interimdisagreement between the DMrsquos belief and the threshold typersquos private belief Naturally dis-closure produces no such disagreement Now add a second sender j to the picture Our IVPtheorem implies that regardless of jrsquos behavior the threshold type of i predicts that jrsquos mes-sage will on average make the DMrsquos posterior less favorable to i as compared to the DMrsquosinterim belief following irsquos nondisclosure Consequently concealment is now less attractiveto sender i IVP further implies that jrsquos effect on i is stronger when j disclosure more iewhen j is more informative In sum sendersrsquo disclosures are strategic complements under aconcealment cost4

The logic reverses under a disclosure cost In a single-sender setting the DMrsquos interpre-tation of nondisclosure is now less favorable than irsquos private belief at the thresholdmdashthe gainfrom disclosing information must compensate i for the direct cost Reasoning analogously toabove IVP now implies that disclosure becomes less attractive when the other sender is moreinformative the threshold type of i expects the other senderrsquos message to make the DMrsquosposterior more favorable to i reducing the gains from disclosure Consequently sendersrsquodisclosures are strategic substitutes under a disclosure cost

These results have notable welfare implications In the case of concealment cost a DMalways benefits from an additional sender not only because of the information this sender

4 In a different model Bourjade and Jullien (2011) find an effect related to that we find under concealmentcost Loosely speaking ldquoreputation lossrdquo in their model plays a similar role to concealment cost in ours

4

provides but also because it improves disclosure from other senders In the case of disclosurecost however the strategic substitution result implies that while a DM gains some directbenefit from consulting an additional sender the indirect effect on other sendersrsquo behavioris deleterious to the DM In general the net effect is ambiguous it is not hard to constructexamples in which the DM is made strictly worse off by adding a sender even if this senderhas an opposite bias to that of an existing sender Thus competition between senders neednot increase information revelation nor benefit the DM5 The DM can even be made worseoff when disclosure costs become lower or a sender is more likely to be informed althougheither change would help the DM in a single-sender setting

Costly signaling games Our IVP theorem is also useful in games of asymmetric informa-tion even when the information asymmetry is eliminated in equilibrium We develop suchan application in Section 4 For concreteness we consider information transmission with ly-ing costs (Kartik 2009) but we also discuss how our themes apply to canonical signalingapplications like education signaling (Spence 1973)

Suppose a sender has imperfect information about a binary state and can falsify or ma-nipulate his information by incurring costs A receiver makes inferences about the state basedon both the senderrsquos signal and some other exogenous information6 The sender has linearpreferences over the receiverrsquos belief about the state Due to a bounded signal space thereis incomplete separation across sender types we focus on equilibria in which ldquolowrdquo typesseparate and ldquohighrdquo types pool What is the effect of better exogenous information (in theBlackwell sense) on the senderrsquos signaling

Even sender types who separate in equilibrium can induce an incorrect interim belief (iethe receiverrsquos belief based only on the senderrsquos signal before incorporating the exogenousinformation) by deviating We use IVP to establish that better exogenous information reducesthe senderrsquos expected benefit from falsifying his information to appear more favorable Intu-itively the sender expects any favorable but incorrect interim belief to get neutralized more

5 In quite different settings Milgrom and Roberts (1986) Dewatripont and Tirole (1999) Krishna and Morgan(2001) and Gentzkow and Kamenica (2017) offer formal analyses supporting the viewpoint that competitionbetween senders helpsmdashor at least cannot hurtmdasha DM Carlin Davies and Iannaccone (2012) present a resultin which increased competition leads to less voluntary disclosure Their model can be viewed as one in whichsenders bear a concealment cost that is assumed to decrease in the amount of disclosure by other senders ElliottGolub and Kirilenko (2014) show how a DM can be harmed by ldquoinformation improvementsrdquo in a cheap-talksetting but the essence of their mechanism is not the strategic interaction between senders

6 Weiss (1983) is an early paper that introduces exogenous information into a signaling environment his isnot however a model in which signaling has direct costs See also Feltovich Harbaugh and To (2002) Moresimilar to our setting are Daley and Green (2014) and Truyts (2015) we elaborate on the connections with thesepapers at the end of Section 4

5

by better exogenous information This means that better exogenous information makes costlydeviations less attractive to the sender which relaxes his incentive constraints Consequentlybetter exogenous information reduces wasteful signaling and enlarges the set of separatingtypesmdashevery type of the sender is better off and so is the receiver because there is more infor-mation revealed by the sender We explain how this result can be also viewed as a strategiccomplementary between multiple senders

Other applications While our paper applies IVP to two models of strategic communicationwe hope that the result will also be useful in other contexts Indeed the logic of IVP underliesand unifies the mechanisms in a few existing papers that study models with heterogeneouspriors under specific information structures In particular see the strategic ldquopersuasion mo-tiverdquo that generates bargaining delays in Yildiz (2004) motivational effects of difference ofopinion in Che and Kartik (2009) and Van den Steen (2010 Proposition 5) and a rationale fordeference in Hirsch (2016 Proposition 8) in a non-strategic setting see why minorities expectlower levels of intermediate bias in Sethi and Yildiz (2012 Proposition 5) We ourselves haveused IVPmdashspecifically invoking Theorem 1 belowmdashto study information acquisition prior todisclosure (Kartik Lee and Suen 2017) We briefly touch on some implications of IVP forBayesian persuasion with heterogeneous priors in the current paperrsquos conclusion Section 5

2 Information Validates the Prior

The backbone of our analysis is a pair of theorems below that relate the informativeness ofan experiment to the expectations of individuals with different beliefs Theorem 1 is a specialcase of Theorem 2 but for expositional clarity we introduce them separately

Throughout we use the following standard definitions concerning information structures(Blackwell 1953) Fix any finite state space Ω with generic element ω An experiment is E equiv(SS PωωisinΩ) where S is a measurable space of signals S is a σ-algebra on S and each Pω

is a probability measure over the signals in state ω An experiment E equiv (S S PωωisinΩ) is agarbling of experiment E if there is a Markov kernel from (SS) to (S S) denoted Q(middot|s)7 suchthat for each ω isin Ω and every set Σ isin S it holds that

Pω(Σ) =

983133

S

Q(Σ|s) dPω(s)

This definition captures the statistical notion that on a state-by-state basis the distribution

7 Ie (i) the map s 983041rarr Q(Σ|s) is S-measurable for every Σ isin S and (ii) the map Σ 983041rarr Q(Σ|s) is a probabilitymeasure on (S S) for every s isin S

6

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

provides but also because it improves disclosure from other senders In the case of disclosurecost however the strategic substitution result implies that while a DM gains some directbenefit from consulting an additional sender the indirect effect on other sendersrsquo behavioris deleterious to the DM In general the net effect is ambiguous it is not hard to constructexamples in which the DM is made strictly worse off by adding a sender even if this senderhas an opposite bias to that of an existing sender Thus competition between senders neednot increase information revelation nor benefit the DM5 The DM can even be made worseoff when disclosure costs become lower or a sender is more likely to be informed althougheither change would help the DM in a single-sender setting

Costly signaling games Our IVP theorem is also useful in games of asymmetric informa-tion even when the information asymmetry is eliminated in equilibrium We develop suchan application in Section 4 For concreteness we consider information transmission with ly-ing costs (Kartik 2009) but we also discuss how our themes apply to canonical signalingapplications like education signaling (Spence 1973)

Suppose a sender has imperfect information about a binary state and can falsify or ma-nipulate his information by incurring costs A receiver makes inferences about the state basedon both the senderrsquos signal and some other exogenous information6 The sender has linearpreferences over the receiverrsquos belief about the state Due to a bounded signal space thereis incomplete separation across sender types we focus on equilibria in which ldquolowrdquo typesseparate and ldquohighrdquo types pool What is the effect of better exogenous information (in theBlackwell sense) on the senderrsquos signaling

Even sender types who separate in equilibrium can induce an incorrect interim belief (iethe receiverrsquos belief based only on the senderrsquos signal before incorporating the exogenousinformation) by deviating We use IVP to establish that better exogenous information reducesthe senderrsquos expected benefit from falsifying his information to appear more favorable Intu-itively the sender expects any favorable but incorrect interim belief to get neutralized more

5 In quite different settings Milgrom and Roberts (1986) Dewatripont and Tirole (1999) Krishna and Morgan(2001) and Gentzkow and Kamenica (2017) offer formal analyses supporting the viewpoint that competitionbetween senders helpsmdashor at least cannot hurtmdasha DM Carlin Davies and Iannaccone (2012) present a resultin which increased competition leads to less voluntary disclosure Their model can be viewed as one in whichsenders bear a concealment cost that is assumed to decrease in the amount of disclosure by other senders ElliottGolub and Kirilenko (2014) show how a DM can be harmed by ldquoinformation improvementsrdquo in a cheap-talksetting but the essence of their mechanism is not the strategic interaction between senders

6 Weiss (1983) is an early paper that introduces exogenous information into a signaling environment his isnot however a model in which signaling has direct costs See also Feltovich Harbaugh and To (2002) Moresimilar to our setting are Daley and Green (2014) and Truyts (2015) we elaborate on the connections with thesepapers at the end of Section 4

5

by better exogenous information This means that better exogenous information makes costlydeviations less attractive to the sender which relaxes his incentive constraints Consequentlybetter exogenous information reduces wasteful signaling and enlarges the set of separatingtypesmdashevery type of the sender is better off and so is the receiver because there is more infor-mation revealed by the sender We explain how this result can be also viewed as a strategiccomplementary between multiple senders

Other applications While our paper applies IVP to two models of strategic communicationwe hope that the result will also be useful in other contexts Indeed the logic of IVP underliesand unifies the mechanisms in a few existing papers that study models with heterogeneouspriors under specific information structures In particular see the strategic ldquopersuasion mo-tiverdquo that generates bargaining delays in Yildiz (2004) motivational effects of difference ofopinion in Che and Kartik (2009) and Van den Steen (2010 Proposition 5) and a rationale fordeference in Hirsch (2016 Proposition 8) in a non-strategic setting see why minorities expectlower levels of intermediate bias in Sethi and Yildiz (2012 Proposition 5) We ourselves haveused IVPmdashspecifically invoking Theorem 1 belowmdashto study information acquisition prior todisclosure (Kartik Lee and Suen 2017) We briefly touch on some implications of IVP forBayesian persuasion with heterogeneous priors in the current paperrsquos conclusion Section 5

2 Information Validates the Prior

The backbone of our analysis is a pair of theorems below that relate the informativeness ofan experiment to the expectations of individuals with different beliefs Theorem 1 is a specialcase of Theorem 2 but for expositional clarity we introduce them separately

Throughout we use the following standard definitions concerning information structures(Blackwell 1953) Fix any finite state space Ω with generic element ω An experiment is E equiv(SS PωωisinΩ) where S is a measurable space of signals S is a σ-algebra on S and each Pω

is a probability measure over the signals in state ω An experiment E equiv (S S PωωisinΩ) is agarbling of experiment E if there is a Markov kernel from (SS) to (S S) denoted Q(middot|s)7 suchthat for each ω isin Ω and every set Σ isin S it holds that

Pω(Σ) =

983133

S

Q(Σ|s) dPω(s)

This definition captures the statistical notion that on a state-by-state basis the distribution

7 Ie (i) the map s 983041rarr Q(Σ|s) is S-measurable for every Σ isin S and (ii) the map Σ 983041rarr Q(Σ|s) is a probabilitymeasure on (S S) for every s isin S

6

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

by better exogenous information This means that better exogenous information makes costlydeviations less attractive to the sender which relaxes his incentive constraints Consequentlybetter exogenous information reduces wasteful signaling and enlarges the set of separatingtypesmdashevery type of the sender is better off and so is the receiver because there is more infor-mation revealed by the sender We explain how this result can be also viewed as a strategiccomplementary between multiple senders

Other applications While our paper applies IVP to two models of strategic communicationwe hope that the result will also be useful in other contexts Indeed the logic of IVP underliesand unifies the mechanisms in a few existing papers that study models with heterogeneouspriors under specific information structures In particular see the strategic ldquopersuasion mo-tiverdquo that generates bargaining delays in Yildiz (2004) motivational effects of difference ofopinion in Che and Kartik (2009) and Van den Steen (2010 Proposition 5) and a rationale fordeference in Hirsch (2016 Proposition 8) in a non-strategic setting see why minorities expectlower levels of intermediate bias in Sethi and Yildiz (2012 Proposition 5) We ourselves haveused IVPmdashspecifically invoking Theorem 1 belowmdashto study information acquisition prior todisclosure (Kartik Lee and Suen 2017) We briefly touch on some implications of IVP forBayesian persuasion with heterogeneous priors in the current paperrsquos conclusion Section 5

2 Information Validates the Prior

The backbone of our analysis is a pair of theorems below that relate the informativeness ofan experiment to the expectations of individuals with different beliefs Theorem 1 is a specialcase of Theorem 2 but for expositional clarity we introduce them separately

Throughout we use the following standard definitions concerning information structures(Blackwell 1953) Fix any finite state space Ω with generic element ω An experiment is E equiv(SS PωωisinΩ) where S is a measurable space of signals S is a σ-algebra on S and each Pω

is a probability measure over the signals in state ω An experiment E equiv (S S PωωisinΩ) is agarbling of experiment E if there is a Markov kernel from (SS) to (S S) denoted Q(middot|s)7 suchthat for each ω isin Ω and every set Σ isin S it holds that

Pω(Σ) =

983133

S

Q(Σ|s) dPω(s)

This definition captures the statistical notion that on a state-by-state basis the distribution

7 Ie (i) the map s 983041rarr Q(Σ|s) is S-measurable for every Σ isin S and (ii) the map Σ 983041rarr Q(Σ|s) is a probabilitymeasure on (S S) for every s isin S

6

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

of signals in E can be generated by taking signals from E and transforming them throughthe state-independent kernel Q(middot) In a sense E does not provide any information that is notcontained in E Indeed E is also said to be more informative than E because every expected-utility decision maker prefers E to E

21 Binary states

In this subsection let Ω = 0 1 and take all beliefs to refer to the probability of state ω = 1There are two individuals A and B with respective prior beliefs βA βB isin (0 1) Given anyexperiment the individualsrsquo respective priors combine with Bayes rule to determine theirrespective posteriors after observing a signal s denoted βi(s) for i isin AB Let EE

A[βB(middot)]denote the ex-ante expectation of individual A over the posterior of individual B under ex-periment E If βA = βB then because the individualsrsquo posteriors always agree it holds thatfor any E EE

A[βB(middot)] = βA this is the martingale or iterated expectations property of Bayesianupdating For individuals with different priors we have

Theorem 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i j isinAB

βi le βj =rArr EEi [βj(middot)] le EE

i [βj(middot)]

βi ge βj =rArr EEi [βj(middot)] ge EE

i [βj(middot)]

and minβA βB le EEi [βj(middot)] le maxβA βB

Suppose that individual A is less optimistic than B ie βA lt βB If an experiment E isuninformativemdashno signal realization would change any individualrsquos beliefsmdashthen EE

A[βB(middot)] =βB gt βA On the other hand if an experiment E is fully informativemdashevery signal reveals thestatemdashthen for any signal s βA(s) = βB(s) and hence βA = EE

A[βA(middot)] = EEA[βB(middot)] lt βB

where the first equality is the previously-noted property of Bayesian updating under anyexperiment In other words individual A believes a fully informative experiment will onaverage bring individual Brsquos posterior perfectly in line with Arsquos own prior whereas an un-informative experiment will obviously entail no such convergence (Of course in turn Bexpects A to update to Brsquos prior under a fully informative experiment) Theorem 1 general-izes this idea to monotonicity among Blackwell-comparable experiments A anticipates thata more informative experiment will on average bring Brsquos posterior closer to Arsquos prior Forshort we will say the theorem shows that information validates the prior or IVP

There is a simple proof for Theorem 1 owing to binary states The key is to recognizethat because the individuals agree on the experiment and only disagree in their priors over

7

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

the state each individualrsquos posterior can be written as a function of both their priors and theother individualrsquos posterior this point does not rely on binary states This observation is alsoused by Gentzkow and Kamenica (2014) Alonso and Camara (2016) and Galperti (2018) todifferent ends For simplicity consider an experiment with a discrete signal space in whichevery signal is obtained with positive probability in both states For any signal realization sBayes rule implies that the posterior belief βi(s) for individual i isin AB satisfies

βi(s)

1minus βi(s)=

βi

1minus βi

P1(s)

P0(s)

where Pω(s) is the probability of observing s in state ω Eliminating the likelihood ratioP1(s)P0(s) yields βB(s) = T (βA(s) βB βA) where

T (βA βB βA) =βA

βB

βA

βAβB

βA+ (1minus βA)

1minusβB

1minusβA

(2)

It is straightforward to verify that this transformation mapping T (middot βB βA) is strictly con-cave (resp convex) in Arsquos posterior when βA lt βB (resp βA gt βB) Theorem 1 followsas an application of Blackwell (1953) who showed that a garbling increases (resp reduces)an individualrsquos expectation of any concave (resp convex) function of his posterior see alsoRothschild and Stiglitz (1970)

The second part of Theorem 1 is a straightforward consequence of combining the first partwith the facts that any experiment is a garbling of a fully informative experiment while anuninformative experiment is a garbling of any experiment (and using the properties of theseextreme experiments noted right after the theorem)

Remark 1 The inequalities in the conclusions of Theorem 1 hold strictly if the garbling isldquostrictrdquo In particular minβA βB lt EE

i [βj(middot)] lt maxβA βB for any experiment E that isneither uninformative nor fully informative

Theorem 1 has an interesting corollary A canonical measure of how much agents A andB disagree when they hold beliefs βA and βB is |βA minus βB| Under this measure an experimentcan generate signals for which posterior disagreement is larger than prior disagreement8 Putdifferently Bayesian agents who agree on the information-generating process can disagree

8 For example let βA = 12 and consider a binary-signal experiment (S = l h) with Pr(l|0) = Pr(h|1) = pFor any p isin (12 1) the posterior disagreement after signal h is larger than the prior disagreement if βB lt 1minuspMore generally it can be shown that whenever βA ∕= βB there is an experiment such that posterior disagreementafter some signal is larger than prior disagreement

8

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

more after new information9 But can they expect ex-ante to disagree more Theorem 1provides a sharp answer captured by the following corollary

Corollary 1 Let Ω = 0 1 and experiment E be a garbling of experiment E Then for any i isinAB

EEi [|βA(middot)minus βB(middot)|] le EE

i [|βA(middot)minus βB(middot)|] (3)

Corollary 1 says that both agents expect a more informative experiment to reduce theirdisagreement by more In particular both predict that any experiment will reduce their dis-agreement on average relative to their prior disagreement This is the case even thoughthey generally hold different expectations about each otherrsquos posteriors for example given afully informative experiment both anticipate zero posterior disagreement even though eitheragent i expects both agentsrsquo posteriors to be βi

The proof of Corollary 1 is simple For any priors βA and βB and any signal s in anyexperiment it is a consequence of Bayesian updating (eg using (2)) that

sign[βA(s)minus βB(s)] = sign[βA minus βB]

In other words any information preserves the prior ordering of the agentsrsquo beliefs Thus if(say) βA ge βB then (3) is equivalent to

EEi [βA(middot)minus βB(middot)] le EE

i [βA minus βB]

which is implied by Theorem 1 because EEi [βi(middot)] = EE

i [βi(middot)] = βi

22 Many states

Now consider an arbitrary finite set of states Ω equiv ω1 ωL sub R with L gt 1 and ω1 lt

middot middot middot lt ωL We write β(ωl) as the probability ascribed to state ωl by a belief β with the notationin bold emphasizing that a belief over Ω is now a vector We assume that the two individualsrsquopriors βA and βB assign strictly positive probability to every state We say that a belief βprime

likelihood-ratio dominates belief β written βprime geLR β if for all ωprime gt ω

βprime(ωprime)β(ω) ge β(ωprime)βprime(ω)

We denote posterior beliefs given a signal s as β(s) equiv (β(ω1|s) β(ωL|s))9 But agentsrsquo posteriors always move in the same direction from their priors ie after any signal s in any

experiment sign[βA(s) minus βA] = sign[βB(s) minus βB ] as can be confirmed using (2) In a sense there cannot bepolarization a point that is established more generally by Baliga et al (2013 Theorem 1)

