Information Transmission When the Informed Party Is Confused

GAMES AND ECONOMIC BEHAVIOR 12, 143–161 (1996)ARTICLE NO. 0010

Information Transmission When the InformedParty Is Confused

JOEL WATSON*

Department of Economics, University of California, San Diego, La Jolla, California 92093

Received June 29, 1993

It is generally accepted within the literature on information transmission that insender–receiver games little or no information will be revealed by the sender aslong as the parties have divergent preferences and certifiable claims cannot bemade. I show that this conclusion does not hold when there is two-sided incompleteinformation. In fact, fully revealing equilibria often exist. I motivate the study oftwo-sided incomplete information by considering that the sender is confused aboutthe meaning of his private information and I discuss the implications for firms, oneof which is that some decisions are optimally delayed. Journal of Economic LiteratureClassification Numbers: C72, L20. 1996 Academic Press, Inc.

Decision makers often rely on second parties to provide them with infor-mation. In a firm, for example, an employee may gather information in thecourse of business that his manager would like to use in making decisions.In that the employee has private information that is of value to the manager,it is in the manager’s interest to motivate the employee to disclose whathe knows. The employee may be asked to simply report his informationto the manager, with his report being unverifiable. However, if the employeedoes not share the preferences of the manager, then it may not be rationalfor the employee to tell the truth.

Such interaction is explored by Crawford and Sobel (1982), who examinea ‘‘sender–receiver’’ game in which the sender has private information thatthe receiver, who must make a decision that affects them both, would liketo know. Crawford and Sobel determine the amount of information that

* E-mail: [email protected]. This research was conducted while I was a researchfellow at Nuffield College, Oxford. I appreciate the comments of Meg Meyer, especially withregard to the intuition behind my main result. I am also grateful to an anonymous referee,whose valuable comments led to substantial improvements in this paper.

1430899-8256/96 $12.00

Copyright 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

144 JOEL WATSON

the sender can be induced to reveal in equilibrium. Roughly, they showthat the amount of information revealed in equilibrium increases as thepreferences of the sender and receiver become aligned. This leads to ageneral conclusion about situations of asymmetric information—one whichis widely accepted in the literature: when parties have divergent preferencesand when announcements are not verifiable, no information will be trans-mitted in equilibrium. For some common situations, though, this conclusionis far off the mark. As I show herein, the conclusion relies on there beingone-sided incomplete information. When both parties have private informa-tion, there are equilibria in which a great deal of information is revealed.In fact, fully revealing equilibria can arise.

I examine a sender–receiver game in which both parties have privateinformation. The sender submits a report to the receiver, who then makesa decision that affects them both. In general, the parameter observed by thesender and that observed by the receiver are correlated, and the receiver’soptimal decision depends on both her information and the sender’s informa-tion. The sender cares only about the receiver’s decision.1 I demonstratethat the sender can be induced to reveal his information honestly as longas, conditional on his observation, it is likely that the receiver’s informationis favorable to the sender (in that it will lead to a decision that the senderlikes). In the next section I develop this simple model and state the necessaryconditions for the existence of a fully revealing equilibrium. I then illustratethe analysis with a job market example.

In Section 2, I provide a specific motivation for studying two-sided incom-plete information, in which the sender is confused about his own informa-tion. I model that the sender receives a private signal but does not knowexactly how the signal is labeled. That is, he does not quite understand whatthe signal means. The receiver understands the labeling convention—whichidentifies her private information—but she does not observe the signalherself, and so she must rely on the sender’s report. I show that this modelis covered by the analysis of Section 1. I prove that, under certain conditions,if the sender is sufficiently confused then there is an equilibrium in which thesender reports truthfully. Roughly stated, the existence of a fully revealingequilibrium requires that the sender not expect too low a payoff on average,relative to his feasible payoffs in the game.

Because confusion may lead to a fully revealing equilibrium, in whichthe receiver obtains her most desired outcome, the receiver may preferthat the sender be confused. The implications of this result for the em-ployee–manager relationship in firms is the topic of Section 3. There Ihighlight the idea that an employee’s confusion may be beneficial to the

1 In Crawford and Soebel’s model, the sender also cares about her own type. My resultscan be extended to this case.

INFORMATION TRANSMISSION AND CONFUSION 145

firm and I provide some related conclusions. I also identify the role thatconfusion can play in determining when decisions are made in a firm. Idiscuss a two-period version of the communication game in which themanager might find it worthwhile to delay a decision that would optimally(in a narrow sense) be made in the first period. She does so to perpetuatethe employee’s confusion and encourage truthful reporting in the secondperiod. This result suggests a rationale for postponing decisions and cluster-ing them in time.

Before plunging into the analysis, I must mention a few papers in theliterature that address the topic of information transmission. Two papersstand out as quite relevant to my work in that they deal with variants ofthe Crawford and Sobel (1982) game in which much information is revealedin equilibrium despite the divergence of the agents’ preferences. First,Seidmann (1990) provides an example of a sender–receiver game with two-sided incomplete information for which there is a fully revealing equilib-rium. All types of the sender share the same ordinal preferences over thedecision of the receiver, but the types differ in their preferences overlotteries. Because the receiver has private information that influences heraction, the sender effectively faces a lottery that is defined by the informa-tion she reveals. Under some conditions the types of the sender can besorted in equilibrium. Seidmann’s example shows that fully revealing equi-libria may arise if the receiver has private information and the sender caresabout his own type directly. This setting should be contrasted with the oneI study, in which the sender does not directly value his own informationand fully revealing equilibria arise due to correlation between the sender’sand receiver’s types. Second, Pitchik and Schotter (1987) formulate a simple(binary) model of information transmission with one-sided uncertainty,which admits a unique mixed strategy equilibrium. The model’s comparativestatics are a bit different from what would be expected from Crawford andSobel (1982).

There is also some interesting work on sender–receiver games in whichthe receiver can commit to a decision rule. Melumad and Shibano (1991)examine both the case in which the receiver can commit and the case inwhich she cannot commit and derive results about payoffs and decisionrules when the parties substantially disagree about what action the receivershould take. Melumad and Reichelstein (1989) study a principal–agentgame in which the agent has private information and the principal can offera menu of contracts—potentially to screen the agent’s types. (I brieflyaddress commitment by the receiver in Section 2.) Other relevant papersexamine the case in which announcements are verifiable, which is the keydifference between the analysis in these papers and my work. Milgrom andRoberts (1986) study persuasion games with a verifiability assumption andinvestigate the roles of competition between interested parties, the sophisti-

146 JOEL WATSON

cation of the decision maker, and how well the decision maker is informedabout the strategic situation. They stress the importance of ‘‘skepticism.’’Fishman and Hagerty (1990), Lipman and Seppi (1992), and Shin (1992)roughly examine variations of the Milgrom and Roberts model—limits ondisclosure, partial provability, and discretion in disclosure are the maintopics. Dye (1986), Jovanovic (1982), and Verrecchia (1983) examine situa-tions in which verifiable information can be disclosed at a cost. Finally,Okuno-Fujiwara et al. (1990) study games of asymmetric information whichare preceded by a phase in which the players can certifiably reveal infor-mation.

1. A SENDER–RECEIVER GAME

Consider a simple communication game with two players, the receiverand the sender, in which the sender has private information. The senderobserves a parameter s (the state) and then submits, without cost, anunverifiable report r to the receiver, who then must make a decision d.The receiver does not observe the state directly, but may learn about thestate through the sender’s report. Suppose s [ S, r [ S, and d [ D, whereS is the set of possible states and D is the set of actions available to thereceiver. For simplicity I will focus on pure strategies, although all of theresults below also hold when mixed strategies are considered. Assume thatS and D are finite and let n 5 #S and l 5 #D, where # denotes ‘‘numberof elements.’’ The state is realized according to some probability distributionwhich is common knowledge between the players.

The receiver cares about both the state and her decision, while the sendercares only about the receiver’s decision.2 Let v: D R R be the sender’svon Neumann–Morgenstern utility function and assume that v(d) ? v(d9)whenever d ? d9. Obviously the receiver would like the sender to revealthe state to her, except in the degenerate case in which knowledge of thestate would not affect her decision. However, since the sender does notshare the receiver’s preferences, it may not be in the sender’s interest totell the truth. In fact, in no Nash equilibrium of the game will the senderdisclose useful information.

THEOREM 1. In the game with one-sided incomplete information, there isno equilibrium in which the receiver’s decision depends on the sender’s report.

Proof. Suppose that in some equilibrium the sender, depending on the

2 One can make this more general, so that the sender also cares about her own information.However, the current framework best allows me to demonstrate the significance of two-sidedincomplete information. My results can be extended to the more general case.


state revealed to him, sends at least two different reports. Let ra and rb betwo of the reports sent and let da and db be the receiver’s decisions inresponse to these two reports, in equilibrium. It must be that da 5 db . Tosee this, presume that da ? db and note that v(da) ? v(db). Without lossof generality, take the case in which v(da) . v(db). Then it is not rationalfor the sender to send the report rb because ra induces a more favorabledecision (keeping in mind that the sender cares only about the decision ofthe receiver). This contradicts that both ra and rb are sent in equilibrium.

Q.E.D.

This result is expected given the work of Crawford and Sobel (1982),who study a similar communication game in which the preferences of thereceiver and sender are at odds to some variable degree. Their analysissuggests a general conclusion: if the sender and receiver have divergentpreferences and certifiable claims cannot be made, then no worthwhileinformation will be revealed by the sender in equilibrium. However,whereas this conclusion is valid in settings of one-sided incomplete informa-tion, it is not legitimate when there is two-sided incomplete information.

To explore the possibility that information is transferred when both thesender and the receiver have private information, expand the model slightly.Suppose the state is represented by a pair (s, z), where s is observed bythe sender and z is observed by the receiver. That is, the receiver and thesender observe separate components of the state. Suppose that z [ Z forsome finite Z and let m 5 #Z. The sender, as before, cares only about thedecision d of the receiver, but the receiver cares about s, z, and d. Let p:S 3 Z R [0, 1] be the joint distribution function for the parameters s andz and assume that it is common knowledge between the players. That is,p(s, z) is the probability that (s, z) is the state in the game.