9

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

Definition 1 An experiment E equiv (SS PωωisinΩ) is an MLRP-experiment if there is a totalorder on S denoted ≽ (with asymmetric relation ≻) such that the monotone likelihood ratioproperty holds sprime ≻ s and ωprime gt ω =rArr p(sprime|ωprime)p(s|ω) ge p(sprime|ω)p(s|ωprime)

As is well known the monotone likelihood ratio property (in the non-strict version above)is without loss of generality when there are only two states any experiment is an MLRP-experiment when L = 2

Let M(β) =983123

ωisinΩ h(ω)β(ω) represent the expectation of an arbitrary non-decreasingfunction h Ω rarr R under belief β The following result generalizes Theorem 1

Theorem 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i j isin AB

M(βi) le M(βj) =rArr EEi [M(βj(middot))] le EE

i [M(βj(middot))]

M(βi) ge M(βj) =rArr EEi [M(βj(middot))] ge EE

i [M(βj(middot))]

and min983051M(βA)M(βB)

983052le EE

i [M(βj(middot))] le max983051M(βA)M(βB)

983052

With many states we can still represent the posterior belief βi(s) of individual i as a trans-formation mapping of βj(s) but this mapping is neither concave nor convex even whenthe prior beliefs of i and j are likelihood-ratio ordered The proof of Theorem 2 (and of allsubsequent formal results) is provided in Appendix A For an illustration suppose M(middot) isthe expected state and Bayesian updating takes the canonical linear form of a convex com-bination of the prior mean and the signal as is the case for any exponential family of sig-nals with conjugate prior (eg normal-normal) Given a signal s isin R under experiment E EE

j [ω|s] = (1minus αE)Ej[ω] + αEs for some αE isin [0 1] Hence

EEi

983045EE

j [ω|s]983046= (1minus αE)Ej[ω] + αEEi[ω]

The weight αE is larger the more informative is the experiment E which implies the conclu-sions of Theorem 2

Theorem 2 says that either agent i expects the otherrsquos posterior to be closer to irsquos priorunder a more informative experiment when closeness of beliefs is measured by the distancebetween their M(middot) statistics Theorem 1 obtains when Ω = 0 1 and M(β) is simply theexpected state under β (ie h(ω) = ω) because both likelihood-ratio ordering assumptions inTheorem 2 are without loss of generality with only two states10 More generally since M(middot)

10 Indeed the likelihood-ratio ordering of the priors can always be viewed as without loss of generalitymdashthe

10

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

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ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

is the expectation of any non-decreasing function of the state Theorem 2 captures the notionthatmdashsubject to the ordering assumptionsmdashinformation validates the prior not only in thesense of the expected state but also for instance in the probability of any ωl ωL

The likelihood-ratio ordering assumptions in Theorem 2 are tight in the following respect(i) there exist priors βB gtLR βA and a non-MLRP-experiment E such that EE

A[M(βB(middot))] gtM(βB) gt M(βA) and (ii) there exist priors βA ≱LR βB and an MLRP-experiment E such thatEE

A[M(βB(middot))] gt M(βA) gt M(βB) See Appendix B for examples demonstrating these pointsthe examples also establish that the likelihood-ratio ordering hypothesis on the priors cannotbe weakened to first-order stochastic dominance

Theorem 2 has an implication about expected disagreement that generalizes Corollary 1

Corollary 2 Consider any two likelihood-ratio ordered priors βA and βB and any two MLRP-experiments E and E with E a garbling of E Then for any i isin AB

EEi [|M(βi(middot))minusM(βj(middot))|] le EE

i [|M(βi(middot))minusM(βj(middot))|]

Corollary 2 says thatmdashsubject to the ordering assumptionsmdashmore information reducesexpected disagreement when the disagreement between beliefs βA and βB is measured by|M(βA) minusM(βB)|11 The corollary follows from Theorem 2 for reasons analogous to our dis-cussion of why Corollary 1 follows from Theorem 1 In particular information from an MLRP-experiment preserves the prior likelihood-ratio ordering of the agentsrsquo beliefs and thus alsothe ordering in terms of their M(middot) statistics moreover since M(middot) is a linear function it holdsfor any i isin AB and any experiment E that EE

i [M(βi(middot)] = M(βi)

In the following sections we use the logic of information validates the prior to studystrategic communication games in which agents begin with a common prior

3 Multi-Sender Disclosure Games

Our main application is to multi-sender voluntary disclosure of verifiable information Webuild on the classic disclosure models mentioned in the Introduction and use IVP to derivenew insights on when competition promotes disclosure

states can be re-labeled so that it holdsmdashso long as (i) the MLRP property of an experiment and (ii) the non-decreasing property of the h(middot) function are both understood to be with respect to the ordering of states thatensures likelihood-ratio ordering of the priors

11 The distance between expected statesmdashmore precisely the expectations of any non-decreasing function ofthe statesmdashis an interesting measure of disagreement but there are of course others See Zanardo (2017) for anaxiomatic approach to measuring disagreement he provides a result in the spirit of Corollary 2 for a family ofdisagreement measures that includes the Kullback-Leibler divergence

11

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

31 The model

Players There is an unknown state of the world ω isin 0 1 A decision maker (DM) willform a belief βDM that the state is ω = 1 For much of our analysis all that matters is thebelief that the DM holds For welfare evaluation however it is useful to view the DM as tak-ing an action a with von-Neumann Morgenstern utility function uDM(aω) such that the DMis strictly better off when there is strictly more information about the state in the Blackwellsense There are two senders indexed by i (Subsection 34 generalizes to many senders) In areduced form each sender i has state-independent preferences over the DMrsquos belief given bythe von-Neumann Morgenstern utility function u(βDM bi) = biβDM where bi isin minus1 1 cap-tures a senderrsquos bias That is each sender has linear preferences over the DMrsquos expectation ofthe state bi = 1 means that sender i is biased upward (ie prefers a higher DMrsquos expectation)and conversely for bi = minus1 Sendersrsquo biases are common knowledge We say that two sendershave similar biases if their biases have the same sign and opposing biases otherwise

Information The DM relies on the senders for information about the state All playersshare a common prior π over the state Each sender may exogenously obtain some privateinformation about the state Specifically with independent probability pi isin (0 1) a sender i isinformed and receives a signal si isin S with probability 1minus pi he is uninformed in which casewe denote si = φ If informed sender irsquos signal is drawn independently from a distributionthat depends upon the true state Without loss we equate an informed senderrsquos signal withhis private belief ie a senderrsquos posterior on state ω = 1 given only his own signal s ∕= φ (asderived by Bayesian updating) is s For convenience we assume the cumulative distributionof an informed senderrsquos signals in each state F (s|ω) for ω isin 0 1 have common supportS = [s s] sube [0 1] and admit respective densities f(s|ω) It is straightforward to allow F (middot) tobe different for different senders but we abstract from such heterogeneity to reduce notation

Communication Signals are ldquohard evidencerdquo a sender with signal si isin S cup φ can send amessage mi isin siφ In other words an uninformed sender only has one message availableφ while an informed sender can either report his true signal or feign ignorance by sendingthe message φ12 We refer to any message mi ∕= φ as disclosure and the message mi = φ

as nondisclosure When an informed sender chooses nondisclosure we say he is concealinginformation That senders must either tell the truth or conceal their information is standard

12 Due to the sendersrsquo monotonic preferences standard ldquoskeptical posturerdquo arguments imply that our resultswould be unaffected if we were to allow for a richer message space for example if an informed sender couldreport any subset of the signal space that contains his true signal Likewise allowing for cheap talk would notaffect our results as cheap talk cannot be influential in equilibrium

12

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

a justification is that signals are verifiable and large penalties will be imposed on a sender if areported signal is discovered to be untrue Note that being uninformed is not verifiable

Message costs A sender i who sends message mi ∕= φ bears a known utility cost c isin RWe refer to the case of c gt 0 as one of disclosure cost and c lt 0 as one of concealment cost Adisclosure cost captures the idea that costly resources may be needed to certify or make veri-fiable the information that one has A concealment cost captures a resource-related or psychicdisutility from concealing information or the expectation of a penalty from ex-post detectionof having withheld information (Daughety and Reinganum (2018) and Dye (2017) considerspecific versions) As is well known a disclosure cost precludes full disclosure (Jovanovic1982 Verrecchia 1983) For this reason our conclusions under c gt 0 do not require the as-sumption that the sender may be uninformed (pi lt 1) We make that assumption to providea unified treatment of both disclosure costs (c gt 0) and no costsconcealment costs (c le 0)13

In the latter cases there would be full disclosure or ldquounravelingrdquo were pi = 1

Timing The game is the following nature initially determines the state ω and then in-dependently (conditional on the realized state) draws each sender irsquos private informationsi isin S cup φ all senders then simultaneously send their respective messages mi to the DM(whether messages are public or privately observed by the DM is irrelevant) the DM thenforms her belief βDM according to Bayes rule whereafter each sender irsquos payoff is realized as

biβDM minus c middot 11mi ∕=φ (4)

All aspects of the game except the state and sendersrsquo signals (or lack thereof) are commonknowledge Our solution concept is the natural adaptation of perfect Bayesian equilibriumwhich we will refer to simply as ldquoequilibriumrdquo14 The notion of welfare for any player isex-ante expected utility

13 When c = 0 our setting is related to Jackson and Tan (2013) and Bhattacharya and Mukherjee (2013) Jacksonand Tan (2013) assume a binary decision space which makes their sendersrsquo payoffs non-linear in the DMrsquos pos-terior and shifts the thrust of their analysis They are ultimately interested in comparing different voting ruleswhich is effectively like changing the pivotal DMrsquos preferences we instead highlight how the presence of disclo-sure or concealment costs affects the strategic interaction between senders holding fixed the DMrsquos preferencesBhattacharya and Mukherjee (2013) assume that informed sendersrsquo signals are perfectly correlated but allow forsenders to have non-monotonic preferences over the DMrsquos posterior

14 That is (1) the receiver forms her belief using Bayes rule on path (note that the message mi = φ is necessarilyon path) and treating any off-path message mi isin S as proving signal si = mi and (2) each sender chooses hismessage optimally given his information the other sendersrsquo strategies and the receiverrsquos belief updating

13

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

32 A single-sender benchmark

As a preliminary step and to set a benchmark begin by considering a (hypothetical) game be-tween a single sender i and the DM Our analysis in this subsection generalizes some existingresults in the literature For concreteness suppose the sender is upward biased straightfor-ward analogs of the discussion below apply if the sender is downward biased

For any β isin [0 1] define fβ(s) = βf(s|1) + (1minus β)f(s|0) as the unconditional density ofsignal s given a belief that puts probability β on state ω = 1 Let Fβ be the cumulative dis-tribution of fβ Since the sender has private belief s upon receiving signal s disclosure ofsignal s will lead to the DM also holding belief s It follows that given any nondisclosure be-lief ie the DMrsquos posterior belief when there is nondisclosure the optimal strategy for thesender (if informed) is a threshold strategy of disclosing all signals above some disclosurethreshold say s and concealing all signals below it If the sender is uninformed his onlyavailable message is φ Suppose the sender uses a disclosure threshold s Define the functionη [0 1]times (0 1)times [0 1] rarr [0 1] by

η(s p π) =1minus p

1minus p+ pFπ(s)π +

pFπ(s)

1minus p+ pFπ(s)Eπ[s|s lt s] (5)

where Eπ[middot] refers to an expectation taken with respect to the distribution Fπ The functionη(s p π) is the posterior implied by Bayes rule in the event of nondisclosure given a conjec-tured disclosure threshold s a probability p of the sender being informed and prior π

An increase in the senderrsquos disclosure threshold has two effects on the DMrsquos nondisclosurebelief first it increases the likelihood that nondisclosure is due to the sender concealing hissignal rather than being uninformed second conditional on the sender in fact concealinghis signal it causes the DM to expect a higher signal As the DMrsquos belief conditional onconcealment is lower than the prior (since the sender is using a threshold strategy) these twoeffects work in opposite directions Moreover the second effect is stronger than the first if andonly if s gt η(s p π) On the other hand holding the disclosure threshold fixed an increase inthe probability that the sender is informed has an unambiguous effect because it increases theprobability that nondisclosure is due to concealed information rather than no information

Lemma 1 The nondisclosure belief function η(s p π) has the following properties

1 It is strictly decreasing in s when s lt η(s p π) and strictly increasing when s gt η(s p π) Con-sequently there is a unique solution to s = η(s p π) which is the interior point argmins(s p π)

2 It is weakly decreasing in p strictly if s isin (s s)

14

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

All but the first sentence of part 1 has appeared previously in Acharya DeMarzo and Kre-mer (2011 Proposition 1 and subsequent discussion) see also Demarzo Kremer and Skrzy-pacz (2018 Proposition 1) The strict quasi-convexity of η(middot p π) in part 1 of the lemma willprove crucial for comparative statics

It follows from the foregoing discussion that any equilibrium is fully characterized bythe senderrsquos disclosure threshold If this threshold is interior the sender must be indifferentbetween disclosing the threshold signal and concealing it As sender irsquos payoff from disclosingany signal si is si minus c we obtain the following equilibrium characterization

Proposition 1 Assume there is only one sender i and this sender is biased upward

1 Any equilibrium has a disclosure threshold s0i such that (i) s0i is interior and η(s0i pi π) = s0iminuscor (ii) s0i = s and π le sminus c or (iii) s0i = s and π ge sminus c Conversely for any s0i satisfying (i)(ii) or (iii) there is an equilibrium with disclosure threshold s0i

2 There is a unique equilibrium if there is no message cost or a concealment cost (c le 0) Moreoverthe equilibrium disclosure threshold is interior if there is no message cost (c = 0)

3 There can be multiple equilibria if there is a disclosure cost (c gt 0)

Part 1 of Proposition 1 is straightforward parts 2 and 3 build on Lemma 1 Multipleequilibria can arise under a disclosure cost because in the relevant domain (to the right of thefixed point of η(middot pi π)) the DMrsquos nondisclosure belief is increasing in the senderrsquos disclosurethreshold In such cases we will focus on properties of the highest and lowest equilibriain terms of the disclosure threshold As an equilibrium with a higher disclosure thresholdis less (Blackwell) informative these extremal equilibria are respectively the worst and bestequilibria in terms of the DMrsquos welfare On the other hand the ranking of these equilibria isreversed for the senderrsquos welfare To see this note that because the senderrsquos preferences arelinear in the DMrsquos belief and we evaluate welfare at the ex-ante stage the senderrsquos welfarein an equilibrium with threshold s0i is π minus pi(1 minus Fπ(s

0i ))c Thus when c gt 0 the senderrsquos

welfare is higher when the disclosure threshold is higher he cannot affect the DMrsquos belief inexpectation and prefers a lower probability of incurring the disclosure cost

When c = 0 the senderrsquos belief if he receives the threshold signal is identical to the DMrsquosequilibrium nondisclosure belief When c ∕= 0 these two beliefs will differ in any equilib-rium if c gt 0 the senderrsquos threshold belief s0i is higher than the DMrsquos nondisclosure beliefη(s0i pi π) and the opposite is true when c lt 0 This divergence of equilibrium beliefs whenthe sender withholds informationmdashwhich for brevity we shall refer to as disagreementmdashwillprove crucial Note that when there is disagreement the sender knows the DMrsquos belief but not

15

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

vice-versa consistent with Aumann (1976) the beliefs under disgareement are not commonknowledge in this game with a common prior

Proposition 1 is stated for an upward biased sender If the sender is instead downwardbiased he reveals all signals below some threshold The DMrsquos nondisclosure belief functionchanges from (5) to

1minus p

1minus p+ p(1minus Fπ(s))π +

p(1minus Fπ(s))

1minus p+ p(1minus Fπ(s))Eπ[s|s gt s]

This expression is single-peaked in s The condition for an interior equilibrium is that it mustequal s + c since the senderrsquos payoff is now minusβDM minus c As with an upward biased senderequilibrium is unique when c le 0 while there can be multiple equilibria when c gt 0 Note thatfor any c ∕= 0 the direction of disagreement between the sender and the DM is now reversedfor a downward biased sender c gt 0 implies the senderrsquos threshold belief is lower than theDMrsquos nondisclosure belief and conversely for c lt 0 Furthermore a lower threshold nowcorresponds to revealing less information

The following comparative statics hold with an upward biased sender the modificationsfor a downward biased sender are straightforward in light of the above discussion

Proposition 2 Assume there is only one sender and this sender is upward biased

1 A higher probability of being informed leads to more disclosure the highest and lowest equilib-rium disclosure thresholds (weakly) decrease

2 An increase in disclosure cost or a reduction in concealment cost leads to less disclosure thehighest and lowest equilibrium disclosure thresholds (weakly) increase

The logic for the first part follows from Lemma 1 given any conjectured threshold ahigher pi leads to a lower nondisclosure belief which increases the senderrsquos gain from disclo-sure over nondisclosure of any signal For the case of c = 0 this comparative static has alsobeen noted by other authors eg Jung and Kwon (1988) and Acharya et al (2011) The secondpart of Proposition 2 is straightforward as a higher c makes disclosure less attractive Sincescaling the magnitude of c is equivalent to scaling the agentrsquos bias parameter bi (cf expression(4)) an equivalent interpretation is that an agent with a stronger persuasion motive disclosesmore when c gt 0 but less when c lt 0

Figure 1 summarizes the results of this subsection15

15 In the figure η(middot) has slope less than one when it crosses si minus c at the highest crossing point This makes

16

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

si

si minus csi

π si minus c

Figure 1 The single-sender game with an upward biased sender illustrated with parameters c gt 0 gt cprime andpprimei gt pi Equilibrium disclosure thresholds are given by the intersections of η(si middot) and si minus c or si minus cprime

Although we postpone a formal argument to Subsection 33 it is worth observing nowthat the comparative statics on disclosure have direct welfare implications Since the DM al-ways prefers more disclosure a lower message cost andor a higher probability of the senderbeing informed (weakly) increases the DMrsquos welfare in a single-sender setting subject to anappropriate comparison of equilibria in particular focussing on the extremal equilibria

33 Strategic substitutes and complements

We are now ready to study the two-sender disclosure game For concreteness we will sup-pose that both senders are upward biased the modifications needed when one or both sendersare downward biased are straightforward

Lemma 2 Any equilibrium is a threshold equilibrium ie both senders use threshold strategies

In light of Lemma 2 we focus on threshold strategies A useful simplification affordedby the assumption of conditionally independent signals is that the DMrsquos belief updating isseparable in the sendersrsquo messages In other words we can treat it as though the DM firstupdates from either sender irsquos message just as in a single-sender model and then uses this

transparent that an increase in pi leads to a reduction in the highest equilibrium threshold If the slope of η(middot)were larger than one at the highest crossing point then the highest equilibrium threshold would be s and asmall increase in pi would not alter this threshold We also note that the η(middot) depicted in the figure is valid itcan be shown that for any continuously differentiable function ψ [s s] rarr [0 1] with ψ(s) = ψ(s) isin (0 1) andsign[sminusψ(s)] = sign[ψprime(s)] (where ψprime denotes the derivative) there are parameters of the modelmdashviz π isin (0 1)pi isin (0 1) f(s|0) and f(s|1)mdashsuch that the nondisclosure belief η(si pπ) = ψ(si) for all si

17

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

updated belief as an interim prior to update again from the other sender jrsquos message withoutany further attention to irsquos message Thus given any conjectured pair of disclosure thresh-olds (s1 s2) there are three relevant nondisclosure beliefs for the DM if only one sender i

discloses his signal si while sender j sends message φ the DMrsquos belief is η(sj pj si) if there isnondisclosure from both senders the DMrsquos belief is η(sj pj η(si pi π))

Suppose the DM conjectures that sender i is using a disclosure threshold si As discussedearlier if i discloses his signal then his expectation of the DMrsquos beliefmdashviewed as a randomvariable that depends on jrsquos messagemdashis si no matter what strategy j is using16 On the otherhand if i conceals his signal then he views the DM as updating from jrsquos message based ona prior of η(si pi π) that may be different from si Denote jrsquos disclosure threshold by sj andany particular message sent by j as mj isin S cup φ Let β(I q) denote the posterior belief givenan arbitrary information set I and prior belief q Then sender irsquos posterior belief about thestate given jrsquos message and his own signal can be written as β(mj si) The transformation (2)implies that irsquos expected payoffmdashequivalently his expectation of the DMrsquos posterior beliefmdashshould he conceal his signal is given by

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

where Esj pj denotes that the expectation is taken over mj using the distribution of beliefs thatsj and pj jointly induce in i about mj (given si)

It is useful to study the ldquobest responserdquo of sender i to any disclosure strategy of senderj More precisely let sBR

i (sj pi pj) represent the equilibrium disclosure threshold in a (hypo-thetical) game between sender i and the DM when sender j is conjectured to mechanicallyadopt disclosure threshold sj we call this sender irsquos best response The threshold si is a bestresponse if and only if it satisfies any one of the following

983099983105983105983105983103

983105983105983105983101

U(si pi sj pj) = si minus c and si isin (s s) or

U(s pi sj pj) le sminus c and si = s or

U(s pi sj pj) ge sminus c and si = s

(6)

The necessity of condition (6) is clear sufficiency follows from the argument given in theproof of Lemma 2 In any equilibrium of the overall game (slowast1 slowast2) condition (6) must hold foreach sender i with si = slowasti when his opponent uses sj = slowastj

16 The distribution of the DMrsquos beliefs as a function of jrsquos message depends both on jrsquos strategy and the DMrsquosconjecture about jrsquos strategy As the two must coincide in equilibrium we bundle them to ease exposition

18

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

sisi minus c

π si minus c

si

Figure 2 The ldquobest responserdquo of sender i to sender j (both upward biased) illustrated with parameters c gt 0 gtcprime pprimej gt pj and sprimej lt sj lt s

Lemma 3 For any sj and pj

si = η(si pi π) =rArr U(si pi sj pj) = si

si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) lt si

si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) gt si

Moreover both weak inequalities above are strict if and only if sj lt s

Lemma 3 is a direct consequence of Theorem 1 (and Remark 1) it reflects that i expects jrsquosinformation disclosure to bring the DMrsquos posterior belief closer to si which is irsquos belief basedon his own signal Graphically this is seen in Figure 2 by comparing the red (short dashed)curve that depicts U(middot) with the black (solid) curve depicting η(middot) The former is a rotation ofthe latter curve around its fixed point toward the diagonal

It is evident from Figure 2 that jrsquos information disclosure has very different consequencesfor the best response of i depending on whether there is a cost of disclosure or a cost ofconcealment If c gt 0 (disclosure cost) then the smallest and largest solutions to (6) will berespectively larger than the smallest and largest single-sender disclosure thresholds If c lt 0

(concealment cost depicted as cprime in Figure 2) the largest solution will be smaller than theunique single-sender threshold17 If c = 0 the unique solution is the same as the single-sender threshold These contrasting effects are due to the different nature of disagreement

17 Although not seen in the figure there can be multiple solutions to (6) even when c lt 0

19

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

induced by the sender When there is a disclosure cost the threshold type in any single-senderequilibrium has a higher belief than the DM upon nondisclosure and hence an expected shiftof the DMrsquos posterior toward the threshold belief makes concealment more attractive Bycontrast when there is a concealment cost the threshold type has a lower belief than the DMupon nondisclosure and hence an expected shift of the DMrsquos posterior toward the thresholdbelief makes concealment less attractive

Theorem 1 implies that these insights are not restricted to comparisons with the single-sender setting More generally the same points hold whenever we compare any (sj pj) with(sprimej p

primej) such that sprimej le sj and pprimej ge pj The latter experiment is more informative than the

former because sender j is more likely to be informed and discloses more conditional onbeing informed More precisely notice that for any message mprime

j under (sprimej pprimej) one can garbleit to produce message mj as follows

mj =

983099983103

983101φ if mprime

j = φ or mprimej isin [sprimej sj)

mprimej with prob pjpprimej or φ with prob 1minus pjp

primej if mprime

j ge sj

In each state of the world the distribution of mj as just constructed is the same as the distri-bution of sender jrsquos message under (sj pj) Thus the message produced under (sj pj) is agarbling of that produced under (sprimej pprimej)

The effect of sender jrsquos message becoming more informative is depicted in Figure 2 asa shift from the red (short dashed) curve to the blue (long dashed) curve Whether senderirsquos best response is to disclose less or more of his own information turns on whether c gt 0

or c lt 0 the logic is the same as discussed earlier For any given sj there can be multiplesolutions to (6) hence sBR

i (middot) is a best-response correspondence We say that sender irsquos bestresponse increases if the largest and the smallest element of sBR

i (middot) both increase (weakly) wesay that the best response is strictly greater than some s written si

BR(middot) gt s if the smallestelement of sBR

i (middot) is strictly greater than s

Proposition 3 Assume both senders are upward biased Any sender irsquos best-response disclosurethreshold sBR

i (sj pi pj) is decreasing in pi Furthermore let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 (Independence) If c = 0 then sBRi (sj pi pj) = s0i is independent of sj and pj

2 (Strategic complements) If c lt 0 then (i) sBRi (sj pi pj) le s0i with equality if and only if

s0i = s or sj = s and (ii) sBRi (sj pi pj) increases in sj and decreases in pj

3 (Strategic substitutes) If c gt 0 then (i) sBRi (sj pi pj) ge s0i with equality if and only if s0i = s

20

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

or sj = s and (ii) sBRi (sj pi pj) decreases in sj and increases in pj

Since each senderrsquos best response is monotone existence of an equilibrium in the two-sender game follows from Tarskirsquos fixed point theorem When c lt 0 (concealment cost) thestrategic complementarity in disclosure thresholds implies that there is a largest equilibriumwhich corresponds to the highest equilibrium thresholds for both senders Each senderrsquosmessage in the largest equilibrium is a garbling of his message in any other equilibrium Itfollows that the largest equilibrium is the least informative and the worst in terms of the DMrsquoswelfare Similarly the smallest equilibriummdashthat with the lowest equilibrium thresholds forboth sendersmdashis the most informative and the best in terms of the DMrsquos welfare On the otherhand when c gt 0 (disclosure cost) the two disclosure thresholds are strategic substitutesThere is an i-maximal equilibrium that maximizes sender irsquos threshold and also minimizessender jrsquos threshold across all equilibria Likewise there is a j-maximal equilibrium that mini-mizes sender irsquos threshold and also maximizes sender jrsquos threshold across all equilibria Thesetwo equilibria are not ranked in terms of (Blackwell) informativeness and in general cannotbe welfare ranked for the DM moreover neither of these equilibria may correspond to eitherthe best or the worst equilibrium for the DM18

The following result is derived using standard monotone comparative statics argumentsAlthough the formal statement refers to extremal equilibria the comparative statics also ob-tain for any equilibrium that is stable in the sense of adaptive dynamics see Echenique (2002)

Proposition 4 Assume both senders are upward biased For any i isin 1 2

1 If c le 0 then an increase in pi or a decrease in c (a higher concealment cost) weakly lowers thedisclosure thresholds of both senders in both the worst and the best equilibria

2 If c gt 0 then an increase in pi weakly lowers sender irsquos disclosure threshold and weakly raisessender jrsquos disclosure threshold in both the i-maximal and the j-maximal equilibria A decreasein c (a lower disclosure cost) has ambiguous effects on the two sendersrsquo equilibrium disclosurethresholds in both the i- and j-maximal equilibria

One can view the single-sender game with i as a two-sender game where sender j is neverinformed ie pj = 0 With this in mind a comparison of the single-sender game and thetwo-sender game can be obtained as a corollary to Proposition 3 and Proposition 4

18 As explained after Proposition 1 the sendersrsquo welfare ranking across equilibria just depends on the prob-ability of disclosure When c lt 0 both sendersrsquo welfare is lowest in the largest equilibrium and highest in thesmallest equilibrium When c gt 0 sender irsquos welfare is highest in the i-maximal equilibrium and lowest in thej-maximal equilibrium

21

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

Corollary 3 Assume both senders are upward biased and let s0i denote the unique (resp smallest)equilibrium threshold in the single-sender game with i when c le 0 (resp c gt 0)

1 If c = 0 equilibrium in the two-sender game is unique and is equal to (s01 s02) The DMrsquos welfareis strictly higher in the two-sender game than in a single-sender game with either sender

2 If c lt 0 every equilibrium in the two-sender game is weakly smaller than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare is strictly higher in any equilibrium of the two-sender game than in a single-sender game with either sender

3 If c gt 0 every equilibrium in the two-sender game is weakly larger than (s01 s02) with equality

if and only if s01 = s02 = s The DMrsquos welfare in the best equilibrium of the two-sender game maybe higher or lower than in the best equilibrium of the single-sender game with sender i or senderj alone

Part 1 of Corollary 3 follows from part 1 of Proposition 3 When c = 0 the best response ofeach sender is to use the same disclosure threshold as in the single-sender setting regardlessof the other senderrsquos strategy Since the DM receives two messages instead of just one andthe probability distribution of these messages remain the same as in the single-sender gameshe is better off when facing both senders than when facing either sender alone

Part 2 of Corollary 3 can be obtained by considering the worst equilibrium of the two-sender game For the case of concealment cost (c lt 0) let slowast(pi pj) represent the vector of dis-closure thresholds in the worst equilibrium Proposition 4 implies that slowasti (pi pj) le slowasti (pi 0) =

s0i and slowastj(pi pj) le slowastj(0 pj) = s0j Thus slowast(pi pj) is weakly smaller than (s0i s0j) and hence every

equilibrium is weakly smaller than (s0i s0j) It follows that the DMrsquos welfare is higher than in

the unique equilibrium of the single-sender game with either sender This higher welfare isdue to both a direct effect of receiving information from an additional sender and an indirecteffect wherein each sender is now disclosing more than in the single-sender setting

Finally for the case of disclosure cost (c gt 0) let silowast(pi pj) represent the i-maximal equi-librium and sjlowast(pi pj) represent the j-maximal equilibrium In any equilibrium irsquos thresh-old is at least as large as sjlowasti (pi pj) ge sjlowasti (pi 0) = s0i where the inequality is by part 2 ofProposition 4 Analogously sender jrsquos threshold in any equilibrium is at least as large assilowastj (pi pj) ge silowastj (0 pj) = s0j Thus both senders are (weakly) less informative than in theDMrsquos best equilibrium of the single-sender game The overall welfare comparison betweenthe two-sender game and the single-sender game is generally ambiguous While adding asecond sender has a direct effect of increasing the DMrsquos information there is an adverse indi-rect effect due to the strategic substitution in disclosure of the other sender It is possible the

22

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

net effect can (strictly) reduce the DMrsquos welfaremdasheven when the two senders have oppositebiases which is often thought to particularly promote information disclosure An examplewith a familiar quadratic loss function for the DM is available from the authors on request

It is appropriate to compare our welfare results with Bhattacharya and Mukherjee (2013)They study a related model to ours allowing senders to have single-peaked preferences overthe DMrsquos posterior but they assume c = 0 (no message costs) and perfectly correlated signalsThey show that an increase in the probability of a sender being informed can reduce the DMrsquoswelfare However a necessary condition for this to happen in their model is that at least onesender must have non-monotonic preferences over the DMrsquos posterior which in turn implies(because senders have single-peaked preferences) that the senders share the same rankingover decisions on a subset of the decision space In this sense their result requires that sendersare not in ldquopure conflictrdquo whereas our setup allows senders to have diametrically opposingpreferences More broadly our results on equilibrium behavior and welfare for c ∕= 0 areorthogonal and complementary to their treatment of non-monotonic preferences

The assumption that sendersrsquo signals are conditionally independent is clearly importantfor our analytical methodology as without it we cannot apply Theorem 1 If the signals areconditionally correlated then upon nondisclosure sender i and the DM disagree not only onthe probability assessment of the states but also on the experiment corresponding to senderjrsquos message Relaxing the conditional independence assumption to obtain a general analysisappears intractable We illustrate in Supplementary Appendix C1 how some of our substan-tive economic conclusions would change under a significantly different information structureperfectly correlated signals Another assumption that is important in applying Theorem 1 isthat each sender has linear preferences Supplementary Appendix C2 discuses how our re-sult under c = 0 extends to non-linear preferences and how our results under c ∕= 0 may ormay not hold under non-linear preferences

34 Many senders

Our results readily generalize to any finite number of senders Suppose in addition to sendersi and j there are K other senders all of whom simultaneously send messages to the DM Letm represent the collection of these K messages Then sender irsquos posterior belief given hisown signal si sender jrsquos message mj and the K other sendersrsquo messages m is β(mjm si) =

β(mj β(m si)) The DMrsquos belief given the K sendersrsquo messages m and given nondisclosureby sender i is η(si pi β(m π)) Thus the transformation mapping from (2) and the law of

23

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

iterated expectations imply that the expected payoff for sender i from concealing his signal is

E983045Esj pj [T (β(mj β(m si)) η(si pi β(m π)) β(m si)) | m]

983046

The inside conditional expectation (given m) is taken over the distribution of mj while theoutside expectation is taken over the distribution of m generated from the equilibrium strate-gies of the K senders Given any m the transformation T (middot) in the multi-sender case is thesame as that in the two-sender case with the common prior π replaced by β(m π) Since ourresults hold for any π the logic of strategic substitution or strategic complementarity contin-ues to apply in the multi-sender case In particular when c lt 0 Esj pj [T (middot) | m] increases in sj

and decreases in pj for any m Consequently sender irsquos expected payoff from nondisclosureE983045Esj pj [T (middot) | m]

983046also increases in sj and decreases in pj Thus disclosure by any two senders

are strategic complements Similarly in the case of disclosure cost (ie c gt 0) disclosure byany two senders are strategic substitutes

It follows from these observations that when there is either no message cost or a conceal-ment cost (c le 0) the DM always benefits from having more senders to supply her withinformation When there is disclosure cost (c gt 0) on the other hand an increase in the num-ber of senders has ambiguous effects on each senderrsquos disclosure threshold and can lead toeither an increase or decrease in the DMrsquos welfare

35 Sequential reporting

The key insight from our analysis of simultaneous disclosure extends to sequential disclosureFor concreteness consider a two-sender game in which both senders are upward biased butdisclosure is sequential sender 1 reports first and his message m1 is made public to both theDM and sender 2 before sender 2 submits his report Sender 2 now effectively faces a single-sender problem where he and the DM share a common prior say β(m1 π) which is a functionto be determined in equilibrium Proposition 2 implies that sender 2 will adopt a disclosurethreshold s02 which depends negatively on p2

Consider now the disclosure decision of sender 1 when the DM conjectures that he isusing a disclosure threshold s1 with corresponding nondisclosure belief η(s1 p1 π) If sender1 discloses his signal s1 his expectation of the DMrsquos posterior belief is simply s1 If he choosesnondisclosure his expectation is Es02p2

[T (β(m2 s1) η(s1 p1 π) s1)] Since sender 2 disclosesmore when he is better informed a higher p2 makes the message m2 more informative bothdirectly through a higher probability of sender 2 getting a signal and indirectly through alower disclosure threshold s02 IVP implies that the DMrsquos belief is expected to move away

24

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

from η(s1 p1 π) toward s1 The same logic that establishes Proposition 4 therefore gives thefollowing result whose proof is omitted

Proposition 5 Consider sequential disclosure and assume sender 1 the first mover is upward biasedIf c gt 0 (resp c lt 0) a higher p2 weakly increases (resp weakly lowers) the equilibrium disclosurethreshold of sender 1 in the 1-maximal and 2-maximal equilibria

An immediate corollary to Proposition 5 is that in the case of concealment cost the firstsender discloses more than he does in a single-sender setting As a result the DM is alwaysbetter off in a sequential game than with sender 1 alone On the other hand a welfare compar-ison between the sequential game and the simultaneous move game is generally ambiguous

We also note that if c = 0 the irrelevance result still holds for sender 1 the first senderadopts the same disclosure threshold under sequential reporting as the disclosure thresholdin the single-sender problem The disclosure threshold chosen by the second sender howeverdepends on the message sent by sender 1 it may be higher or lower than sender 2rsquos disclosurethreshold were he the only sender

4 Costly Signaling

A key feature of the multi-sender disclosure game of Section 3 is that message costs induceequilibrium disagreement between a sender and the receiver Since a sender needs to predicthow the receiver reacts to the message from another sender whose informativeness is itself anequilibrium object our IVP result is particularly pertinent because it provides a useful generaltool regarding expectations about beliefs when individuals disagree

In this section we illustrate in a costly-signaling application how IVP is useful even whenthere may be no disagreement in equilibrium (because of separation) rather its usefulnesscomes from considerations of how disagreement off the equilibrium path would be affected bynew information The application also illustrates how IVP is applicable even with just onesender and an exogenous information source

41 The model

We consider a communication game with lying costs a variation of Kartik (2009) A senderand a receiver share a common prior belief about a state ω isin Ω = 0 1 The sender has typet isin [0 1] which is drawn from a distribution F (middot|ω) with corresponding density f(middot|ω) As inSection 3 we assume without loss of generality that t is the senderrsquos private belief that ω = 1For simplicity we assume f(t|ω) gt 0 for all t isin (0 1) and ω We sometimes refer to t as the

25

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

ldquotruthrdquo The sender sends a message m isin [0 1] that entails a cost c(m t) elaborated belowThe receiver forms a belief based on both m and an additional signal s isin [0 1] that conditionalon the state ω is drawn independently of t or m from a distribution G(middot|ω) with densityg(middot|ω) Without loss we assume that the signal s satisfies the weak monotone likelihood ratioproperty The signal s can either be the receiverrsquos private information or it can be publiclyobserved but only after the sender has chosen his message m We assume that no realizationof s perfectly reveals ω Denote the receiverrsquos posterior expectation of the state by E[ω|m s]The senderrsquos payoff is linear in the receiverrsquos expectation specifically his payoff is

E[ω|m s]minus c(m t)

That is the sender is upward biased and prefers a higher posterior belief in the receiver (Ouranalysis carries over to a downward bias with straightforward modifications)

The receiverrsquos belief updating process can be analyzed in two steps based on the messagem from the sender she forms an interim belief π(m) about the state (all beliefs about the staterefer to Pr[ω = 1]) and then uses this interim belief to further update from the signal s to forma posterior belief β(s π(m)) By Bayes rule

β(s π) =πg(s|1)

πg(s|1) + (1minus π)g(s|0)

The expected payoff of a type-t sender from sending message m is thus

Es|t[β(s π(m))]minus c(m t)

where Es|t denotes the type-t senderrsquos expectation over the signal s An important observationis that the first term in the above display is strictly increasing in π(m) the sender prefers thereceiver to hold a higher interim belief

As in Kartik (2009) assume the cost function c(middot) is smooth with partc(t t)partm = 0 for all tie the marginal cost of lying when one is telling the truth is zero Furthermore the marginalcost of sending a higher message is increasing in m and decreasing in t ie part2c(m t)partm2 gt

0 gt part2c(m t)partmpartt

A single-crossing condition Unlike in a setting without exogenous information for the re-ceiver the above cost assumptions do not assure a suitable single-crossing property of thesenderrsquos indifference curves The indifference curves of interest are those in the space of thesenderrsquos message m and the receiverrsquos interim belief π These indifference curves are upward-

26

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

sloping for m gt t sending a higher message than the truth requires compensation through ahigher interim belief The single-crossing property we require is that these indifference curvesare flatter for higher types meaning that higher types are more willing to send higher mes-sages (among those above the truth) to induce a given increase in interim belief It turns outthat this property is assured by the following assumption which we will maintain

For m gt tpart2c(m t)partmpartt

partc(m t)partmle minus 1

1minus t (7)

Condition (7) can be interpreted as saying that higher types have a sufficiently largemarginal lying cost advantage relative to marginal lying costs19 An example of a lying costfunction that satisfies all our requirements is c(m t) = (mminus t)2