Upon observing s, and before sending his report to the receiver, thesender forms a conditional belief about z which is given by the marginaldistribution p(s, ?). Given the sender’s report and her own information z,the receiver forms a conditional belief about the state and selects d tomaximize her expected payoff. Let BR: S 3 Z R D be the receiver’s best-response correspondence over states. (It will not be necessary to explicitlydefine the receiver’s payoffs.) In other words, if the receiver were tolearn s and z then it would be optimal for her to select an action fromBR(s, z) , D. Assume that BR is single-valued and is thus a function.

It is well known that this type of game always features a ‘‘babbling’’equilibrium in which the sender always selects the same report, regardlessof s, and the receiver chooses d guided only by her own private information,z. However, I wish to explore the possibility that a fully revealing, or truthtelling, equilibrium exists in this game—one in which the sender reportshis information honestly (r 5 s) and the receiver selects her best-response

148 JOEL WATSON

with knowledge of the state (d 5 BR(r, z)). Note that there are other typesof fully revealing equilibria, in which the receiver and sender coordinateon a different meaning for each report (like signaling ‘‘orange’’ with ‘‘apple’’and vice versa). But if any fully revealing equilibrium exists, then there isone in which the sender sets r 5 s. (This is a version of the revelationprinciple.) Therefore, without loss of generality, I will focus on this naturaltype of equilibrium.

In a fully revealing equilibrium, the sender, after observing s and updatinghis belief about z, must have no incentive to misrepresent what he knows.Truthful reporting can thus be supported in equilibrium if and only if

Oz[Z

p(s, z)v(BR(s, z)) $ Oz[Z

p(s, z)v(BR(s9, z)) for all s, s9 [ S. (1)

The left side of this inequality is the sender’s expected utility from tellingthe truth when he observes s, while the right side is the expected utility ofreporting s9 instead of s.

To see that fully revealing equilibria exist, and to construct a simpleexample, take the binary case in which n 5 m 5 l 5 2. Let S 5 hs1 , s2j,D 5 hd1 , d2j, and Z 5 hz1 , z2j. For simplicity, define pij ; p(si , zj) for i 51, 2 and j 5 1, 2, so that the probability distribution over the state is givenby the matrix

z1 z2

s1 p11 p12.

s2 p21 p22

Exclude from consideration the degenerate and uninteresting case in whichBR is constant. For i, j 5 1, 2, let vij ; v(BR(si , zj)) be the sender’s payoffwhen the receiver selects her most preferred action for state (si , zj). Thencondition 1 above becomes

p11(v11 2 v21) 1 p12(v12 2 v22) $ 0 andp21(v21 2 v11) 1 p22(v22 2 v12) $ 0.

A necessary condition for these inequalities to hold is that BR(s1 , z1) 5BR(s2 , z2) ? BR(s1 , z2) 5 BR(s2 , z1). That is, a fully revealing equilibriumcan exist only in situations in which the receiver wishes to select one actionif s and z ‘‘match’’ and another action otherwise.3

3 This, of course, is specific to the 2 3 2 case, but the intuition extends to the general case.


Suppose that BR(s1 , z1) 5 BR(s2 , z2) 5 d1 and BR(s1 , z2) 5BR(s2 , z1) 5 d2 and take the case in which v(d1) . v(d2). In this setting,there is a fully revealing equilibrium if and only if

p11 $ p12 and p22 $ p21 .

The intuition is clear. Knowing that the receiver selects the sender’s pre-ferred action when s and z match, the sender must ascribe at least probabilityAs to this event, conditional on seeing s, in order for him to tell the truth.For instance, if the sender observes that s 5 s1 then he must believe thatz1 is conditionally more likely than z2 . Note that, in general, revealingequilibria rely on correlation between s and z.4

A Job Market Example

Consider a job market in which workers apply for positions at two differ-ent firms. There are two types of workers—decisive and congenial—withthe decisive workers representing a proportion f of the population. Further-more, one of the firms is in need of decisive workers, while the other firmwould like to hire congenial workers. Firms wish to hire workers only ofthe type they individually need. For example, a congenial worker is useless(in fact, costly) to the firm that needs decisive workers. For their part, theworkers would rather be hired by either firm than be unemployed.

The workers cannot distinguish between the two firms and the firmscannot distinguish between the two types of workers. However, the firmsand workers participate in a conference which is designed to match themby type. At the conference, each worker is assigned to one of the two firmsaccording to a screening process that functions independently betweenworkers and that assigns each worker to the ‘‘correct’’ firm with a probabilityq [ [As, 1). For instance, q is the probability that a given decisive workeris assigned to the firm that needs decisive workers. After the screening iscomplete, each worker is interviewed by a representative of the firm withwhich he is matched. Then the firms decide which workers to hire. Thereis no secondary market for workers who are not offered jobs.