Lemma 4partc(m t)partm

partEs|t[β(s π)]partπstrictly decreases in t for t lt m

Another interpretation Although our model is posed as a communication game we mayalso interpret it as a variation of the standard Spence (1973) signaling model In this interpre-tation a worker possesses a private traitcharacteristic his type t (eg intelligence) which isindicative of whether his job productivity will be high or low but a potential employer mayalso observe a signal s about his productivity (eg through the job interview process) Thewage of a worker depends on the employerrsquos posterior of the workerrsquos productivity giventhe schooling level m chosen by the worker and the employerrsquos own signal s The marginalcost of schooling is decreasing in the characteristic t (part2c(m t)partmpartt lt 0) The constraint thatm isin [0 1] introduces an upper bound on the signaling level m but this assumption can berelaxed Further while we suppose that higher-type workers intrinsically prefer acquiringmore education (partc(t t)partm = 0 for all t) our analysis also applies under the more traditionalassumption that all workers prefer less education (partc(m t)partm gt 0 for all m t) See the dis-cussion after Proposition 6 We also note that the model can be reformulated with discretetypes with little change in the subsequent analysis except for possible mixing by some type

42 Equilibrium and comparative statics

A fully separating outcome is not supportable as a (weak perfect Bayesian) equilibrium forreasons similar to those discussed in Kartik (2009) We focus on low types separate and high

19 Similarly in Daley and Green (2014) a marginal cost advantage of the higher type does not guarantee singlecrossing of ldquobelief indifference curvesrdquo in the action-interim belief space rather single crossing requires themarginal cost advantage to be sufficiently large

27

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

types poolmdashhereafter LSHPmdashequilibria That is we will look for equilibria in which there is acutoff t isin [0 1] such that the senderrsquos (pure) strategy micro [0 1] rarr [0 1] satisfies

1 Riley condition micro(0) = 02 Separation at bottom micro(middot) is strictly increasing on [0 t)3 Pooling at top micro(t) = 1 for all t gt t

This class of equilibria extends the standard least cost separating equilibrium logic to a boundedsignal space See Kartik (2009) for more detailed discussion and justification of LSHP equilib-ria albeit when there is no exogenous information for the receiver

In an LSHP equilibrium the receiver can infer the senderrsquos type if his message m belongsto the interval [0 micro(t)) For such m her interim belief upon observing m is simply π(m) =

microminus1(m) If m = 1 the receiver infers that t gt t so π(1) = πP (t) = E[ω|t gt t] where thesuperscript P is mnemonic for ldquopooledrdquo We note that πP (t) isin (t 1)

The equilibrium strategy micro(middot) for t lt t is determined by the following differential equation

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ (8)

with boundary condition micro(0) = 0 Equation 8 is obtained from the binding local upwardincentive compatibility constraints The left-hand-side is the marginal cost for type t of mim-icking a slightly higher type the right-hand-side is the marginal benefit which comes frominducing a higher interim belief in the receiver Note that the benefit is affected by the senderrsquosbelief about the exogenous signal s Since microprime(t) gt 0 and the right-hand-side is strictly positivewe must have micro(t) gt t for all t gt 0 Thus the solution micro(middot) to the boundary-value problemwill hit 1 at some interior t ie micro(t) = 1 for some t isin (0 1) The cutoff t isin [0 t) must satisfy

t gt 0 =rArr Es|t983045β(s πP (t))

983046minus c(1 t) = Es|t [β(s t)]minus c(micro(t) t) (9)

t = 0 =rArr Es|0983045β(s πP (0))

983046minus c(1 0) ge Es|0 [β(s 0)]minus c(0 0) (10)

Condition (9) requires that if t is interior then type t must be indifferent between sendingmessage m = 1 (pooling with types above him) and sending message micro(t) (revealing his truetype) In this case micro(middot) is discontinuous at t = t If (10) holds then every type prefers to poolthan separate so there is complete pooling

We have explained why local incentive compatibility in an LSHP equilibrium implies thedifferential equation (8) with boundary condition micro(0) = 0 and conditions (9) and (10) forthe cutoff type t The assumption of (7) which yields the single-crossing property stated in

28

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

Lemma 4 guarantees that these conditions are also sufficient for global incentive compatibil-ity We can thus use these conditions to establish

Lemma 5 There is a unique LSHP equilibrium

In proving Lemma 5 we assign the off-path belief π(m) = t for m isin (micro(t) 1) and showthat no type t has incentive to deviate from micro(t) It can be shown that this off-path beliefsatisfies the natural adaptation of the D1 refinement (Cho and Sobel 1990) to our setting

Lemma 5 identifies a unique cutoff type t above which pooling occurs The informationthe receiver gets about the senderrsquos type is fully characterized by the cutoff a higher cut-off corresponds to a more (Blackwell) informative equilibrium The following result will becrucial in understanding how changes in the receiverrsquos exogenous information affects thesenderrsquos equilibrium signaling strategy

Lemma 6 If s is drawn from a more informative experiment than s then for any t lt 1

1 partEs|t [β(s t)] partπ le partEs|t [β(s t)] partπ

2 Es|t983045β(s πP (t))

983046le Es|t

983045β(s πP (t))

983046

Roughly Lemma 6 says that the senderrsquos gain from inducing a higher interim belief thanthe truth is lower when the exogenous signal is more informative For part 1 of the lemmanote that for any type t lt 1 and any small ε gt 0 inducing an interim belief t + ε createsdisagreement type t views this interim belief as higher than the truth IVP implies that texpects a more informative exogenous signal to correct this interim belief more so that trsquosexpected utility gain from inducing that interim belief is lower under a more informativeexperiment More precisely when s is more informative than s Theorem 1 implies that

Es|t [β(s t+ ε)]minus Es|t [β(s t)] le Es|t [β(s t+ ε)]minus Es|t [β(s t)]

where we have used Es|t[β(s t)] = Es|t[β(s t)] = t Dividing both sides of the above inequalityby ε and taking ε rarr 0 proves part 1 of Lemma 6 Part 2 of the lemma also follows fromTheorem 1 since πP (t) gt t

Part 1 of Lemma 6 implies that a more informative exogenous signal reduces the right-hand side of Equation 8 Since each type expects a smaller marginal benefit from inducinga higher belief in the receiver the solution micro to the differential equation (8) with boundarycondution micro(0) = 0 is pointwise lower Furthermore part 2 of Lemma 6 implies that a moreinformative signal reduces the left-hand side of the consequent in (9) while the right-hand

29

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

side is increased because micro(t) is lower Thus type t of the sender now strictly prefers revealinghis type to pooling with higher types Consequently the equilibrium cutoff type increaseswhen the receiver has access to a more informative signal In summary

Proposition 6 Let s be drawn from a more informative experiment than s The LSHP equilibrium ismore informative under s than s ts ge ts Furthermore in the LSHP equilibrium every type bears alower signaling cost under s than under s

Since Proposition 6 applies to arbitrary experiments we can readily adapt the foregoinganalysis to study a multi-sender signaling game in which each sender gets a conditionallyindependent signal about the state For concreteness suppose there are two upward biasedsenders and each adopts an LSHP strategy20 Then from one senderrsquos perspective the othersenderrsquos message is an endogenous experiment Furthermore the informativeness of thatexperiment is increasing in the other senderrsquos cutoff The logic of Proposition 6 can be usedto deduce that the two sendersrsquo cutoffs are strategic complements each sender reveals moreand bears a lower signaling cost when the other sender reveals more If any senderrsquos costmdashrelative to the persuasion incentivemdashincreases in a suitable sense (for example if sender irsquoscost is kic(m t) and ki gt 0 increases) then all senders become more informative

It bears emphasis that the second part of Proposition 6 holds even when there is no poolingat the top Consider for example a canonical model a la Spence (1973) with an unboundedsignal space m isin [0infin) and differentiable cost function kC(m t) where k gt 0 partCpartm gt 0part2Cpartm2 gt 0 and part2Cpartmpartt lt 0 In this setting (the analog of) the familiar least cost separat-ing equilibrium is given by the solution to Equation 8 with the boundary condition micro(0) = 0Given condition (7) Lemma 6 (in particular part 1) still holds the benefit from mimickinga marginally higher type is lower when the receiver has access to better information Theright-hand side of Equation 8 is thus lower for each type Consequently there will still be fullseparation but every type bears a lower signaling cost

Although it is intuitive that more informative exogenous information reduces the senderrsquosgain from misrepresenting his type and therefore reduces the senderrsquos incentive to incur mis-representation costs we emphasize that Proposition 6 relies on the logic of IVP Supplemen-tary Appendix C3 contains examples showing that if the senderrsquos payoff is not linear in thereceiverrsquos belief or if the receiverrsquos exogenous information is not from an MLRP-experiment(in a multi-state extension of this sectionrsquos model) then the senderrsquos marginal benefit from

20 Any combination of biases can be accommodated so long as one focus on appropriate equilibria in partic-ular a downward biased sender deflates rather than inflates his messages

30

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

mimicking a higher type can be higher when the receiver has access to more informationwhich leads to higher equilibrium signaling costs when the receiver is better informed

Frank (1985 Section III) also suggests that better exogenous information can reduce dis-sipative signaling Weiss (1983) studies when exogenous information allows for separatingequilibria even when there is no heterogeneity in the direct costs of signaling his focus isnot comparative statics in the quality of exogenous information Daley and Green (2014) em-phasize the stability of non-separating equilibria when the marginal cost advantage of highertypes is insufficiently large relative to the accuracy of exogenous information this leads to aldquodouble crossingrdquo of appropriate indifference curves contrary to our single-crossing assump-tion (cf fn 19) Truyts (2015) shows that under noisy signalingmdashrather than the noiseless casemore commonly studied including heremdashbetter exogenous information can exacerbate dis-sipative signaling

5 Conclusion

To recap this paper makes three contributions First it provides a fundamental result thatldquoinformation validates the priorrdquo (IVP Theorem 2) in an appropriate sense Anne expectsmore information to bring Bobrsquos posterior closer to Annersquos prior This result can also be in-terpreted in terms of their expected disagreement (Corollary 2) Second it demonstrates howthis statistical result concerning agents with heterogenous priors can be fruitfully applied tofamiliar common-prior environments with asymmetric informationmdashspecifically to disclo-sure and costly signaling games (Section 3 and Section 4 respectively) Third it offers newsubstantive insights into these economic problems For example in the disclosure contextthe nature of message costs are critical disclosure costs induce strategic substitutes betweensendersrsquo disclosure concealments costs induce strategic complements In the former scenariobut not the latter a receiver can be worse off when there are more senders

While we have focussed on two applications with asymmetric information in this paperwe believe IVP is instructive more broadly Consider sender-receiver persuasion via informa-tion design (Kamenica and Gentzkow 2011) Alonso and Camara (2016) develop an elegantgeneral analysis under heterogeneous priors see also Galperti (2018) When the senderrsquospreferences are state-independent and concave in the receiverrsquos expectation of the state thereis no scope for beneficial persuasion under common priors Alonso and Camara (2016 Sec-tion 43) show that this observation does not hold generically with heterogenous priors whenthere are more than two states Our Theorem 2 delivers additional insights For instanceif the priors are likelihood-ratio ordered and only MLRP-experiments are available then the

31

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

sender does not benefit from persuasion if the receiverrsquos prior is favorable in the sense thatthe receiverrsquos prior dominates (resp is dominated by) the senderrsquos when the senderrsquos utilityis increasing (resp decreasing) in the receiverrsquos posterior expectation

We close by highlighting that IVP only speaks to the ex-ante expectation of the posteriormdashmore precisely certain statistics of the posterior such as the posterior mean The ex-anteexpectation is of course a salient property and also relevant in various economic problemsNevertheless it does restrict the direct applicability of IVP in particular our applicationsassumed certain linearity of utility functions Future research might explore how more in-formation affects other statistics of one agentrsquos ex-ante belief about another agentrsquos posteriorand how to deploy such results

32

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

Appendices

A Proofs

Proof of Theorem 1 This is a special case of Theorem 2 below because with two states anypair of priors are likelihood-ratio ordered and any experiment is an MLRP-experiment

Proof of Theorem 2 For simplicity we write the proof assuming that experiments have sig-nal distributions with (Radon-Nikodym) densitiesmdashsubsuming probability mass functionsmdashin each state

Suppose M(βB) ge M(βA) Since the priors are likelihood-ratio ordered the previousinequality is equivalent to βB dominating βA in the likelihood-ratio order Let p(s|ω) representthe density function of s in state ω under experiment E and let p(s|ω) represent the densityfunction under experiment E By the definition of garbling there exists a non-negative kerneldensity q(s|s) with

983125sq(s|s) ds = 1 for all s such that for any state ω

p(s|ω) =983133

s

q(s|s)p(s|ω) ds

In the following for k = AB we let pk(s) =983123

ω p(s|ω)βk(ω) and pk(s) =983123

ω p(s|ω)βk(ω)The posterior density function over S conditional on s and with prior belief βk is given by

qk(s|s) =q(s|s)pk(s)

pk(s)

ThereforeqB(s|s)qA(s|s)

=pA(s)

pB(s)

pB(s)

pA(s)

Because E is an MLRP-experiment and βB likelihood-ratio dominates βA the ratio pB(s)pA(s)

increases in s Therefore for any s qB(middot|s) likelihood-ratio dominates qA(middot|s)

Since M (βB(s)) increases in s the likelihood-ratio dominance of qB(middot|s) over qA(middot|s) im-plies

EEB [M (βB(s)) | s] ge EE

A [M (βB(s)) | s] (11)

For any realization s from experiment E

EEB [M (βB(s)) | s] = M (βB(s)) (12)

because by the law of iterated expectation Brsquos expectation of any linear function of Brsquos pos-

33

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

terior generated by experiment E is Brsquos belief prior to that experiment Combining (11) and(12) and taking the expectation over s (using the prior βA) gives

EEA [M (βB(s))] ge EE

A

983045EE

A [M (βB(s)) | s]983046

=

983133

s

983061983133

s

M (βB(s)) qA(s|s) ds983062pA(s) ds

=

983133

s

M (βB(s))

983061983133

s

q(s|s)pA(s)

pA(s) ds

983062pA(s) ds

= EEA [M (βB(s))]

Given that the priors are likelihood-ratio ordered the above inequality would be reversed ifand only if M(βA) ge M(βB)

Finally any experiment E is a garbling of the fully informative experiment F that revealsthe true state ω whereas the uninformative experiment U that provides no information is agarbling of E It holds that

EFA [M (βB(s))] = M

983043βA

983044and EU

A [M (βB(s))] = M983043βB

983044

For M(βB) ge M(βA) the experiments F and U respectively provide the lower bound and theupper bound in the second part of the theorem for M(βB) le M(βA) the experiment F givesthe upper bound and U gives the lower bound

Proof of Lemma 1 Partially differentiating (5) with respect to the first argument yields

partη(s p π)

parts=

minusp(1minus p)fπ(s)

(1minus p+ pFπ(s))2(π minus Eπ[s|s lt s]) +

pFπ(s)

1minus p+ pFπ(s)

fπ(s)

Fπ(s)(sminus Eπ[s|s lt s])

=pfπ(s)

1minus p+ pFπ(s)

983061minus(1minus p)

1minus p+ pFπ(s)(π minus Eπ[s|s lt s]) + (sminus Eπ[s|s lt s])

983062

=pfπ(s)

1minus p+ pFπ(s)(sminus η(s p π))

Hence sign [partη(s p π)parts] = sign [sminus η(s p π)] Part 1 of the lemma follows from the obser-vation that for any p and π η(s p π) = η(s p π) = π

Partially differentiating with respect to the second argument and simplifying yields

partη(s p π)

partp=

Fπ(s) (Eπ[s|s lt s]minus π)

(1minus p+ pFπ(s))2

34

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

which proves part 2 of the lemma because Eπ[s|s lt s] lt π if and only if s lt s and Fπ(s) gt 0

if and only if s gt s

Proof of Proposition 1 The first part is straightforward and omitted The second part fol-lows from the fact that for any p isin (0 1) and π η(middot p π) is strictly decreasing on the domain[0 s] where s is the fixed point of η(middot p π) which is interior (Lemma 1) The third part fol-lows because parameters can be chosen so that c gt 0 and there are multiple solutions in s tos minus c = η(s pi π) as depicted in Figure 1 This can be seen by fixing all parameters except cand pi and then considering pi rarr 1 with a suitable choice of c details are available from theauthors on request

Proof of Proposition 2 Fix any pi gt pi and let s0i and s0i denote the corresponding highestequilibrium disclosure thresholds Suppose by way of contradiction that s0i gt s0i Since s0i isthe highest equilibrium threshold at pi it follows that for any s gt s0i η(s pi π) lt sminusc But thefact that η(s p π) is weakly decreasing in p (Lemma 1) implies that for any s gt s0i we haveη(s pi π) lt sminus c which implies that s0i le s0i a contradiction A similar argument can be usedto establish the result for the lowest equilibrium threshold We omit the proof of the secondpart as it follows the same logic as the first part

Proof of Lemma 2 Fix any equilibrium and any sender i and sender j ∕= i It suffices toshow that the difference in the expected payoff for i from disclosing versus concealing isstrictly increasing in si Denote the expected payoff from concealing as E[βDM(mjmi = φ)]where βDM(mjmi = φ) denotes the DMrsquos equilibrium belief following any message mj andnondisclosure by i and the expectation is taken over mj given irsquos beliefs under si Because mj

is uncorrelated with si conditional on the state and irsquos belief about the state given si is just si

E[βDM(mjmi = φ)] = siE[βDM(mjmi = φ)|ω = 1] + (1minus si)E[βDM(mjmi = φ)|ω = 0]

The derivative of the right-hand-side of the above equation with respect to si is strictly lessthan one because E[βDM(mjmi = φ)|ω = 1] lt 1 as beliefs lie in [0 1] and mj cannot perfectlyreveal the state Therefore the payoff difference E[βDM(mjmi = φ)] minus (si minus c) is single-crossing in si

Proof of Lemma 3 For any pj pi si and si U(si pi sj pj) is irsquos expectation of the DMrsquos belief(viewed as random variable whose realization depends on jrsquos message) under a prior si for iand η(si pi π) for the DM It follows immediately from Theorem 1 that

35

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

1 si = η(si pi π) =rArr U(si pi sj pj) = si

2 si gt η(si pi π) =rArr η(si pi π) le U(si pi sj pj) le si

3 si lt η(si pi π) =rArr η(si pi π) ge U(si pi sj pj) ge si

Because pj lt 1 the last inequalities in items 2 and 3 above are in fact strict as an equalityin either case requires jrsquos message to be fully informative of the state (cf Remark 1) Finallythe other inequalities in items 2 and 3 are also strict if and only if sj lt s as jrsquos message isuninformative if and only if sj = s

Proof of Proposition 3 The transformation T (βA βB βA) defined in (2) is increasing in βBLemma 1 shows that η(si pi π) is decreasing in pi Hence sender irsquos payoff from concealingsignal si given a correct conjecture by the DM

U(si pi sj pj) = Esj pj [T (β(mj si) η(si pi π) si)]

decreases in pi for any signal si while his payoff from disclosure does not depend on piFollowing the same argument as in the proof of Proposition 2 the largest and smallest best-response disclosure thresholds must decrease in pi

Let s0i be the smallest equilibrium threshold in the single-sender with i recall that this isthe unique equilibrium threshold if c le 0

Consider first c = 0 It follows from Part 1 of Lemma 3 that sender i is indifferent be-tween nondisclosure and disclosure when his signal is s0i hence s0i isin sBR

i (sj pi pj) Nextwe claim there exists no other best-response disclosure threshold Suppose to contradictionthat sprime gt s0i and sprime isin sBR

i (sj pi pj) By Lemma 1 η(sprime pi π) lt sprime Lemma 3 then impliesthat U(sprime pi sj pj) lt sprime Therefore sender i strictly prefers disclosure to nondisclosure whenhis signal is sprime contradicting sprime isin sBR

i (sj pi pj) A similar argument establishes that sprime lt s0i

implies sprime isin sBRi (sj pi pj)