Imagine a worker (the sender) being interviewed by a firm’s representa-tive (the receiver). The representative has private information about thetype, z, of her firm and the worker has private information about histype, s. Let s1 and s2 signify the decisive and congenial types of worker,respectively, let z1 be the type of firm that seeks decisive workers, and let

4 Correlated information is also valuable in some mechanism design problems. Correlationbetween the agents’ types can improve the designer’s ability to extract surplus from the agents.On this point, see Cremer and McLean (1985, 1988) and McAfee and Reny (1992).

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z2 be the firm that demands congenial workers. Conditional on the workerbeing matched with the firm, the joint distribution of s and z is given bythe matrix

z1 z2

s1 fq f(1 2 q).

s2 (1 2 f)(1 2 q) (1 2 f)q

The firm’s representative would like the worker to reveal his true type.However, the worker wishes to convince the representative that his typematches that of the firm, so their preferences are not aligned. Since q . As,though, a fully revealing equilibrium exists, which leads to the best possibleoutcome for the firm. That is, there is an equilibrium in which the workerhonestly reports his type during the interview.

One might wonder if the coordination problem in this example wouldbe solved if the firms simply announced their types and let the workersself-sort, without the help of the screening process. In fact, this is the caseas long as workers are indifferent between working at the two types of firm(as I assumed above). However, suppose that decisive workers would ratherwork at firms that need congenial workers— perhaps because they are morelikely to find other decisive workers like themselves at those firms—and viceversa. Then announcing its type is not in the best interest of each firm. Butfully revealing equilibria of the interview game still exist for large enoughq. (Such a game is not formally covered here, though, as it involves a senderwhose payoff depends not only on the receiver’s action but on the receiver’stype as well.)

2. CONFUSION

The analysis of the last section demonstrates that in order for a fullyrevealing equilibrium to exist, the sender cannot be certain how his informa-tion will affect the decision of the receiver. In other words, truth telling relieson two-sided incomplete information. One interpretation of the receiver’sprivate information is that it captures the sender’s lack of understandingabout the information with which the sender is dealing. It is to this interpre-tation that I now turn. As I show below, fully revealing equilibria exist inmany settings, as long as the sender is sufficiently confused.

For example, suppose the manager of a firm can invest in one of twoprojects—call them A and B. The manager, who wishes to invest in the


more profitable project, employs a worker whose duties include gatheringinformation about the projects and reporting this information back to themanager. The employee will participate in whatever project is chosen andknows that he would enjoy project A more than project B. Therefore, theemployee, regardless of the results of his research, would like the managerto select project A. As I showed at the beginning of the last section, if thiswere the whole story then the employee has no incentive to tell the truth.

However, what if the employee does not understand the informationwith which he is dealing? Suppose the employee learns which project hasthe higher risk-adjusted expected net present value (RAENPV), yet theemployee has no clue what ‘‘risk-adjusted expected net present value’’means. Imagine that the employee conjectures that with probability As

RAENPV is synonymous with ‘‘profitability’’ and with probability As

RAENPV signals a loss. Finally, suppose the manager gives the employeethese instructions: ‘‘Please report your findings to me; I am going to takeyour report at face value and base my decision on it. By the way, I thinkproject A is more likely to be the profitable one.’’ Then there is an equilib-rium in which the employee reports his information truthfully. Honestreporting leads to the selection of project A with a probability that exceedsAs, whatever the employee observes, while lying leads to the selection ofproject A with a probability of less than As.

The sender’s confusion can be easily modeled in a general setting andthen shown to map into the framework developed in the last section.Suppose that the underlying uncertainty for the receiver is represented by aparameter y, which is selected from a finite set Y according to the probabilitydistribution a. Assume that a(y) . 0 for all y [ Y and let n 5 #Y. Thereceiver must make some decision d, as before, and her payoff is determinedby both y and d. Assume that the receiver’s optimal decision given y isunique, and let B: Y R D be the receiver’s best-response function. Thatis, if the receiver knew that y were the realized parameter value, then shewould select B(y). Of course, the receiver does not observe y directly, butrelies on the report of the sender, who cares only about the receiver’sdecision. As in the last section, let v: D R R be the sender’s utility function.Also, assume that B(y) is not a constant function, so that knowledge of yis beneficial to the receiver.

The sender obtains some information about y but may not know what theinformation means. This uncertainty is modeled by introducing a function z:Y R Y (the labeling convention) which encodes the information that thesender learns. Let Z be the set of one-to-one functions from Y onto Y.(The set Z will soon take on the same meaning as it did in the last section.)The labeling convention z is selected from Z according to the probabilitydistribution b and is then revealed to the receiver. That is, the receiverknows how the information about y is encoded but does not learn z(y).

152 JOEL WATSON

The sender observes s 5 z(y); he receives the encoded version of y butnot the code. Note that z21 is a well-defined function and so the receivercould infer y, and make her optimal decision, if the sender were to revealhis information s. But under what conditions will the sender be willing totell the truth?

It is not difficult to show that this model is covered by the analysis ofthe last section. As before, s is the sender’s private information and z isthe receiver’s private information. Note that m 5 #Z 5 n! since there aren! one-to-one functions from Y to Y. The receiver’s best response as afunction of s and z is given by BR(s, z) 5 B(z21(s)), for all s and z. Allthat is left to do in translating the model into the framework of the lastsection is to find the joint probability distribution of s and z. It’s easy tosee that p(s, z) 5 b(z)a(z21(s)) for all s and z.