Now consider c lt 0 If si gt s0i then si minus c gt η(si pi π) Since Lemma 3 implies thatU(si pi sj pj) le maxsi η(si pi π) it follows that U(si pi sj pj) lt si minus c and hence si isinsBRi (sj pi pj) Furthermore if sj lt s then s0i lt η(s0i pi π) and Lemma 3 together implyU(s0i pi sj pj) gt η(s0i pi π) and hence s0i isin sBR

i (sj pi pj) if and only if s0i = s Converselyit is obvious that s0i = sBR

i (sj pi pj) if sj = s because sender jrsquos message is uninformativeThis proves part (i) of the result for c lt 0 To prove part (ii) we first note from part (i) thatif si isin sBR

i (sj pi pj) then si le s0i and hence η(si pi π) gt si Theorem 1 then implies that anygarbling of sender jrsquos message decreases U(si pi sj pj) Thus an increase in sj or a decrease

36

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

in pjmdashboth of which represent garblingsmdashlowers sender irsquos nondisclosure payoff withoutaffecting his disclosure payoff at signal si Following the same argument as in the proof ofProposition 2 the largest and smallest best-response disclosure thresholds must increase

We omit the proof for c gt 0 as it follows a symmetric argument to that for c lt 0 theonly point to note is that here the definition of s0i as the smallest equilibrium threshold in thesingle-sender game is used to ensure that si lt s0i implies si minus c lt η(si pi π)

Proof of Proposition 4 Consider first the case c le 0 For each sender i define the function

wi(si sj pi pj) = infsi | U(si pi sj pj) le si minus c

That is wi(middot) gives the smallest element of sBRi (sj pi pj) Let w = (wi wj) and define

slowast(pi pj) = inf(si sj) | w(si sj pi pj) le (si sj)

By Proposition 3 w(middot pi pj) is monotone increasing Hence slowast(pi pj) is its smallest fixedpoint It remains to be shown that slowast(pi pj) is the smallest fixed point of the best-responsecorrespondence (sBR

i sBRj ) Let s be any other fixed point of the correspondence Since w is

monotone we have w(slowast and s) le w(s) le s and w(slowast and s) le w(slowast) le slowast (We follow the notationthat s and sprime equiv (mins1 sprime1mins2 sprime2)) Thus w(slowast and s) le slowast and s By the definition of slowast this inturn implies that slowast le slowast and s which is possible only if slowast le s as required Thus this argumentestablishes that the smallest fixed point of the minimal best response is also the smallest fixedpoint among all best responses In other words slowast(pi pj) is the smallest equilibrium Proposi-tion 3 establishes that w is decreasing in pi It follows from standard monotone comparativestatics that the smallest fixed point of w decreases in pi It is also straightforward to see fromthe definition of wi that w is increasing in c Hence the best equilibrium increases in c Aparallel argument shows that the worst equilibrium also decreases in pi and increases in c

For the case c gt 0 we keep the sign of si but flip the sign of sj in the definition of w so thatit is monotone in (siminussj) The smallest fixed point of w then corresponds to the j-maximalequilibrium By Proposition 3 a higher pi decreases sender irsquos best response but increasessender jrsquos best response Hence in a j-maximal equilibrium (siminussj) is decreasing in pi Thesame conclusion holds for an i-maximal equilibrium

Proof of Lemma 4 The senderrsquos marginal rate of substitution between m and π is given by

partc(m t)partm

partEs|t[β(s π)]partπ= π(1minus π)

partc(m t)partm

Es|t[β(s π)(1minus β(s π))]

37

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

For t lt m the marginal rate of substitution is strictly decreasing in senderrsquos type t if

part2c(m t)partmpartt

partc(m t)partmlt

Es|1[β(s π)(1minus β(s π))]minus Es|0[β(s π)(1minus β(s π))]

tEs|1[β(s π)(1minus β(s π))] + (1minus t)Es|0[β(s π)(1minus β(s π))]

Since Es|1[β(s π)(1minusβ(s π))] gt 0 the right-hand side of the above inequality is strictly greaterthan

0minus Es|0[β(s π)(1minus β(s π))]

0 + (1minus t)Es|0[β(s π)(1minus β(s π))]= minus 1

1minus t

So if condition (7) holds the marginal rate of substitution strictly decreases in t for t lt m

Proof of Lemma 5 For brevity we denote α(π t) = Es|t[β(s π)]

We first establish uniqueness of the cutoff t Suppose by way of contradiction that t lt tprime

are both equilibrium cutoffs From (9) and (10)

α(πP (t) t)minus α(t t) ge c(1 t)minus c(micro(t) t)

Since πP (tprime) gt πP (t) we obtain

α(πP (tprime) t)minus α(t t) gt c(1 t)minus c(micro(t) t)

This implies that type t would deviate to m = 1 from the tprime equilibrium a contradiction

Next we show that no type has incentive to deviate from micro(middot)

Case (i) t le t For any tprime isin (t t] Equation 8 and Lemma 4 imply

microprime(tprime) =partα(tprime tprime)partπ

partc(micro(tprime) tprime)partmgt

partα(tprime t)partπ

partc(micro(tprime) t)partm

The inequality is due to Lemma 4 which applies because micro(tprime) gt tprime gt t The above inequalitycan be written as

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) lt 0

This inequality is true for any tprime isin (t t] For any t in the same interval integrating the inequal-ity over tprime from t to t gives

α(t t)minus c(micro(t) t) lt α(t t)minus c(micro(t) t)

38

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

Thus type t has no incentive to deviate upward to mimic type t isin (t t]

Further by (9) and Lemma 4 for any t lt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) lt α(t t)minus c(micro(t) t)

Since we have already shown that type t lt t has no incentive to mimic type t type t has noincentive to mimic types higher than t by deviating to m = 1 either

Now let tprime lt t be such that tprime ge microminus1(t) Then an analogous argument establishes that

partα(tprime t)

partπminus partc(micro(tprime) t)

partmmicroprime(tprime) gt 0

for any tprime isin [microminus1(t) t) We can apply Lemma 4 for tprime gt microminus1(t) because micro(tprime) gt t The abovealso holds for tprime = microminus1(t) because partc(tt)

partm= 0 For any t in the same interval integrating the

inequality over tprime from t to t gives

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

Thus type t le t has no incentive to deviate downward to mimic type t isin [microminus1(t) t)

For any t lt microminus1(t) we have α(t t) lt α(microminus1(t) t) and c(micro(t) t) gt c(micro(microminus1(t)) t) = 0 Sincetype t has no incentive to mimic type microminus1(t) type t has no incentive to mimic type t lt microminus1(t)

either

Case (ii) t isin (t micro(t)] By (9) and Lemma 4 for any t gt t

α(πP (t) t)minus c(1 t) = α(t t)minus c(micro(t) t) =rArr α(πP (t) t)minus c(1 t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t)

Take any t isin [microminus1(t) t) In case (i) we have shown that type t has no incentive to deviatedownward to mimic t

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

By condition (7) this inequality implies

α(t t)minus c(micro(t) t) gt α(t t)minus c(micro(t) t)

for t isin (t micro(t)] Since type t has no incentive to deviate to mimic type t type t has no incentiveto mimic type t isin [microminus1(t) t) either

39

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

The same argument as in case (i) shows that type t isin (t micro(t)] has no incentive to deviateto any type lower than microminus1(t)

Case (iii) t gt micro(t) Let π gt t be the interim belief such that type t is indifferent betweenm = t and m = 1

α(πP (t) t)minus c(1 t) = α(π t)minus c(t t)

Since t gt micro(t) gt tα(πP (t) t)minus c(1 t) gt α(πP (t) t)minus c(1 t)

By condition (7) this implies

α(πP (t) t)minus c(1 t) gt α(π t)minus c(t t) gt α(t t)minus c(micro(t) t)

Thus type t strictly prefers m = 1 to m = micro(t) The same argument as in case (i) shows thattype t has no incentive to deviate to types lower than t either

Finally it remains to be shown that no type has an incentive to deviate to some m isin(micro(t) 1) which is not used in equilibrium We assign the off-equilibrium belief π(m) = t forsuch m In cases (i) and (ii) we show that type t le micro(t) has no incentive to deviate to mimict Since c(micro(t) t) lt c(m t) for off-equilibrium m this type has no incentive to deviate to m

either In case (iii) we show that type t gt micro(t) prefers belief-message pair (πP (t) 1) to (π t)where π gt t This type must prefer (π t) to (tm) because c(t t) le c(m t)

Proof of Proposition 6 We first show that the function micro(t) defined by Equation 8 is point-wise (weakly) decreasing in the informativeness of the experiment

If we show that micro(t) decreases pointwise when the right-hand-side of Equation 8 decreasesfor all t the result then follows from Lemma 6 Accordingly let micro(t) and micro(t) be two solutionsto Equation 8 with micro(0) = micro(0) = 0 where micro solves Equation 8 with a pointwise lower right-hand-side (These are defined over some respective domains [0 t] and [0 t] The argumentestablishes that t ge t) For any t gt 0 if micro(t) = micro(t) then microprime(t) ge microprime(t) gt 0 This implies that atany touching point micro must touch micro from below Consequently by continuity

micro(tprime) ge micro(tprime) for tprime gt 0 =rArr micro(t) ge micro(t) for all t ge tprime

Now suppose by way of contradiction that for some t gt 0 micro(t) gt micro(t) Then it must bethe case that micro(t) gt micro(t) for all t isin (0 t) Since part2cpartm2 gt 0 and micro corresponds to a lowerright-hand-side of Equation 8 it follows from Equation 8 that microprime(t) lt microprime(t) for all t isin (0 t)

40

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

But then

micro(t)minus micro(t) =

983133 t

0

[microprime(t)minus microprime(t)] dt lt 0

a contradiction

Next we prove that ts ge ts The result is trivial if ts = 0 so assume ts gt 0 Under the moreinformative experiment s and the corresponding function micro type ts will (weakly) prefer toseparate than pool with the top There are two reasons both working in the same direction(i) separation cost is lower with micro than with micro (as shown above) ie the right-hand-side ofthe consequent in (9) increases and (ii) the benefit of pooling is lower (part 2 of Lemma 6) iethe left-hand-side of the consequent in (9) decreases Since ts ge ts ge ts and ts strictly prefersto pool with [ts 1] rather than separate (same message cost better inference) by continuitythere will be some cutoff ts ge ts and this cutoff is unique by Lemma 5

The second statement of the proposition follows because all types are sending lower sig-nals in the new equilibrium (and each type trsquos signal is larger than t in both equilibria)

B Discussion of Theorem 2

This appendix establishes that the conclusion of Theorem 2mdashviz that agent i expects agentjrsquos posterior to be closer to irsquos prior in the sense of their M(middot) statistics under a more infor-mative experimentmdashcan fail with a non-MLRP experiment (Example 1) or if the priors areordered by first-order stochastic dominance (FOSD) rather than by likelihood ratio (Exam-ple 2) In both examples M(middot) denotes the simple expectation operator ie we take h(ω) = ω

Example 1 Let Θ = 0 1 2 Consider the following experiment E with a binary signal spacesl sh

983077Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 2)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 2)

983078=

9830770 1 0

1 0 1

983078

This experiment does not satisfy MLRP Consider the priors βB = (0 02 08) and βA =

(09 01 0) the former likelihood-ratio dominates the latter A direct calculation gives

M(βA) = 01 lt M(βB) = 18 lt EEA[M(βB(middot))] = 19

contrary to Theorem 2rsquos conclusion that EEA[M(βB(middot))] isin [M(βA)M(βB)] when M(βA) lt

M(βB)

41

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

The priors in Example 1 violate our maintained assumption of full support but the exam-ple would go through (with more elaborate calculations) if we replaced βB(0) and βA(2) witha small ε gt 0 instead of 0 This example is similar to Alonso and Camara (2016 pp 674ndash675)who use their example to illustrate how a ldquoskepticrdquo can design information to persuade aldquobelieverrdquo

Example 2 Let Θ = 0 1 8 Consider the following MLRP experiment E with signal spacesl sm sh 983093

983097983095Pr(sl|θ = 0) Pr(sl|θ = 1) Pr(sl|θ = 8)

Pr(sm|θ = 0) Pr(sm|θ = 1) Pr(sm|θ = 8)

Pr(sh|θ = 0) Pr(sh|θ = 1) Pr(sh|θ = 8)

983094

983098983096 =

983093

983097983095

12

0 014

12

014

12

1

983094

983098983096

Consider the priors βB = (23 0 13) and βA = (0 23 13) The prior βA dominates βB inthe sense of FOSD but not in likelihood ratio Direct calculations establish the following

1 The posteriors afters signal sh βB(middot|sh) and βA(middot|sh) are not ordered by FOSD becauseβA(0|sh) = 0 lt βB(0|sh) = 13 while

983123θisin01

βA(θ|sh) = 12 gt983123

θisin01βB(θ|sh) = 13

2 M(βB) = 83 lt M(βA) = 103 lt EEA[M(βB(middot))] = 329

The first point shows that posteriors may not be ranked by FOSD even when priors are andthe experiment satisfies MLRP The second point is contrary to Theorem 2rsquos conclusion thatEE

A[M(βB(middot))] isin [M(βB)M(βA)] when M(βB) lt M(βA)

The priors in Example 2 violate our maintained assumption of full support but the ex-ample would go through (with more elaborate calculations) if we replaced βB(1) and βA(0)

with a small ε gt 0 instead of 0 This example implies that even when priors are first-order-stochastically ordered a sender with a linearmdashor by continuity even strictly convexmdashutility function over the receiverrsquos posterior mean could prefer persuading a ldquoskepticrdquo via apartially-revealing experiment than a fully-revealing one Theorem 2 implies that this can-not arise under likelihood-ratio ordered priors as also noted by Alonso and Camara (2016Proposition 61)

42

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

References

ACHARYA V V P DEMARZO AND I KREMER (2011) ldquoEndogenous Information Flows andthe Clustering of Announcementsrdquo American Economic Review 101 2955ndash79

ALONSO R AND O CAMARA (2016) ldquoBayesian Persuasion with Heterogeneous PriorsrdquoJournal of Economic Theory 165 672ndash706

AUMANN R J (1976) ldquoAgreeing to Disagreerdquo Annals of Statistics 4 1236ndash1239

BALIGA S E HANANY AND P KLIBANOFF (2013) ldquoPolarization and Ambiguityrdquo AmericanEconomic Review 103 3071ndash3083

BHATTACHARYA S AND A MUKHERJEE (2013) ldquoStrategic Information Revelation when Ex-perts Compete to Influencerdquo RAND Journal of Economics 522ndash544

BLACKWELL D (1951) ldquoComparison of Experimentsrdquo in Proceedings of the Second BerkeleySymposium on Mathematical Statistics and Probability University of California Press vol 193ndash102

mdashmdashmdash (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of Mathematical Statistics24 265ndash272

BOURJADE S AND B JULLIEN (2011) ldquoThe Roles of Reputation and Transparency on theBehavior of Biased Expertsrdquo RAND Journal of Economics 42 575ndash594

CARLIN B I S W DAVIES AND A IANNACCONE (2012) ldquoCompetition Comparative Per-formance and Market Transparencyrdquo American Economic Journal Microeconomics 4 202ndash37

CHE Y-K AND N KARTIK (2009) ldquoOpinions as Incentivesrdquo Journal of Political Economy 117815ndash860

CHO I-K AND J SOBEL (1990) ldquoStrategic Stability and Uniqueness in Signaling GamesrdquoJournal of Economic Theory 50 381ndash413

DALEY B AND B GREEN (2014) ldquoMarket Signaling with Gradesrdquo Journal of Economic Theory151 114ndash145

DAUGHETY A AND J REINGANUM (2018) ldquoEvidence Suppression by Prosecutors Viola-tions of the Brady Rulerdquo Journal of Law Economics and Organization 34 475ndash510

DEMARZO P I KREMER AND A SKRZYPACZ (2018) ldquoTest Design and Disclosurerdquo Forth-coming in the American Economic Review

DEWATRIPONT M AND J TIROLE (1999) ldquoAdvocatesrdquo Journal of Political Economy 107 1ndash39

DRANOVE D AND G Z JIN (2010) ldquoQuality Disclosure and Certification Theory and Prac-ticerdquo Journal of Economic Literature 48 935ndash63

43

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

DYE R A (1985) ldquoDisclosure of Nonproprietary Informationrdquo Journal of Accounting research23 123ndash145

mdashmdashmdash (2017) ldquoOptimal Disclosure When There are Penalties for Nondisclosurerdquo RAND Jour-nal of Economics 48 704ndash732

ECHENIQUE F (2002) ldquoComparative Statics by Adaptive Dynamics and the CorrespondencePrinciplerdquo Econometrica 70 833ndash844

ELLIOTT M B GOLUB AND A KIRILENKO (2014) ldquoHow Better Information Can GarbleExpertsrsquo Advicerdquo American Economic Review 104 463ndash468

FELTOVICH N R HARBAUGH AND T TO (2002) ldquoToo Cool for School Signaling andCountersignalingrdquo RAND Journal of Economics 33 630ndash649

FRANK R H (1985) ldquoThe Demand for Unobservable and Other Nonpositional GoodsrdquoAmerican Economic Review 75 101ndash116

GALPERTI S (2018) ldquoPersuasion The Art of Changing Worldviewsrdquo Forthcoming in theAmerican Economic Review

GENTZKOW M AND E KAMENICA (2014) ldquoCostly Persuasionrdquo American Economic Review104 457ndash462

mdashmdashmdash (2017) ldquoCompetition in Persuasionrdquo Review of Economic Studies 84 300ndash322

GROSSMAN S J AND O D HART (1980) ldquoDisclosure Laws and Takeover Bidsrdquo Journal ofFinance 35 323ndash334

HIRSCH A V (2016) ldquoExperimentation and Persuasion in Political Organizationsrdquo AmericanPolitical Science Review 110 68ndash84

JACKSON M O AND X TAN (2013) ldquoDeliberation Disclosure of Information and VotingrdquoJournal of Economic Theory 148 2ndash30

JOVANOVIC B (1982) ldquoTruthful Disclosure of Informationrdquo Bell Journal of Economics 13 36ndash44

JUNG W-O AND Y K KWON (1988) ldquoDisclosure When the Market is Unsure of InformationEndowment of Managersrdquo Journal of Accounting Research 26 146ndash153

KAMENICA E AND M GENTZKOW (2011) ldquoBayesian Persuasionrdquo American Economic Re-view 101 2590ndash2615

KARTIK N (2009) ldquoStrategic Communication with Lying Costsrdquo Review of Economic Studies76 1359ndash1395

KARTIK N F X LEE AND W SUEN (2017) ldquoInvestment in Concealable Information byBiased Expertsrdquo RAND Journal of Economics 48 24ndash43

44

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

KRISHNA V AND J MORGAN (2001) ldquoA Model of Expertiserdquo Quarterly Journal of Economics116 747ndash775

MILGROM P (2008) ldquoWhat the Seller Wonrsquot Tell You Persuasion and Disclosure in MarketsrdquoJournal of Economic Perspectives 22 115ndash131

MILGROM P AND J ROBERTS (1986) ldquoRelying on the Information of Interested PartiesrdquoRAND Journal of Economics 17 18ndash32

MILGROM P R (1981) ldquoGood News and Bad News Representation Theorems and Applica-tionsrdquo Bell Journal of Economics 12 380ndash391

ROTHSCHILD M AND J STIGLITZ (1970) ldquoIncreasing Risk I A Definitionrdquo Journal of Eco-nomic Theory 2 225ndash243

SETHI R AND M YILDIZ (2012) ldquoPublic Disagreementrdquo American Economic Journal Microe-conomics 4 57ndash95

SHIN H S (1994a) ldquoThe Burden of Proof in a Game of Persuasionrdquo Journal of EconomicTheory 64 253ndash264

mdashmdashmdash (1994b) ldquoNews Management and the Value of Firmsrdquo RAND Journal of Economics 2558ndash71

SPENCE M (1973) ldquoJob Market Signalingrdquo Quarterly Journal of Economics 87 355ndash374

THE NEW YORK TIMES (2015) ldquoHonda Fined for Violations of Safety Lawrdquo httpnytims1yFNVUa

TRUYTS T (2015) ldquoNoisy Signaling with Exogenous Informationrdquo Unpublished

VAN DEN STEEN E (2010) ldquoCulture Clash The Costs and Benefits of Homogeneityrdquo Man-agement Science 56 1718ndash1738

VERRECCHIA R E (1983) ldquoDiscretionary Disclosurerdquo Journal of Accounting and Economics 5179ndash194