In order to state results about the existence of a fully revealing equilib-rium when the sender is confused, I will need to analyze this model withvarious distributions over the labeling convention. Given a distribution b, let

c(b) ; S n!n! 2 1D max

z[ZUb(z) 2

1n!U

define the distance between b and the distribution that assigns eachz [ Z equal probability. Total confusion on the part of the sender, withregard to the code z, is represented by b such that c(b) 5 0. If the senderknows z then c(b) 5 1.

THEOREM 2. Suppose oy[Ya(y)v(B(y)) $ oy[Y(1/n)v(B(y)). Thenthere is a number c $ 0 such that for all b with c(b) # c the game has afully revealing equilibrium. Furthermore, if the condition holds strictly thenc . 0.

Proof. Given in the Appendix.

This theorem establishes a condition under which, with enough confusionon the part of the sender, a fully revealing equilibrium exists. Note thatoy[Ya(y)v(B(y)) is the sender’s expected utility if the receiver were toobserve y directly, with y distributed according to a. Furthermore,oy[Y(1/n)v(B(y)) is the sender’s expected utility if the receiver were toobserve y directly, with each y [ Y assigned equal probability. Thus, thecondition of the theorem is as follows: if the receiver were to observe ydirectly then the sender would fare at least as well under the distributiona as he would if each y [ Y were selected with equal probability.

The condition is roughly that realizations of the parameter y which thesender finds more attractive occur with greater probability than those whichthe sender finds less attractive. In the next section I will speak of this setting


as that in which ‘‘the sender’s favorite states are most probable.’’ A sufficientcondition for the criterion of the theorem (but certainly not a necessarycondition) is that v(B(y)) $ v(B(y9)) implies that a(y) $ a(y9). Whenthinking about this model and Theorem 2 remember that the sender doesnot care about the parameter y per se. Rather, he cares about the receiver’soptimal decision given the parameter, B(y).

To build intuition for the theorem, think of the sender as being uncertainabout two things: the parameter y and the labeling convention z. Whenthe sender observes s he simultaneously updates his beliefs regarding y andz, with the precision of the conditional beliefs determined by the precisionof the prior beliefs. For instance, suppose that a priori the sender is quitecertain about z but not certain about y. In the extreme case, he may knowz exactly, yet ascribe equal probability to each y [ Y. Then, upon observings and updating his beliefs, the sender’s confidence about y will improvesignificantly. On the other hand, and trivially perhaps, the sender’s assess-ment of z will not change much. Likewise, if a priori the sender is veryunsure about the labeling convention, yet more certain of the parametery, his confidence about y will not be significantly altered upon observing s.

The theorem requires that the sender be very confused, so that he is moreuncertain about the labeling convention than he is about the parameter y.This guarantees that upon observing s the sender does not update hisassessment of y very much, although his confidence regarding z may improveprofoundly. The theorem also requires that the sender’s favorite states bemost probable. Therefore, given that the sender’s belief about y does notchange much after observing s, he believes ex post that his favorite statesare most probable, which gives him an incentive to tell the truth.5

In a fully revealing equilibrium the receiver fares as best as she can,because she chooses d knowing y. Since a confused sender can often beinduced to report his information truthfully, the receiver benefits from thesender’s confusion. In the next section I will outline a few implications of thisresult. However, before doing so I must prove that when the requirements ofTheorem 2 are not met, fully revealing equilibria cannot be supported, tothe chagrin of the receiver.

THEOREM 3. Fix a communication game as defined in this section, withb as yet unspecified. Then (a) there is a number c , 1 such that for all b

5 The analysis of Meyer (1991) is similar in that there are two dimensions of information,one of which is more important to the decision maker. Meyer addresses the problem ofdesigning a sequence of two contests between two workers; the two pieces of informationare the identity of the better worker and the accuracy of each contest. Meyer shows that itis often optimal in the second contest to favor the winner of the first because this allows thedecision maker to learn as much as possible about the identity of the better worker (the moreimportant bit of information).

154 JOEL WATSON

with c(b) . c, the game has no fully revealing equilibrium and (b) ifoy[Ya(y)v(B(y)) , oy[Y(1/n)v(B(y)) then, regardless of b, the game pos-sesses no fully revealing equilibrium.

Proof. Given in the Appendix.

I have two comments on Theorems 2 and 3. First, one cannot take c 5c, although the equality holds in the binary case. In the Appendix I presentan example of a communication game such that for some b and b9 withc(b) . c(b9), the game has a fully revealing equilibrium under b but hasno fully revealing equilibrium under b9. That c may be different from c isa product of the definition of c and the interaction between the payoffs,a, and b. It would not be difficult to prove that there exists some alternativedefinition of c for which the two bounds coincide, but the definition wouldbe valid only if a and the payoffs were fixed.

Second, the theorems say nothing about the existence of partially reveal-ing equilibria. In general there are equilibria in which the sender revealssome, but not all, of the information he has. It would be difficult to character-ize fully the set of partially revealing equilibria without studying a morespecific class of sender–receiver games, but it is not difficult to see thatpartially revealing equilibria exist. An easy way to construct one is toconsider two independent sender–receiver games, such as two identical2 3 2 games, that each have fully revealing equilibria. We know that thegames also have noninformative (babbling) equilibria. Imagine the gameformed by the product of these two games, in which the payoffs are definedas the sum of the payoffs of the component games. There is a partiallyrevealing equilibrium for this game which consists of full revelation in onecomponent game and babbling in the other.

Remember that in the sender–receiver game neither party can committo play a specific strategy. In particular, the receiver cannot commit to adecision rule. In some settings, though, it may be reasonable to assumethat the receiver can commit to play a certain strategy. The opportunityto commit makes the receiver weakly better off in this game, because shealways has the option of commiting to play her part of one of the equilibriastudied above. To the extent that commitment improves the receiver’spayoff in the game, one can view commitment as an alternative to confusionfor the receiver. However, note that the receiver always prefers the fullyrevealing equilibrium under confusion to the outcome of the game in whichshe can commit to a decision rule. To see this, realize that in a fully revealingequilibrium, the receiver obtains her best feasible payoff. If she is to achievethis payoff through commitment (and without confusion), the receiver mustcommit to using in her best interest the information the sender transmits.But then the sender has an incentive to misrepresent what he knows and,


as in Theorem 1, the sender will not fully reveal his information. Therefore,the receiver cannot obtain her best possible payoff through commitment.

In some settings, like those modelled in the classical principal–agentframework, the receiver may also be able to contract with the sender andoffer transfer payments as a function of what the sender reports. As withcommitment, this generally works to the receiver’s advantage. It is notimmediately clear, though, whether transfer payments enable the receiverto reach her greatest possible expected payoff. To answer this question,one would have to address, among other things, risk aversion. I will notattempt to study transfer payments here. It is important to note, however,that both transfer payments and commitment may be options for the re-ceiver in some settings, in particular, the manager–employee relationshipin firms, which is the application to which I now turn.

3. IMPLICATIONS FOR FIRMS

In most firms one can find instances in which an employee/worker hasinformation that a manager could use in making a decision. Furthermore,in such situations the manager may not be able to perfectly monitor theemployee, nor might the employee be able to make certifiable claims. Theseare situations in which the analysis of the last two sections applies. Theorems2 and 3 then define conditions under which the manager would prefer thather employee is confused. The theorems also have implications for theamount of influence that the manager wishes to give the employee and forthe value of delaying some decisions.

To derive some conclusions from the model, I shall assume that themanager (receiver) and the employee (sender) coordinate on a Nash equi-librium of the communication game that the manager most prefers. Forexample, if a fully revealing equilibrium exists then I will assume that it isplayed, rather than, say, the babbling, uninformative equilibrium. In defenseof this assumption, note that in most firms one can find many coordinatingdevices that are controlled by management. For instance, the manager ofa firm might provide instructions to her employees, detailing how theyare supposed to perform in various situations. Such instructions might betranslated by the game theorist (with some liberties taken) as, ‘‘dear worker,A is how I expect you to behave, and B is how you should expect me tobehave; notice that A and B constitute a Nash equilibrium.’’ Since theequilibrium is selected by the manager, one would expect that she picksthe equilibrium which she most prefers.

Assume that the manager can, at some positive cost, learn the statedirectly. That is, there is a costly technology by which the manager canobserve s. This technology might represent a source of information that is

156 JOEL WATSON

separate from the employee, it might be a monitoring technology thatinduces the worker to report truthfully, or it could simply refer to themanager duplicating the employee’s work in order to discover s herself.Obviously, if the employee can be motivated to report honestly thenthe manager has no need for the alternative technology. However, ifno fully revealing equilibrium exists in the communication game, thenit may be worthwhile for the manager to invest in the alternativetechnology. One can refer to the case in which the manager relies onthe worker for information as that in which ‘‘the manager gives theemployee influence over her decision.’’ In the setting in which thealternative technology functions to monitor the worker, one might replace‘‘influence’’ with ‘‘discretion.’’

Conclusion 1. When the employee is sufficiently confused about thelabeling convention and the employee’s favorite states are most probable,the manager prefers to give the employee much discretion (and influ-ence).

Conclusion 2. If the employee’s favorite states are not most probablethen the manager prefers to give the employee little discretion (and in-fluence).

Conclusion 3. Only when the employee’s favorite states are mostprobable does the manager prefer that the employee be confused.

Theorem 3 also has implications for the timing of decisions in the firm.Consider a scenario in which the manager and the employee interact overtwo periods. The communication game is played twice—once in each pe-riod—with the two games independent except that they share the samelabeling convention. In the first period y1 is realized, the worker observess1 5 z(y1), and the worker sends a report r1 to the manager, who can thenchoose d1 or delay the decision until the second period. In the secondperiod y2 is realized, the worker observes s2 5 z(y2), the worker sends areport r2 to the manager, and the manager selects d2 (as well as d1 if thefirst period decision was delayed). The parameters y1 and y2 are indepen-dently drawn from Y according to the distribution a. Assume that decisionsmade by the manager are immediately observed by the worker.