WEISS A (1983) ldquoA Sorting-cum-Learning Model of Educationrdquo Journal of Political Economy91 420ndash442

YILDIZ M (2004) ldquoWaiting to Persuaderdquo Quarterly Journal of Economics 119 223ndash248

ZANARDO E (2017) ldquoHow to Measure Disagreementrdquo Unpublished

45

C Supplementary Appendix (Not for Publication)

The first two parts of this supplementary appendix provide additional results and discussionfor the disclosure application in Section 3 The final part deals with the role of linearity andMLRP in the signaling application of Section 4

C1 Perfectly correlated signals in the disclosure application

Consider the extreme case where informed sendersrsquo signals are perfectly correlated and forsimplicity c = 0 In other words there is a single signal s drawn from a distribution F (s|ω)and each sender i is independently either informed of s with probability pi or remains unin-formed This setting is effectively identical to the ldquoextreme agendardquo case of Bhattacharya andMukherjee (2013)21 If both senders are biased in the same direction then this model can bemapped to a single-sender problem where the sender is informed with probability pi+pjminuspipj which is larger than maxpi pj22 Proposition 2 then implies that each sender discloses morewhen there is an additional sender hence the DM is always better off with two senders thanone

It is instructive to consider why the irrelevance result no longer holds For simplicity sup-pose both senders are upward biased and symmetric (pi = pj = p) Let s0 denote the commonsingle-sender threshold so that the nondisclosure belief satisfies η(s0 p π) = s0 The essentialobservation is that when sender j is added to the picture say with the hypothesis that he toodiscloses all signals weakly above s0 type s0 of sender i no longer expects the DMrsquos belief tobe s0 should he conceal his signal in contrast to the case of conditionally independent signalsRather he expects the DMrsquos belief to be strictly lower if j is informed the DMrsquos belief will bes0 and if j is uninformed the DMrsquos belief will be strictly lower because of nondisclosure fromtwo senders rather than just one This makes type s0 of sender i strictly prefer disclosureFrom the perspective of Theorem 1 the point is that under conditionally correlated signalswhen an informed sender i does not disclose his signal i and the DM do not agree on theexperiment generated by jrsquos message thus even if the DMrsquos nondisclosure belief agrees withirsquos belief (over the state) irsquos expectation of the DMrsquos posterior belief can be different

Interestingly welfare conclusions under perfectly correlated signals are very differentwhen the senders are opposite biased For simplicity continue to consider c = 0 The follow-ing proposition shows that when senders are opposite biased each sender discloses strictlyless than he does in his single-sender game

Proposition 7 Assume perfectly correlated signals c = 0 and that the two senders are oppositebiased Then each sender discloses strictly less than what he does in his single-sender game

21 Note that they allow for the sendersrsquo utility functions to be non-linear the ensuing discussion does notdepend on linearity either because our single-sender analysis does not require linearity

22 Perfect correlation implies that there is only one relevant nondisclosure belief viz when both senders donrsquotdisclose So senders who are biased in the same direction must use the same equilibrium disclosure thresholdGiven any such threshold the nondisclosure belief is then computed just as in (5) (assuming the bias is upward)but with 1minus p replaced by the probability that both senders are uninformed ie (1minus pi)(1minus pj)

46

Proof of Proposition 7 We prove it for the upward biased sender the argument is symmetricfor the other sender Let sender 1 be upward biased and sender 2 be downward biased Lets01 denote the single-sender threshold ie η(s01) = s01 Write η(s1 s2) as the nondisclosurebelief (in the event both senders do not disclose) in the two-sender game when the respectivethresholds are s1 and s2 Even though the DMrsquos updating is not separable as in our baselinemodel it is clear that η(s1 s2) ge η(s1) with equality if and only if s2 = s This follows fromthe simple observation that the nondisclosure event can be viewed as the union of two events(i) m1 = m2 = φ and s2 = φ and (ii) m1 = m2 = φ and s2 gt s2 Conditional on the first eventthe DMrsquos posterior is η(s1) whereas conditional on the second event the posterior is largerthan η(s1) (strictly if and only if s2 gt s) It follows that for any s2 gt s because η(s1) ge s1 forall s1 le s01 if the DM conjectures thresholds (s1 s2) sender 1 with signal s1 = s1 will strictlyprefer nondisclosure to disclosure given that if sender 2 discloses he necessarily discloses s1Therefore since sender 2 will use a threshold strictly larger than s in any equilibrium anyequilibrium involves sender 1rsquos threshold being strictly larger than s01

Thus despite the increased availability of information the overall disclosure of informa-tion in the two-sender setting is not more informative than under either single-sender prob-lem Consequently for either sender i there exist preferences of the DM such that she wouldstrictly prefer to face sender i alone rather than the two senders simultaneously An implica-tion is that the welfare conclusion in Corollary 2 of Bhattacharya and Mukherjee (2013) can bereversed under alternative DM preferences

In general for an arbitrary c an interior equilibrium (s1 s2) requires

Pr[m2 ∕= φ|s1 = s1 s2]s1 + Pr[m2 = φ|s1 = s1 s2]η(s1 s2) = s1 minus c

or(s1 minus η(s1 s2)) =

c

Pr[m2 = φ|s1 = s1 s2]

Thus when c ge 0 using η(s1 s2) gt η(s1) (strictness by interiority of s2) it follows that s1 gtη(s1) Furthermore for any c ge 0 there is an equilibrium in which s1 is weakly larger thanthe largest single-sender equilibrium When c lt 0 the comparison is ambiguous

C2 Non-linear utility functions in the disclosure application

Another assumption that is important in applying Theorem 1 is that each sender has linearpreferences Suppose more generally that a sender irsquos utility is given by some functionVi(βDM) The comparative statics of sender irsquos disclosure depends on the comparative staticsof

E[Vi(T (βi βDM βi))]minus E[Vi(βi)] (13)

across Blackwell-comparable experiments where the expectation is taken over the posteriorβi using irsquos beliefs and T (middot) is the transformation of (2) When Vi(middot) is linear one can ignore thesecond term in expression (13) as it is constant across experiments and Theorem 1 tells us thatthe sign of the change in the first term is determined by the sign of βi minus βDM Unambiguous

47

comparative statics of expression (13) cannot be obtained for arbitrary specifications of Vi(middot)However because T (βi x x) = βi for any βi and x the logic behind our irrelevance resultextends generally In particular

Proposition 8 If c = 0 and Vi(middot) is strictly monotone then no matter jrsquos disclosure strategy the bestresponse disclosure threshold for i is the same as when he is a single sender

Proof of Proposition 8 Denote r equiv 1minusβDM

βDM

βi

1minusβi and define W (β r) = Vi(T (β r)) minus Vi(β)

where T (β r) = ββ+(1minusβ)r

is a shorthand for the T (β βDM βi) transformation defined in (2)When Vi(middot) is strictly monotone and c = 0 irsquos best response threshold must be such thatE[W (middot r)] = 0 when r is determined by irsquos threshold type and the DMrsquos nondisclosure belief

Observe that when r = 1 then for any β T (β 1) = β and hence W (β 1) = 0 Furthermorebecause T (β r) is strictly decreasing in r for all interior β it follows that for any non-perfectly-informative experiment E[W (middot r)] = 0 if and only if r = 1 Thus no matter jrsquos disclosurestrategy (so long as it is not perfectly informative of the state which it cannot be since pj lt 1)irsquos best response threshold is such that r = 1 ie the DMrsquos nondisclosure belief is the same asirsquos threshold type But this is the same condition as in the single-sender game

When c ∕= 0 there are non-linear specifications for Vi(middot) under which our themes aboutstrategic complementarity under concealment cost or substitutability under disclosure costdo extend and there are other specifications which make conclusions ambiguous or evenreversed Below we show through a family of exponential utility functions how our con-clusions are affected by departures from linearity of Vi(middot) To this end define W (β r) =Vi(T (β r)) minus Vi(β) as in the proof of Proposition 8 Thus under a disclosure cost (c gt 0) therelevant case is r gt 1 if the sender is upward biased and r lt 1 if the sender is downwardbiased under a concealment cost (c lt 0) the relevant case is r lt 1 if the sender is upwardbiased and r gt 1 if the sender is downward biased

In the following proposition we say that an upward biased sender irsquos disclosure is astrategic substitute (resp complement) to jrsquos if whenever jrsquos message is more Blackwell-informative irsquos largest and smallest best response disclosure thresholds increase (resp de-crease)

Proposition 9 Assume Vi(β) = γβα where either γα gt 0 or γα lt 0 so that sender i is upwardbiased Then irsquos disclosure is

1 a strategic substitute to jrsquos under disclosure cost if 0 lt α le 1 and γ gt 0

2 a strategic substitute to jrsquos under concealment cost if α lt 0 and γ lt 0

3 a strategic complement to jrsquos under disclosure cost if α le minus1 and γ lt 0

Part 1 of Proposition 9 is a generalization of Part 2 of Proposition 3 to some non-linearpreferences on the other hand Parts 2 and 3 of Proposition 9 show how our main findingsof strategic complementary under concealment cost and strategic substitutability under dis-closure cost can actually be reversed for other non-linear preferences Note that one has to be

48

careful with the analog of Proposition 9 for the case of a downward biased sender becausethe direction of disagreement between i and the DM reverses Thus if Vi(β) = minusγβα then ineach part of Proposition 9 one should replace ldquodisclosure costrdquo with ldquoconcealment costrdquo andvice-versa

Given the discussion preceding Proposition 9 and invoking Blackwellrsquos results as in dis-cussion in the text after Theorem 1 Proposition 9 is a straightforward consequence of thefollowing lemma

Lemma 7 If Vi(β) = βα then G(β r) is

1 convex in β if 0 lt α le 1 and r gt 1

2 concave in β if α lt 0 and r lt 1

3 convex in β if α lt minus1 and r gt 1

Proof of Lemma 7 Denoting partial derivatives with subscripts as usual we obtain that Wββ(middot)is equal to

V primeprimei

983061β

β + (1minus β)r

983062983061r2

(β + (1minus β)r)4

983062+ V prime

i

983061β

β + (1minus β)r

9830629830612r(r minus 1)

(β + (1minus β)r)3

983062minus V primeprime

i (β)

Plugging in Vi(β) = βα and doing some algebra shows that Wββ(middot) has the same sign as

α

983063(1minus α) +

r(2β(r minus 1)minus r(1minus α))

(β + (1minus β)r)α+2

983064= H(βα r)

Observe that H(0α r) = α(1minusα)(1minusrminusα) and hence if α lt 1 and α ∕= 0 then sign[H(0α r)] =sign[r minus 1] Differentiating yields

Hβ(middot) =α(α + 1)(r minus 1)r(αr + 2β(r minus 1))

(β + (1minus β)r)α+3

We now consider four cases

1 Suppose 0 lt α le 1 and r gt 1 Then H(0α r) ge 0 and Hβ(middot) gt 0 and hence H(βα r) gt0 for all β isin (0 1)

2 Suppose minus1 le α lt 0 and 0 le r lt 1 Then H(0α r) lt 0 and Hβ(middot) le 0 and henceH(βα r) lt 0 for all β isin (0 1)

3 Suppose α lt minus1 and r gt 1 Then H(0α r) gt 0 and H(1α r) = α(r minus 1)(α minus 1 + r(1 +α)) gt 0 We will show that Hβ(βα r) = 0 implies H(βα r) gt 0 which combines withthe previous two inequalities to imply that H(middot) gt 0 Accordingly assume Hβ(βα r) =0 which occurs when β = αr

2(1minusr) which implies α isin (minus2minus1) and r ge 2

2+α(because

β le 1 and α lt minus1) Furthermore

H

983061αr

2(1minus r)α r

983062= α

9830771minus αminus r2

9830612

r(α + 2)

983062α+2983078

49

The derivative of the above expression with respect to r is α983043

2α+2

983044α+2rminusαminus1 which is

strictly positive given α isin (minus2minus1) and r gt 2α+2

gt 2 Moreover when evaluated withr = 2 the expression reduces to 1 minus α minus 4

(α+2)2 which is strictly positive given α isin

(minus2minus1) Therefore H983059

αr2(1minusr)

α r983060gt 0 as was to be shown

4 Suppose α lt minus1 and 0 le r lt 1 Then H(0α r) lt 0 and H(1α r) = α(r minus 1)(α minus 1 +r(1 + α)) lt 0 As argued in the previous case Hβ(βα r) = 0 requires α isin (minus2minus1)and r ge 2

2+αgt 2 which is not possible given that we have assumed r lt 1 Thus Hβ(middot)

has a constant sign in the relevant domain which implies that H(middot) lt 0 in the relevantdomain

C3 The role of linearity and MLRP in the signaling application

Recall that in an LSHP equilibrium the strategy micro(middot) for types that separate is determined bythe initial condition micro(0) = 0 and the differential equation (8)

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ

Part 1 of Lemma 6 established that

partEs|t [β(s t)]

partπle

partEs|t [β(s t)]

partπ(14)

when signal s is more informative (ie drawn from a more informative experiment) than sig-nal s Inequality (14) implies that the solution to the aforementioned initial-value problem ispointwise lower under the more informative experiment and hence the equilibrium signalinglevel micro(t) is lower for every type when the receiver has access to s rather than s

We show below how the conclusion can be altered by dropping either linearity of thesenderrsquos payoff in the receiverrsquos posterior (Example 3) or the MLRP property of the receiverrsquosexperiments (Example 4)

Example 3 Letting V (β) equiv β(1minus β) suppose the senderrsquos payoff is

V (β)minus c(m t)

which is convex in β Assumption (7) in Section 4 continues to imply the relevant single-crossing condition for this modified objective Using Bayes rule we compute

Es|t[V (β(s π))] =π

1minus πEs|t

983063g(s|1)g(s|0)

983064

Differentiating and evaluating at π = t

partEs|t[V (β(s t))]

partπ=

1

t(1minus t)Es|t

983063β(s t)

1minus β(s t)

983064

50

The term inside the expectation operator on the right-hand side above is a convex function ofβ(middot) It follows that

partEs|t[V (β(s t))]

partπge

partEs|t[V (β(s t))]

partπ

by contrast to inequality (14) That is the convexity in V (middot) is strong enough to ensure that themarginal benefit from inducing a higher interim belief π (locally at π = t) is higher when theexogenous signal is more informative It follows that in an LSHP equilibrium all types beara higher (at least weakly) signaling cost when the exogenous signal is more informative23

Example 4 To see the role of MLRP-experiments we have to modify the signaling modelof Section 4 by introducing more states because any experiment in a two-state model is anMLRP-experiment

Assume a full-support common prior about the state ω isin Ω = 0 1 2 The sender re-ceives some private information indexed by t isin [0 1] which updates his belief about thestate to (z 1 minus z(1 + t) zt) where each element of this vector is the probability assigned tothe corresponding state The parameter z isin (0 12) is a commonly known constant We referto t as the senderrsquos type Letting M(β) equiv

983123ω ωβ(ω) be the receiverrsquos expectation of the state

when she holds belief β the senderrsquos payoff is

M(β)minus c(m t)

Let s represent the outcome of an uninformative experiment and let β(s t) represent theposterior of the receiver after observing s when she puts probability one on the senderrsquos typet It clearly holds that

Es|t983045M(β(s t))

983046= M(β(s t)) = 1minus z + zt

Now consider an experiment E with a binary signal space s isin sl sh Let g(s|ω) denotethe probability of the signal realization in each state specified as follows

983063g(sl|0) g(sl|1) g(sl|2)g(sh|0) g(sh|1) g(sh|2)

983064=

9830630 1 01 0 1

983064

This experiment is the same as that used in Example 1 of Appendix B it is not an MLRP-experiment Note also that s is more informative than s

Suppose the receiver ascribes probability one to the senderrsquos type t By Bayes rule if thesignal realization is sl the receiverrsquos posterior belief about the state is β(sl t) = (0 1 0) and

23 On the other hand if V (β) equiv log[β(1 minus β)] then the local marginal benefit of inducing a higher interimbelief is independent of the exogenous experiment The reason is that now V (β(sπ)) = log

983059π

1minusπ

983060+log

983059g(s|1)g(s|0)

983060

and hence partEs|t[V (β(s t))]partπ does not vary with the experiment Note that in this example V (middot) is neitherconvex nor concave

51

therefore M(β(sl t)) = 1 For signal realization sh

β(sh t) =

9830611

1 + t 0

t

1 + t

983062

and therefore

M(β(sh t)) =2t

1 + t

The sender of type trsquos expectation is

Es|t983045M(β(s t))

983046= (1minus z(1 + t))M(β(sl t)) + z(1 + t)M(β(sh t))

The derivative of the above expression with respect to t evaluated at t = t is

partEs|t [M(β(s t))]

part t=

2z

1 + tge z =

partEs|t [M(β(s t))]

part t (15)

This inequality is opposite to inequality (14) even though s is more informative than s

In this example Es|t[M(β(s t))] and Es|t[M(β(s t))] are both (weakly) supermodular inthe senderrsquos type t The assumption that cmt(m t) lt 0 ensures that indifference curves in thespace of (m t) for different types are single-crossing As local incentive compatibility thenimplies global incentive compatibility inequality (15) implies that all types incur higher (atleast weakly) signaling costs when the receiver has access to more information

52

Proof of Proposition 7 We prove it for the upward biased sender the argument is symmetricfor the other sender Let sender 1 be upward biased and sender 2 be downward biased Lets01 denote the single-sender threshold ie η(s01) = s01 Write η(s1 s2) as the nondisclosurebelief (in the event both senders do not disclose) in the two-sender game when the respectivethresholds are s1 and s2 Even though the DMrsquos updating is not separable as in our baselinemodel it is clear that η(s1 s2) ge η(s1) with equality if and only if s2 = s This follows fromthe simple observation that the nondisclosure event can be viewed as the union of two events(i) m1 = m2 = φ and s2 = φ and (ii) m1 = m2 = φ and s2 gt s2 Conditional on the first eventthe DMrsquos posterior is η(s1) whereas conditional on the second event the posterior is largerthan η(s1) (strictly if and only if s2 gt s) It follows that for any s2 gt s because η(s1) ge s1 forall s1 le s01 if the DM conjectures thresholds (s1 s2) sender 1 with signal s1 = s1 will strictlyprefer nondisclosure to disclosure given that if sender 2 discloses he necessarily discloses s1Therefore since sender 2 will use a threshold strictly larger than s in any equilibrium anyequilibrium involves sender 1rsquos threshold being strictly larger than s01

Thus despite the increased availability of information the overall disclosure of informa-tion in the two-sender setting is not more informative than under either single-sender prob-lem Consequently for either sender i there exist preferences of the DM such that she wouldstrictly prefer to face sender i alone rather than the two senders simultaneously An implica-tion is that the welfare conclusion in Corollary 2 of Bhattacharya and Mukherjee (2013) can bereversed under alternative DM preferences

In general for an arbitrary c an interior equilibrium (s1 s2) requires

Pr[m2 ∕= φ|s1 = s1 s2]s1 + Pr[m2 = φ|s1 = s1 s2]η(s1 s2) = s1 minus c

or(s1 minus η(s1 s2)) =

c

Pr[m2 = φ|s1 = s1 s2]

Thus when c ge 0 using η(s1 s2) gt η(s1) (strictness by interiority of s2) it follows that s1 gtη(s1) Furthermore for any c ge 0 there is an equilibrium in which s1 is weakly larger thanthe largest single-sender equilibrium When c lt 0 the comparison is ambiguous

C2 Non-linear utility functions in the disclosure application

Another assumption that is important in applying Theorem 1 is that each sender has linearpreferences Suppose more generally that a sender irsquos utility is given by some functionVi(βDM) The comparative statics of sender irsquos disclosure depends on the comparative staticsof

E[Vi(T (βi βDM βi))]minus E[Vi(βi)] (13)

across Blackwell-comparable experiments where the expectation is taken over the posteriorβi using irsquos beliefs and T (middot) is the transformation of (2) When Vi(middot) is linear one can ignore thesecond term in expression (13) as it is constant across experiments and Theorem 1 tells us thatthe sign of the change in the first term is determined by the sign of βi minus βDM Unambiguous