Because the labeling convention is the same in both periods, the outcomeof first-period interaction will provide the employee with information aboutz, and this information may affect behavior in the second period. Theemployee can learn about z in two ways. First, as the analysis in thepreceding section suggests, the worker might learn about z through s1.Second, the worker may learn about z through any decision that the man-ager makes in the first peiod. If the employee were to determine z in thefirst period, and this were common knowledge, then no useful information


would be transferred in the second period (recall Theorem 3). It may be inthe manager’s interest to delay choosing d1, if doing so keeps the employeesufficiently confused about z to induce truth telling in the second period.Of course, delaying a decision will also be costly in that the managerdiscounts the future.

A few simplifying assumptions will help produce a clear resultabout delay. Take the binary case in which Y 5 hy1 , y2j and suppose thatv(B(y1)) . v(B(y2)) so that the worker prefers that the manager takeaction B(y1) over action B(y2). In this setting, the condition of Theorem2 is satisfied if and only if a(y1) $ As, which I assume is the case. Alsoassume that all of the manager’s payoffs are positive, so that delaying adecision is indeed costly. Note that there are two possible labeling conven-tions; assume that they are equally likely.

By Theorem 2 there is a fully revealing equilibrium of the static game,which may describe behavior in the first period. After observing s1 5 s,though, the worker updates his belief about the labeling convention. Letbs, denote this conditional distribution. It is easy to show that there is alsoa fully revealing equilibrium in the static game with the labeling conventiondistributed according to bs , for s 5 y1 , y2 . That is, the employee will beconfused about the labeling convention even after observing s1. Therefore,if the manager were to delay choosing d1 until the second period then theemployee could be motivated to report his information honestly in bothperiods. This would allow the manager to select both d1 and d2 optimally,albeit the former with inefficient delay. To see that such an equilibriumexists, note that it involves fully revealing equilibria of the two communica-tion games, the second defined by bs1.

If the worker reports truthfully in the first period and the managerselects d1 optimally and without delay, then the worker can deduce zfrom d1. In this case z is common knowledge before the second periodbegins, at which point no useful information will be revealed by theemployee, leaving the manager with a poor expected payoff. One mightwonder, then, if the manager could select d1 in the first period, but witha random hand, so that she achieves a close-to-optimal result in thefirst communication game while keeping the employee confused for thesecond period. Unfortunately, given that c , 1 in the static game, themanager cannot achieve arbitrarily close to the first-best by using thistechnique. For this simple example I thus obtain the following result(which is not difficult to prove formally).

Conclusion 4. Consider the two-period game delineated above. If themanager discounts the future sufficiently little then she prefers to delaychoosing d1 until the second period.

Traditional economic theory would suggest that decisions be delayed

158 JOEL WATSON

only when the decision maker anticipates that more useful information willbecome available. Conclusion 4 demonstrates another reason for delay.Deferring decisions can help sustain a favorable strategic environment—one that would collapse if decisions were made promptly because employeeswould learn too much about underlying parameters. It is sometimes optimalfor the manager to delay decisions, even when all of the relevant informationis at hand, in order to perpetuate the employee’s uncertainty. In this case,the benefits of keeping the employee confused outweigh the costs dueto delay.

The conclusion also indicates a rationale for the clustering of decisions.Imagine a firm in which the manager and worker interact many times. Themanager may prefer to delay decisions in order to maintain the worker’sconfusion, but at some point in time the cost of delaying the growingnumber of decisions for which all of the relevant information is availableoutweighs the future benefit of sustaining confusion. At such a time, themanager will make several decisions at once. Perhaps the manager willthen wait for the employee to become confused again (if some labelingconventions to be used in the future are not too correlated with thosealready discovered by the employee) at which point the process might beginanew. Alternatively, the manager may randomly invest in changing thelabeling convention to restore the employee’s confusion. Or perhaps thepoint at which decisions are clustered marks a good time for the managerto rotate employees, in order to replace the learned worker with a fresh,confused one.

4. CONCLUSION

One message from the literature on information transmission is thattruth telling is thwarted by divergent preferences and the inability to makecertifiable claims. I have shown that this conclusion is false in settings inwhich there is two-sided incomplete information. One such setting concernsan employee who is called upon to deal with information that he does notfully understand. The employee’s confusion can be quite beneficial to thefirm. Furthermore, the value of keeping the employee confused may leadthe manager to postpone decisions and cluster them in time. My workrepresents an opposite extreme from most of the studies identified in theIntroduction in that I assume that reports cannot be verified. Perhaps itwould be interesting to explore the middle ground, wherein one mightderive results for optimal task assignment and monitoring. I also think thatthe issue of clustering decisions deserves further attention.


APPENDIX

Proof of Theorem 2. Using the notation of the confusion model, condi-tion (1) can be written as

Oz[Z

b(z)a(z21(s))v(B(z21(s)))

$ Oz[Z

b(z)a(z21(s))v(B(z21(s9))) for all s, s9 [ S.