47

comparative statics of expression (13) cannot be obtained for arbitrary specifications of Vi(middot)However because T (βi x x) = βi for any βi and x the logic behind our irrelevance resultextends generally In particular

Proposition 8 If c = 0 and Vi(middot) is strictly monotone then no matter jrsquos disclosure strategy the bestresponse disclosure threshold for i is the same as when he is a single sender

Proof of Proposition 8 Denote r equiv 1minusβDM

βDM

βi

1minusβi and define W (β r) = Vi(T (β r)) minus Vi(β)

where T (β r) = ββ+(1minusβ)r

is a shorthand for the T (β βDM βi) transformation defined in (2)When Vi(middot) is strictly monotone and c = 0 irsquos best response threshold must be such thatE[W (middot r)] = 0 when r is determined by irsquos threshold type and the DMrsquos nondisclosure belief

Observe that when r = 1 then for any β T (β 1) = β and hence W (β 1) = 0 Furthermorebecause T (β r) is strictly decreasing in r for all interior β it follows that for any non-perfectly-informative experiment E[W (middot r)] = 0 if and only if r = 1 Thus no matter jrsquos disclosurestrategy (so long as it is not perfectly informative of the state which it cannot be since pj lt 1)irsquos best response threshold is such that r = 1 ie the DMrsquos nondisclosure belief is the same asirsquos threshold type But this is the same condition as in the single-sender game

When c ∕= 0 there are non-linear specifications for Vi(middot) under which our themes aboutstrategic complementarity under concealment cost or substitutability under disclosure costdo extend and there are other specifications which make conclusions ambiguous or evenreversed Below we show through a family of exponential utility functions how our con-clusions are affected by departures from linearity of Vi(middot) To this end define W (β r) =Vi(T (β r)) minus Vi(β) as in the proof of Proposition 8 Thus under a disclosure cost (c gt 0) therelevant case is r gt 1 if the sender is upward biased and r lt 1 if the sender is downwardbiased under a concealment cost (c lt 0) the relevant case is r lt 1 if the sender is upwardbiased and r gt 1 if the sender is downward biased

In the following proposition we say that an upward biased sender irsquos disclosure is astrategic substitute (resp complement) to jrsquos if whenever jrsquos message is more Blackwell-informative irsquos largest and smallest best response disclosure thresholds increase (resp de-crease)

Proposition 9 Assume Vi(β) = γβα where either γα gt 0 or γα lt 0 so that sender i is upwardbiased Then irsquos disclosure is

1 a strategic substitute to jrsquos under disclosure cost if 0 lt α le 1 and γ gt 0

2 a strategic substitute to jrsquos under concealment cost if α lt 0 and γ lt 0

3 a strategic complement to jrsquos under disclosure cost if α le minus1 and γ lt 0

Part 1 of Proposition 9 is a generalization of Part 2 of Proposition 3 to some non-linearpreferences on the other hand Parts 2 and 3 of Proposition 9 show how our main findingsof strategic complementary under concealment cost and strategic substitutability under dis-closure cost can actually be reversed for other non-linear preferences Note that one has to be

48

careful with the analog of Proposition 9 for the case of a downward biased sender becausethe direction of disagreement between i and the DM reverses Thus if Vi(β) = minusγβα then ineach part of Proposition 9 one should replace ldquodisclosure costrdquo with ldquoconcealment costrdquo andvice-versa

Given the discussion preceding Proposition 9 and invoking Blackwellrsquos results as in dis-cussion in the text after Theorem 1 Proposition 9 is a straightforward consequence of thefollowing lemma

Lemma 7 If Vi(β) = βα then G(β r) is

1 convex in β if 0 lt α le 1 and r gt 1

2 concave in β if α lt 0 and r lt 1

3 convex in β if α lt minus1 and r gt 1

Proof of Lemma 7 Denoting partial derivatives with subscripts as usual we obtain that Wββ(middot)is equal to

V primeprimei

983061β

β + (1minus β)r

983062983061r2

(β + (1minus β)r)4

983062+ V prime

i

983061β

β + (1minus β)r

9830629830612r(r minus 1)

(β + (1minus β)r)3

983062minus V primeprime

i (β)

Plugging in Vi(β) = βα and doing some algebra shows that Wββ(middot) has the same sign as

α

983063(1minus α) +

r(2β(r minus 1)minus r(1minus α))

(β + (1minus β)r)α+2

983064= H(βα r)

Observe that H(0α r) = α(1minusα)(1minusrminusα) and hence if α lt 1 and α ∕= 0 then sign[H(0α r)] =sign[r minus 1] Differentiating yields

Hβ(middot) =α(α + 1)(r minus 1)r(αr + 2β(r minus 1))

(β + (1minus β)r)α+3

We now consider four cases

1 Suppose 0 lt α le 1 and r gt 1 Then H(0α r) ge 0 and Hβ(middot) gt 0 and hence H(βα r) gt0 for all β isin (0 1)

2 Suppose minus1 le α lt 0 and 0 le r lt 1 Then H(0α r) lt 0 and Hβ(middot) le 0 and henceH(βα r) lt 0 for all β isin (0 1)

3 Suppose α lt minus1 and r gt 1 Then H(0α r) gt 0 and H(1α r) = α(r minus 1)(α minus 1 + r(1 +α)) gt 0 We will show that Hβ(βα r) = 0 implies H(βα r) gt 0 which combines withthe previous two inequalities to imply that H(middot) gt 0 Accordingly assume Hβ(βα r) =0 which occurs when β = αr

2(1minusr) which implies α isin (minus2minus1) and r ge 2

2+α(because

β le 1 and α lt minus1) Furthermore

H

983061αr

2(1minus r)α r

983062= α

9830771minus αminus r2

9830612

r(α + 2)

983062α+2983078

49

The derivative of the above expression with respect to r is α983043

2α+2

983044α+2rminusαminus1 which is

strictly positive given α isin (minus2minus1) and r gt 2α+2

gt 2 Moreover when evaluated withr = 2 the expression reduces to 1 minus α minus 4

(α+2)2 which is strictly positive given α isin

(minus2minus1) Therefore H983059

αr2(1minusr)

α r983060gt 0 as was to be shown

4 Suppose α lt minus1 and 0 le r lt 1 Then H(0α r) lt 0 and H(1α r) = α(r minus 1)(α minus 1 +r(1 + α)) lt 0 As argued in the previous case Hβ(βα r) = 0 requires α isin (minus2minus1)and r ge 2

2+αgt 2 which is not possible given that we have assumed r lt 1 Thus Hβ(middot)

has a constant sign in the relevant domain which implies that H(middot) lt 0 in the relevantdomain

C3 The role of linearity and MLRP in the signaling application

Recall that in an LSHP equilibrium the strategy micro(middot) for types that separate is determined bythe initial condition micro(0) = 0 and the differential equation (8)

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ

Part 1 of Lemma 6 established that

partEs|t [β(s t)]

partπle

partEs|t [β(s t)]

partπ(14)

when signal s is more informative (ie drawn from a more informative experiment) than sig-nal s Inequality (14) implies that the solution to the aforementioned initial-value problem ispointwise lower under the more informative experiment and hence the equilibrium signalinglevel micro(t) is lower for every type when the receiver has access to s rather than s

We show below how the conclusion can be altered by dropping either linearity of thesenderrsquos payoff in the receiverrsquos posterior (Example 3) or the MLRP property of the receiverrsquosexperiments (Example 4)

Example 3 Letting V (β) equiv β(1minus β) suppose the senderrsquos payoff is

V (β)minus c(m t)

which is convex in β Assumption (7) in Section 4 continues to imply the relevant single-crossing condition for this modified objective Using Bayes rule we compute

Es|t[V (β(s π))] =π

1minus πEs|t

983063g(s|1)g(s|0)

983064

Differentiating and evaluating at π = t

partEs|t[V (β(s t))]

partπ=

1

t(1minus t)Es|t

983063β(s t)

1minus β(s t)

983064

50

The term inside the expectation operator on the right-hand side above is a convex function ofβ(middot) It follows that

partEs|t[V (β(s t))]

partπge

partEs|t[V (β(s t))]

partπ

by contrast to inequality (14) That is the convexity in V (middot) is strong enough to ensure that themarginal benefit from inducing a higher interim belief π (locally at π = t) is higher when theexogenous signal is more informative It follows that in an LSHP equilibrium all types beara higher (at least weakly) signaling cost when the exogenous signal is more informative23

Example 4 To see the role of MLRP-experiments we have to modify the signaling modelof Section 4 by introducing more states because any experiment in a two-state model is anMLRP-experiment

Assume a full-support common prior about the state ω isin Ω = 0 1 2 The sender re-ceives some private information indexed by t isin [0 1] which updates his belief about thestate to (z 1 minus z(1 + t) zt) where each element of this vector is the probability assigned tothe corresponding state The parameter z isin (0 12) is a commonly known constant We referto t as the senderrsquos type Letting M(β) equiv

983123ω ωβ(ω) be the receiverrsquos expectation of the state

when she holds belief β the senderrsquos payoff is

M(β)minus c(m t)

Let s represent the outcome of an uninformative experiment and let β(s t) represent theposterior of the receiver after observing s when she puts probability one on the senderrsquos typet It clearly holds that

Es|t983045M(β(s t))

983046= M(β(s t)) = 1minus z + zt

Now consider an experiment E with a binary signal space s isin sl sh Let g(s|ω) denotethe probability of the signal realization in each state specified as follows

983063g(sl|0) g(sl|1) g(sl|2)g(sh|0) g(sh|1) g(sh|2)

983064=

9830630 1 01 0 1

983064

This experiment is the same as that used in Example 1 of Appendix B it is not an MLRP-experiment Note also that s is more informative than s

Suppose the receiver ascribes probability one to the senderrsquos type t By Bayes rule if thesignal realization is sl the receiverrsquos posterior belief about the state is β(sl t) = (0 1 0) and

23 On the other hand if V (β) equiv log[β(1 minus β)] then the local marginal benefit of inducing a higher interimbelief is independent of the exogenous experiment The reason is that now V (β(sπ)) = log

983059π

1minusπ

983060+log

983059g(s|1)g(s|0)

983060

and hence partEs|t[V (β(s t))]partπ does not vary with the experiment Note that in this example V (middot) is neitherconvex nor concave

51

therefore M(β(sl t)) = 1 For signal realization sh

β(sh t) =

9830611

1 + t 0

t

1 + t

983062

and therefore

M(β(sh t)) =2t

1 + t

The sender of type trsquos expectation is

Es|t983045M(β(s t))

983046= (1minus z(1 + t))M(β(sl t)) + z(1 + t)M(β(sh t))

The derivative of the above expression with respect to t evaluated at t = t is

partEs|t [M(β(s t))]

part t=

2z

1 + tge z =

partEs|t [M(β(s t))]

part t (15)

This inequality is opposite to inequality (14) even though s is more informative than s

In this example Es|t[M(β(s t))] and Es|t[M(β(s t))] are both (weakly) supermodular inthe senderrsquos type t The assumption that cmt(m t) lt 0 ensures that indifference curves in thespace of (m t) for different types are single-crossing As local incentive compatibility thenimplies global incentive compatibility inequality (15) implies that all types incur higher (atleast weakly) signaling costs when the receiver has access to more information

52

comparative statics of expression (13) cannot be obtained for arbitrary specifications of Vi(middot)However because T (βi x x) = βi for any βi and x the logic behind our irrelevance resultextends generally In particular

Proposition 8 If c = 0 and Vi(middot) is strictly monotone then no matter jrsquos disclosure strategy the bestresponse disclosure threshold for i is the same as when he is a single sender

Proof of Proposition 8 Denote r equiv 1minusβDM

βDM

βi

1minusβi and define W (β r) = Vi(T (β r)) minus Vi(β)

where T (β r) = ββ+(1minusβ)r

is a shorthand for the T (β βDM βi) transformation defined in (2)When Vi(middot) is strictly monotone and c = 0 irsquos best response threshold must be such thatE[W (middot r)] = 0 when r is determined by irsquos threshold type and the DMrsquos nondisclosure belief

Observe that when r = 1 then for any β T (β 1) = β and hence W (β 1) = 0 Furthermorebecause T (β r) is strictly decreasing in r for all interior β it follows that for any non-perfectly-informative experiment E[W (middot r)] = 0 if and only if r = 1 Thus no matter jrsquos disclosurestrategy (so long as it is not perfectly informative of the state which it cannot be since pj lt 1)irsquos best response threshold is such that r = 1 ie the DMrsquos nondisclosure belief is the same asirsquos threshold type But this is the same condition as in the single-sender game

When c ∕= 0 there are non-linear specifications for Vi(middot) under which our themes aboutstrategic complementarity under concealment cost or substitutability under disclosure costdo extend and there are other specifications which make conclusions ambiguous or evenreversed Below we show through a family of exponential utility functions how our con-clusions are affected by departures from linearity of Vi(middot) To this end define W (β r) =Vi(T (β r)) minus Vi(β) as in the proof of Proposition 8 Thus under a disclosure cost (c gt 0) therelevant case is r gt 1 if the sender is upward biased and r lt 1 if the sender is downwardbiased under a concealment cost (c lt 0) the relevant case is r lt 1 if the sender is upwardbiased and r gt 1 if the sender is downward biased

In the following proposition we say that an upward biased sender irsquos disclosure is astrategic substitute (resp complement) to jrsquos if whenever jrsquos message is more Blackwell-informative irsquos largest and smallest best response disclosure thresholds increase (resp de-crease)

Proposition 9 Assume Vi(β) = γβα where either γα gt 0 or γα lt 0 so that sender i is upwardbiased Then irsquos disclosure is

1 a strategic substitute to jrsquos under disclosure cost if 0 lt α le 1 and γ gt 0

2 a strategic substitute to jrsquos under concealment cost if α lt 0 and γ lt 0

3 a strategic complement to jrsquos under disclosure cost if α le minus1 and γ lt 0

Part 1 of Proposition 9 is a generalization of Part 2 of Proposition 3 to some non-linearpreferences on the other hand Parts 2 and 3 of Proposition 9 show how our main findingsof strategic complementary under concealment cost and strategic substitutability under dis-closure cost can actually be reversed for other non-linear preferences Note that one has to be

48

careful with the analog of Proposition 9 for the case of a downward biased sender becausethe direction of disagreement between i and the DM reverses Thus if Vi(β) = minusγβα then ineach part of Proposition 9 one should replace ldquodisclosure costrdquo with ldquoconcealment costrdquo andvice-versa

Given the discussion preceding Proposition 9 and invoking Blackwellrsquos results as in dis-cussion in the text after Theorem 1 Proposition 9 is a straightforward consequence of thefollowing lemma

Lemma 7 If Vi(β) = βα then G(β r) is

1 convex in β if 0 lt α le 1 and r gt 1

2 concave in β if α lt 0 and r lt 1

3 convex in β if α lt minus1 and r gt 1

Proof of Lemma 7 Denoting partial derivatives with subscripts as usual we obtain that Wββ(middot)is equal to

V primeprimei

983061β

β + (1minus β)r

983062983061r2

(β + (1minus β)r)4

983062+ V prime

i

983061β

β + (1minus β)r

9830629830612r(r minus 1)

(β + (1minus β)r)3

983062minus V primeprime

i (β)

Plugging in Vi(β) = βα and doing some algebra shows that Wββ(middot) has the same sign as

α

983063(1minus α) +

r(2β(r minus 1)minus r(1minus α))

(β + (1minus β)r)α+2

983064= H(βα r)

Observe that H(0α r) = α(1minusα)(1minusrminusα) and hence if α lt 1 and α ∕= 0 then sign[H(0α r)] =sign[r minus 1] Differentiating yields

Hβ(middot) =α(α + 1)(r minus 1)r(αr + 2β(r minus 1))

(β + (1minus β)r)α+3

We now consider four cases

1 Suppose 0 lt α le 1 and r gt 1 Then H(0α r) ge 0 and Hβ(middot) gt 0 and hence H(βα r) gt0 for all β isin (0 1)

2 Suppose minus1 le α lt 0 and 0 le r lt 1 Then H(0α r) lt 0 and Hβ(middot) le 0 and henceH(βα r) lt 0 for all β isin (0 1)

3 Suppose α lt minus1 and r gt 1 Then H(0α r) gt 0 and H(1α r) = α(r minus 1)(α minus 1 + r(1 +α)) gt 0 We will show that Hβ(βα r) = 0 implies H(βα r) gt 0 which combines withthe previous two inequalities to imply that H(middot) gt 0 Accordingly assume Hβ(βα r) =0 which occurs when β = αr

2(1minusr) which implies α isin (minus2minus1) and r ge 2

2+α(because

β le 1 and α lt minus1) Furthermore

H

983061αr

2(1minus r)α r

983062= α

9830771minus αminus r2

9830612

r(α + 2)

983062α+2983078

49

The derivative of the above expression with respect to r is α983043

2α+2

983044α+2rminusαminus1 which is

strictly positive given α isin (minus2minus1) and r gt 2α+2

gt 2 Moreover when evaluated withr = 2 the expression reduces to 1 minus α minus 4

(α+2)2 which is strictly positive given α isin

(minus2minus1) Therefore H983059

αr2(1minusr)

α r983060gt 0 as was to be shown

4 Suppose α lt minus1 and 0 le r lt 1 Then H(0α r) lt 0 and H(1α r) = α(r minus 1)(α minus 1 +r(1 + α)) lt 0 As argued in the previous case Hβ(βα r) = 0 requires α isin (minus2minus1)and r ge 2

2+αgt 2 which is not possible given that we have assumed r lt 1 Thus Hβ(middot)

has a constant sign in the relevant domain which implies that H(middot) lt 0 in the relevantdomain

C3 The role of linearity and MLRP in the signaling application

Recall that in an LSHP equilibrium the strategy micro(middot) for types that separate is determined bythe initial condition micro(0) = 0 and the differential equation (8)

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ

Part 1 of Lemma 6 established that

partEs|t [β(s t)]

partπle

partEs|t [β(s t)]

partπ(14)

when signal s is more informative (ie drawn from a more informative experiment) than sig-nal s Inequality (14) implies that the solution to the aforementioned initial-value problem ispointwise lower under the more informative experiment and hence the equilibrium signalinglevel micro(t) is lower for every type when the receiver has access to s rather than s

We show below how the conclusion can be altered by dropping either linearity of thesenderrsquos payoff in the receiverrsquos posterior (Example 3) or the MLRP property of the receiverrsquosexperiments (Example 4)

Example 3 Letting V (β) equiv β(1minus β) suppose the senderrsquos payoff is

V (β)minus c(m t)

which is convex in β Assumption (7) in Section 4 continues to imply the relevant single-crossing condition for this modified objective Using Bayes rule we compute

Es|t[V (β(s π))] =π

1minus πEs|t

983063g(s|1)g(s|0)

983064

Differentiating and evaluating at π = t

partEs|t[V (β(s t))]

partπ=

1

t(1minus t)Es|t

983063β(s t)

1minus β(s t)

983064

50

The term inside the expectation operator on the right-hand side above is a convex function ofβ(middot) It follows that

partEs|t[V (β(s t))]

partπge

partEs|t[V (β(s t))]

partπ

by contrast to inequality (14) That is the convexity in V (middot) is strong enough to ensure that themarginal benefit from inducing a higher interim belief π (locally at π = t) is higher when theexogenous signal is more informative It follows that in an LSHP equilibrium all types beara higher (at least weakly) signaling cost when the exogenous signal is more informative23

Example 4 To see the role of MLRP-experiments we have to modify the signaling modelof Section 4 by introducing more states because any experiment in a two-state model is anMLRP-experiment

Assume a full-support common prior about the state ω isin Ω = 0 1 2 The sender re-ceives some private information indexed by t isin [0 1] which updates his belief about thestate to (z 1 minus z(1 + t) zt) where each element of this vector is the probability assigned tothe corresponding state The parameter z isin (0 12) is a commonly known constant We referto t as the senderrsquos type Letting M(β) equiv

983123ω ωβ(ω) be the receiverrsquos expectation of the state

when she holds belief β the senderrsquos payoff is

M(β)minus c(m t)

Let s represent the outcome of an uninformative experiment and let β(s t) represent theposterior of the receiver after observing s when she puts probability one on the senderrsquos typet It clearly holds that