I will demonstrate that this is equivalent to the condition of the theoremwhen c(b) 5 0. The equivalence also holds in the strict version of theinequalities. In addition, note that for any r . 0, c(b) # r implies thatub(z) 2 1/n!u # r for all z [ Z. These last two facts prove the secondassertion of the theorem, once I prove the claim for c(b) 5 0.

To prove the equivalence for the case in which c(b) 5 0, note first thatthis implies that b(z) 5 1/n! for all z [ Z, and so condition (1) becomes

Oz[Z

a(z21(s))v(B(z21(s))) $ Oz[Z

a(z21(s))v(B(z21(s9))) for all s, s9 [ S.

Note that for any s and y, there are (n 2 1)! distinct z such that z(y) 5 s.This implies that the left side of the inequality is simply (n 2 1)! oy[Y

a(y)v(B(y)), independent of s. Likewise, for s ? s9 and y ? y9, there are(n 2 2)! distinct z such that z(s) 5 y and z(s9) 5 y9. Thus, the right sideof the inequality is (n 2 2)!oy[Y oy9?ya(y)v(B(y9)) for s ? s9. Condition(1) is therefore

(n 2 1) Oy[Y

a(y)v(B(y)) $ Oy[Y

Oy9?y

a(y)v(B(y9)).

Reorganizing the terms yields

Oy[Y

v(B(y))[(n 2 1)a(y) 2 Oy9?y

a(y9)] $ 0.

Noting that oy9?ya(y9) 5 1 2 a(y) and performing a bit of algebraicmanipulation leads to the condition of the theorem. Q.E.D.

Proof of Theorem 3. To prove (a) recall that the condition for a fullyrevealing equilibrium is that

Oz[Z

b(z)a(z21(s))v(B(z21(s))) $ Oz[Z

b(z)a(z21(s))v(B(z21(s9)))

160 JOEL WATSON

for all s, s9 [ S. For any « . 0 one can find a r , 1 such that for each bwith c(b) . r, the left side of this inequality is within « ofa(z21(s))v(B(z21(s))) and the right side is within « of a(z21(s))v(B(z21(s9))),for some particular z [ Z which may depend on b. (Recall that a(y) . 0for all y and so a(y) $ a for all y and some a . 0. Also recall that c(b)close to one implies that b(z) is close to one, for some z.) Remember thatz is one-to-one. Thus, for every « . 0 there is a r(«) , 1 such that forc(b) . r(«) the existence of a fully revealing equilibrium requires thatv(B(s)) $ v(B(s9)) 2 2«/a, for all s, s9 [ S. Since B is not constant andv(y) ? v(y9) for all y ? y9, we can find a positive number s such thatv(B(s)) 2 v(B(s9)) . s for some s and s9. Letting 2«/a , s, c(b) .r(«) ; c contradicts the existence of a fully revealing equilibrium, provingpart (a).

To prove part (b) presume that a fully revealing equilibrium exists forsome b. For such an equilibrium, let V(s) be the sender’s (interim) expectedutility upon observing s and let e(s) ; o hb(z)a(y) u s 5 z(y)j be theprobability that the sender observes s. Note that

Os[Y

e(s)V(s) 5 Oy[Y

a(y)v(B(y)),

since the equilibrium is fully revealing. Furthermore, let Vr(s) be the send-er’s expected utility, conditional on the sender observing s and selecting rin response. Note that

Or[Y

(1/n) Os[Y

e(s)Vr(s) 5 Oy[Y

(1/n)v(B(y)).

The condition of the theorem implies that

Or[Y

(1/n) Os[Y

e(s)Vr(s) . Os[Y

e(s)V(s),

which means that

Os[Y

e(s)Vr(s) . Os[Y

e(s)V(s)

for some particular r. (There are n elements in Y.) This in turn impliesthat Vr(s) . V(s) for some s, which contradicts equilibrium. Q.E.D.

Counterexample to c 5 c. Let Y 5 hy1 , y2 , y3j and D 5 hd1 , d2 , d3j.Let B(yi) 5 di and vi(di) 5 i for i 5 1, 2, 3. Suppose that a(y1) 5a(y2) 5 0.3 and a(y3) 5 0.4. The labeling convention z can be represented


by a triple (i, j, k); z 5 (i, j, k) means that z(y1) 5 yi , z(y2) 5 yj andz(y3) 5 yk . Take z1 5 (3, 2, 1), z2 5 (1, 2, 3), z3 5 (2, 3, 1), z4 5 (3, 1, 2),z5 5 (1, 3, 2), and z6 5 (2, 1, 3).

Examine the two communication games given by probability distributionsb and b9, defined as follows. Under distribution b, b(z2) 5 0.23 andb(zl) 5 0.154 for l ? 2. Distribution b9 is defined by b9(z2) 5 0.11 andb9(zl) 5 0.178 for l ? 2. It is easy to compute that c(b) 5 aQgQ;R; andc(b9) 5 aQgP;W;, so c(b) . c(b9). One can show (I used a spreadsheet—it wasnot difficult) that a fully revealing equilibrium exists under b but that nofully revealing equilibrium exists under b9. Therefore, it must be the casethat c , c.

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Information Transmission When the Informed Party Is Confused