Es|t983045M(β(s t))

983046= M(β(s t)) = 1minus z + zt

Now consider an experiment E with a binary signal space s isin sl sh Let g(s|ω) denotethe probability of the signal realization in each state specified as follows

983063g(sl|0) g(sl|1) g(sl|2)g(sh|0) g(sh|1) g(sh|2)

983064=

9830630 1 01 0 1

983064

This experiment is the same as that used in Example 1 of Appendix B it is not an MLRP-experiment Note also that s is more informative than s

Suppose the receiver ascribes probability one to the senderrsquos type t By Bayes rule if thesignal realization is sl the receiverrsquos posterior belief about the state is β(sl t) = (0 1 0) and

23 On the other hand if V (β) equiv log[β(1 minus β)] then the local marginal benefit of inducing a higher interimbelief is independent of the exogenous experiment The reason is that now V (β(sπ)) = log

983059π

1minusπ

983060+log

983059g(s|1)g(s|0)

983060

and hence partEs|t[V (β(s t))]partπ does not vary with the experiment Note that in this example V (middot) is neitherconvex nor concave

51

therefore M(β(sl t)) = 1 For signal realization sh

β(sh t) =

9830611

1 + t 0

t

1 + t

983062

and therefore

M(β(sh t)) =2t

1 + t

The sender of type trsquos expectation is

Es|t983045M(β(s t))

983046= (1minus z(1 + t))M(β(sl t)) + z(1 + t)M(β(sh t))

The derivative of the above expression with respect to t evaluated at t = t is

partEs|t [M(β(s t))]

part t=

2z

1 + tge z =

partEs|t [M(β(s t))]

part t (15)

This inequality is opposite to inequality (14) even though s is more informative than s

In this example Es|t[M(β(s t))] and Es|t[M(β(s t))] are both (weakly) supermodular inthe senderrsquos type t The assumption that cmt(m t) lt 0 ensures that indifference curves in thespace of (m t) for different types are single-crossing As local incentive compatibility thenimplies global incentive compatibility inequality (15) implies that all types incur higher (atleast weakly) signaling costs when the receiver has access to more information

52

careful with the analog of Proposition 9 for the case of a downward biased sender becausethe direction of disagreement between i and the DM reverses Thus if Vi(β) = minusγβα then ineach part of Proposition 9 one should replace ldquodisclosure costrdquo with ldquoconcealment costrdquo andvice-versa

Given the discussion preceding Proposition 9 and invoking Blackwellrsquos results as in dis-cussion in the text after Theorem 1 Proposition 9 is a straightforward consequence of thefollowing lemma

Lemma 7 If Vi(β) = βα then G(β r) is

1 convex in β if 0 lt α le 1 and r gt 1

2 concave in β if α lt 0 and r lt 1

3 convex in β if α lt minus1 and r gt 1

Proof of Lemma 7 Denoting partial derivatives with subscripts as usual we obtain that Wββ(middot)is equal to

V primeprimei

983061β

β + (1minus β)r

983062983061r2

(β + (1minus β)r)4

983062+ V prime

i

983061β

β + (1minus β)r

9830629830612r(r minus 1)

(β + (1minus β)r)3

983062minus V primeprime

i (β)

Plugging in Vi(β) = βα and doing some algebra shows that Wββ(middot) has the same sign as

α

983063(1minus α) +

r(2β(r minus 1)minus r(1minus α))

(β + (1minus β)r)α+2

983064= H(βα r)

Observe that H(0α r) = α(1minusα)(1minusrminusα) and hence if α lt 1 and α ∕= 0 then sign[H(0α r)] =sign[r minus 1] Differentiating yields

Hβ(middot) =α(α + 1)(r minus 1)r(αr + 2β(r minus 1))

(β + (1minus β)r)α+3

We now consider four cases

1 Suppose 0 lt α le 1 and r gt 1 Then H(0α r) ge 0 and Hβ(middot) gt 0 and hence H(βα r) gt0 for all β isin (0 1)

2 Suppose minus1 le α lt 0 and 0 le r lt 1 Then H(0α r) lt 0 and Hβ(middot) le 0 and henceH(βα r) lt 0 for all β isin (0 1)

3 Suppose α lt minus1 and r gt 1 Then H(0α r) gt 0 and H(1α r) = α(r minus 1)(α minus 1 + r(1 +α)) gt 0 We will show that Hβ(βα r) = 0 implies H(βα r) gt 0 which combines withthe previous two inequalities to imply that H(middot) gt 0 Accordingly assume Hβ(βα r) =0 which occurs when β = αr

2(1minusr) which implies α isin (minus2minus1) and r ge 2

2+α(because

β le 1 and α lt minus1) Furthermore

H

983061αr

2(1minus r)α r

983062= α

9830771minus αminus r2

9830612

r(α + 2)

983062α+2983078

49

The derivative of the above expression with respect to r is α983043

2α+2

983044α+2rminusαminus1 which is

strictly positive given α isin (minus2minus1) and r gt 2α+2

gt 2 Moreover when evaluated withr = 2 the expression reduces to 1 minus α minus 4

(α+2)2 which is strictly positive given α isin

(minus2minus1) Therefore H983059

αr2(1minusr)

α r983060gt 0 as was to be shown

4 Suppose α lt minus1 and 0 le r lt 1 Then H(0α r) lt 0 and H(1α r) = α(r minus 1)(α minus 1 +r(1 + α)) lt 0 As argued in the previous case Hβ(βα r) = 0 requires α isin (minus2minus1)and r ge 2

2+αgt 2 which is not possible given that we have assumed r lt 1 Thus Hβ(middot)

has a constant sign in the relevant domain which implies that H(middot) lt 0 in the relevantdomain

C3 The role of linearity and MLRP in the signaling application

Recall that in an LSHP equilibrium the strategy micro(middot) for types that separate is determined bythe initial condition micro(0) = 0 and the differential equation (8)

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ

Part 1 of Lemma 6 established that

partEs|t [β(s t)]

partπle

partEs|t [β(s t)]

partπ(14)

when signal s is more informative (ie drawn from a more informative experiment) than sig-nal s Inequality (14) implies that the solution to the aforementioned initial-value problem ispointwise lower under the more informative experiment and hence the equilibrium signalinglevel micro(t) is lower for every type when the receiver has access to s rather than s

We show below how the conclusion can be altered by dropping either linearity of thesenderrsquos payoff in the receiverrsquos posterior (Example 3) or the MLRP property of the receiverrsquosexperiments (Example 4)

Example 3 Letting V (β) equiv β(1minus β) suppose the senderrsquos payoff is

V (β)minus c(m t)

which is convex in β Assumption (7) in Section 4 continues to imply the relevant single-crossing condition for this modified objective Using Bayes rule we compute

Es|t[V (β(s π))] =π

1minus πEs|t

983063g(s|1)g(s|0)

983064

Differentiating and evaluating at π = t

partEs|t[V (β(s t))]

partπ=

1

t(1minus t)Es|t

983063β(s t)

1minus β(s t)

983064

50

The term inside the expectation operator on the right-hand side above is a convex function ofβ(middot) It follows that

partEs|t[V (β(s t))]

partπge

partEs|t[V (β(s t))]

partπ

by contrast to inequality (14) That is the convexity in V (middot) is strong enough to ensure that themarginal benefit from inducing a higher interim belief π (locally at π = t) is higher when theexogenous signal is more informative It follows that in an LSHP equilibrium all types beara higher (at least weakly) signaling cost when the exogenous signal is more informative23

Example 4 To see the role of MLRP-experiments we have to modify the signaling modelof Section 4 by introducing more states because any experiment in a two-state model is anMLRP-experiment

Assume a full-support common prior about the state ω isin Ω = 0 1 2 The sender re-ceives some private information indexed by t isin [0 1] which updates his belief about thestate to (z 1 minus z(1 + t) zt) where each element of this vector is the probability assigned tothe corresponding state The parameter z isin (0 12) is a commonly known constant We referto t as the senderrsquos type Letting M(β) equiv

983123ω ωβ(ω) be the receiverrsquos expectation of the state

when she holds belief β the senderrsquos payoff is

M(β)minus c(m t)

Let s represent the outcome of an uninformative experiment and let β(s t) represent theposterior of the receiver after observing s when she puts probability one on the senderrsquos typet It clearly holds that

Es|t983045M(β(s t))

983046= M(β(s t)) = 1minus z + zt

Now consider an experiment E with a binary signal space s isin sl sh Let g(s|ω) denotethe probability of the signal realization in each state specified as follows

983063g(sl|0) g(sl|1) g(sl|2)g(sh|0) g(sh|1) g(sh|2)

983064=

9830630 1 01 0 1

983064

This experiment is the same as that used in Example 1 of Appendix B it is not an MLRP-experiment Note also that s is more informative than s

Suppose the receiver ascribes probability one to the senderrsquos type t By Bayes rule if thesignal realization is sl the receiverrsquos posterior belief about the state is β(sl t) = (0 1 0) and

23 On the other hand if V (β) equiv log[β(1 minus β)] then the local marginal benefit of inducing a higher interimbelief is independent of the exogenous experiment The reason is that now V (β(sπ)) = log

983059π

1minusπ

983060+log

983059g(s|1)g(s|0)

983060

and hence partEs|t[V (β(s t))]partπ does not vary with the experiment Note that in this example V (middot) is neitherconvex nor concave

51

therefore M(β(sl t)) = 1 For signal realization sh

β(sh t) =

9830611

1 + t 0

t

1 + t

983062

and therefore

M(β(sh t)) =2t

1 + t

The sender of type trsquos expectation is

Es|t983045M(β(s t))

983046= (1minus z(1 + t))M(β(sl t)) + z(1 + t)M(β(sh t))

The derivative of the above expression with respect to t evaluated at t = t is

partEs|t [M(β(s t))]

part t=

2z

1 + tge z =

partEs|t [M(β(s t))]

part t (15)

This inequality is opposite to inequality (14) even though s is more informative than s

In this example Es|t[M(β(s t))] and Es|t[M(β(s t))] are both (weakly) supermodular inthe senderrsquos type t The assumption that cmt(m t) lt 0 ensures that indifference curves in thespace of (m t) for different types are single-crossing As local incentive compatibility thenimplies global incentive compatibility inequality (15) implies that all types incur higher (atleast weakly) signaling costs when the receiver has access to more information

52

The derivative of the above expression with respect to r is α983043

2α+2

983044α+2rminusαminus1 which is

strictly positive given α isin (minus2minus1) and r gt 2α+2

gt 2 Moreover when evaluated withr = 2 the expression reduces to 1 minus α minus 4

(α+2)2 which is strictly positive given α isin

(minus2minus1) Therefore H983059

αr2(1minusr)

α r983060gt 0 as was to be shown

4 Suppose α lt minus1 and 0 le r lt 1 Then H(0α r) lt 0 and H(1α r) = α(r minus 1)(α minus 1 +r(1 + α)) lt 0 As argued in the previous case Hβ(βα r) = 0 requires α isin (minus2minus1)and r ge 2

2+αgt 2 which is not possible given that we have assumed r lt 1 Thus Hβ(middot)

has a constant sign in the relevant domain which implies that H(middot) lt 0 in the relevantdomain

C3 The role of linearity and MLRP in the signaling application

Recall that in an LSHP equilibrium the strategy micro(middot) for types that separate is determined bythe initial condition micro(0) = 0 and the differential equation (8)

partc(micro(t) t)

partmmicroprime(t) =

partEs|t [β(s t)]

partπ

Part 1 of Lemma 6 established that

partEs|t [β(s t)]

partπle

partEs|t [β(s t)]

partπ(14)

when signal s is more informative (ie drawn from a more informative experiment) than sig-nal s Inequality (14) implies that the solution to the aforementioned initial-value problem ispointwise lower under the more informative experiment and hence the equilibrium signalinglevel micro(t) is lower for every type when the receiver has access to s rather than s

We show below how the conclusion can be altered by dropping either linearity of thesenderrsquos payoff in the receiverrsquos posterior (Example 3) or the MLRP property of the receiverrsquosexperiments (Example 4)

Example 3 Letting V (β) equiv β(1minus β) suppose the senderrsquos payoff is

V (β)minus c(m t)

which is convex in β Assumption (7) in Section 4 continues to imply the relevant single-crossing condition for this modified objective Using Bayes rule we compute

Es|t[V (β(s π))] =π

1minus πEs|t

983063g(s|1)g(s|0)

983064

Differentiating and evaluating at π = t

partEs|t[V (β(s t))]

partπ=

1

t(1minus t)Es|t

983063β(s t)

1minus β(s t)

983064

50

The term inside the expectation operator on the right-hand side above is a convex function ofβ(middot) It follows that

partEs|t[V (β(s t))]

partπge

partEs|t[V (β(s t))]

partπ

by contrast to inequality (14) That is the convexity in V (middot) is strong enough to ensure that themarginal benefit from inducing a higher interim belief π (locally at π = t) is higher when theexogenous signal is more informative It follows that in an LSHP equilibrium all types beara higher (at least weakly) signaling cost when the exogenous signal is more informative23

Example 4 To see the role of MLRP-experiments we have to modify the signaling modelof Section 4 by introducing more states because any experiment in a two-state model is anMLRP-experiment

Assume a full-support common prior about the state ω isin Ω = 0 1 2 The sender re-ceives some private information indexed by t isin [0 1] which updates his belief about thestate to (z 1 minus z(1 + t) zt) where each element of this vector is the probability assigned tothe corresponding state The parameter z isin (0 12) is a commonly known constant We referto t as the senderrsquos type Letting M(β) equiv

983123ω ωβ(ω) be the receiverrsquos expectation of the state

when she holds belief β the senderrsquos payoff is

M(β)minus c(m t)

Let s represent the outcome of an uninformative experiment and let β(s t) represent theposterior of the receiver after observing s when she puts probability one on the senderrsquos typet It clearly holds that

Es|t983045M(β(s t))

983046= M(β(s t)) = 1minus z + zt

Now consider an experiment E with a binary signal space s isin sl sh Let g(s|ω) denotethe probability of the signal realization in each state specified as follows

983063g(sl|0) g(sl|1) g(sl|2)g(sh|0) g(sh|1) g(sh|2)

983064=

9830630 1 01 0 1

983064

This experiment is the same as that used in Example 1 of Appendix B it is not an MLRP-experiment Note also that s is more informative than s

Suppose the receiver ascribes probability one to the senderrsquos type t By Bayes rule if thesignal realization is sl the receiverrsquos posterior belief about the state is β(sl t) = (0 1 0) and

23 On the other hand if V (β) equiv log[β(1 minus β)] then the local marginal benefit of inducing a higher interimbelief is independent of the exogenous experiment The reason is that now V (β(sπ)) = log

983059π

1minusπ

983060+log

983059g(s|1)g(s|0)

983060

and hence partEs|t[V (β(s t))]partπ does not vary with the experiment Note that in this example V (middot) is neitherconvex nor concave

51

therefore M(β(sl t)) = 1 For signal realization sh

β(sh t) =

9830611

1 + t 0

t

1 + t

983062

and therefore

M(β(sh t)) =2t

1 + t

The sender of type trsquos expectation is

Es|t983045M(β(s t))

983046= (1minus z(1 + t))M(β(sl t)) + z(1 + t)M(β(sh t))

The derivative of the above expression with respect to t evaluated at t = t is

partEs|t [M(β(s t))]

part t=

2z

1 + tge z =

partEs|t [M(β(s t))]

part t (15)

This inequality is opposite to inequality (14) even though s is more informative than s

In this example Es|t[M(β(s t))] and Es|t[M(β(s t))] are both (weakly) supermodular inthe senderrsquos type t The assumption that cmt(m t) lt 0 ensures that indifference curves in thespace of (m t) for different types are single-crossing As local incentive compatibility thenimplies global incentive compatibility inequality (15) implies that all types incur higher (atleast weakly) signaling costs when the receiver has access to more information

52

The term inside the expectation operator on the right-hand side above is a convex function ofβ(middot) It follows that

partEs|t[V (β(s t))]

partπge

partEs|t[V (β(s t))]

partπ

by contrast to inequality (14) That is the convexity in V (middot) is strong enough to ensure that themarginal benefit from inducing a higher interim belief π (locally at π = t) is higher when theexogenous signal is more informative It follows that in an LSHP equilibrium all types beara higher (at least weakly) signaling cost when the exogenous signal is more informative23

Example 4 To see the role of MLRP-experiments we have to modify the signaling modelof Section 4 by introducing more states because any experiment in a two-state model is anMLRP-experiment

Assume a full-support common prior about the state ω isin Ω = 0 1 2 The sender re-ceives some private information indexed by t isin [0 1] which updates his belief about thestate to (z 1 minus z(1 + t) zt) where each element of this vector is the probability assigned tothe corresponding state The parameter z isin (0 12) is a commonly known constant We referto t as the senderrsquos type Letting M(β) equiv

983123ω ωβ(ω) be the receiverrsquos expectation of the state

when she holds belief β the senderrsquos payoff is

M(β)minus c(m t)

Let s represent the outcome of an uninformative experiment and let β(s t) represent theposterior of the receiver after observing s when she puts probability one on the senderrsquos typet It clearly holds that

Es|t983045M(β(s t))

983046= M(β(s t)) = 1minus z + zt

Now consider an experiment E with a binary signal space s isin sl sh Let g(s|ω) denotethe probability of the signal realization in each state specified as follows

983063g(sl|0) g(sl|1) g(sl|2)g(sh|0) g(sh|1) g(sh|2)

983064=

9830630 1 01 0 1

983064

This experiment is the same as that used in Example 1 of Appendix B it is not an MLRP-experiment Note also that s is more informative than s

Suppose the receiver ascribes probability one to the senderrsquos type t By Bayes rule if thesignal realization is sl the receiverrsquos posterior belief about the state is β(sl t) = (0 1 0) and

23 On the other hand if V (β) equiv log[β(1 minus β)] then the local marginal benefit of inducing a higher interimbelief is independent of the exogenous experiment The reason is that now V (β(sπ)) = log

983059π

1minusπ

983060+log

983059g(s|1)g(s|0)

983060

and hence partEs|t[V (β(s t))]partπ does not vary with the experiment Note that in this example V (middot) is neitherconvex nor concave

51

therefore M(β(sl t)) = 1 For signal realization sh

β(sh t) =

9830611

1 + t 0

t

1 + t

983062

and therefore

M(β(sh t)) =2t

1 + t

The sender of type trsquos expectation is

Es|t983045M(β(s t))

983046= (1minus z(1 + t))M(β(sl t)) + z(1 + t)M(β(sh t))

The derivative of the above expression with respect to t evaluated at t = t is

partEs|t [M(β(s t))]

part t=

2z

1 + tge z =

partEs|t [M(β(s t))]

part t (15)

This inequality is opposite to inequality (14) even though s is more informative than s

In this example Es|t[M(β(s t))] and Es|t[M(β(s t))] are both (weakly) supermodular inthe senderrsquos type t The assumption that cmt(m t) lt 0 ensures that indifference curves in thespace of (m t) for different types are single-crossing As local incentive compatibility thenimplies global incentive compatibility inequality (15) implies that all types incur higher (atleast weakly) signaling costs when the receiver has access to more information

52

therefore M(β(sl t)) = 1 For signal realization sh

β(sh t) =

9830611

1 + t 0

t

1 + t

983062

and therefore

M(β(sh t)) =2t

1 + t

The sender of type trsquos expectation is

Es|t983045M(β(s t))

983046= (1minus z(1 + t))M(β(sl t)) + z(1 + t)M(β(sh t))

The derivative of the above expression with respect to t evaluated at t = t is

partEs|t [M(β(s t))]

part t=

2z

1 + tge z =

partEs|t [M(β(s t))]

part t (15)

This inequality is opposite to inequality (14) even though s is more informative than s

In this example Es|t[M(β(s t))] and Es|t[M(β(s t))] are both (weakly) supermodular inthe senderrsquos type t The assumption that cmt(m t) lt 0 ensures that indifference curves in thespace of (m t) for different types are single-crossing As local incentive compatibility thenimplies global incentive compatibility inequality (15) implies that all types incur higher (atleast weakly) signaling costs when the receiver has access to more information

52