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Evidence Games:Truth and Commitment

Sergiu Hart

This version: September 2016

Evidence Games:Truth and Commitment

Sergiu HartCenter for the Study of Rationality

Dept of Economics Dept of MathematicsThe Hebrew University of Jerusalem

hart@huji.ac.ilhttp://www.ma.huji.ac.il/hart

Joint work with

Ilan KremerMotty Perry

Hebrew University of JerusalemUniversity of Warwick

Papers

"Evidence Games: Truth and Commitment"Center for Rationality DP-684, May 2015Revised September 2016American Economic Review (forthcoming)

www.ma.huji.ac.il/hart/abs/st-ne.html

Papers

"Evidence Games: Truth and Commitment"Center for Rationality DP-684, May 2015Revised September 2016American Economic Review (forthcoming)

www.ma.huji.ac.il/hart/abs/st-ne.html

"Evidence Games with RandomizedRewards"

Center for Rationality 2016

www.ma.huji.ac.il/hart/abs/st-ne-mixed.html

Q: "Do you deserve a pay raise?"

Q: "Do you deserve a pay raise?"A: "Of course."

Q: "Are you guilty and deserve punishment?"

Q: "Are you guilty and deserve punishment?"A: "Of course not."

How can one obtain reliable information?

How can one determine the "right" reward, orpunishment?

How can one "separate" and avoid"unraveling" (Akerlof 70)?

Evidence Games: General Setup

AGENT who is informed

PRINCIPAL who takes decision but isuninformed

Agent TRANSMITS information to Principal(costlessly)

Two Setups

SETUP 1: Principal decides after receivingAgent’s message

Two Setups

SETUP 2: Principal chooses a policy beforeAgent’s message

Two Setups

policy : a function that assigns a decisionof Principal to each message of Agent

Two Setups

policy : a function that assigns a decisionof Principal to each message of Agent(Agent knows the policy when sending hismessage)

Two Setups

policy : a function that assigns a decisionof Principal to each message of Agent(Agent knows the policy when sending hismessage)Principal is committed to the policy

Two Setups

GAME: Principal decides after receivingAgent’s message

MECHANISM: Principal chooses a policybefore Agent’s message

policy : a function that assigns a decisionof Principal to each message of Agent(Agent knows the policy when sending hismessage)Principal is committed to the policy

Main Result

In EVIDENCE GAMES

Main Result

In EVIDENCE GAMES

the GAME EQUILIBRIUM outcome(obtained without commitment )

Main Result

In EVIDENCE GAMES

and the OPTIMAL MECHANISM outcome(obtained with commitment )

Main Result

In EVIDENCE GAMES

COINCIDE

Main Result: Equivalence

In EVIDENCE GAMES

COINCIDE

Evidence Games

AGENT has some pieces of evidenceregarding his "value"(known to him but not to the PRINCIPAL )

Evidence Games

PRINCIPAL decides on the reward

Evidence Games

PRINCIPAL wants the reward to be as closeas possible to the value

Evidence Games

AGENT wants the reward to be as high aspossible (same preference for all types )

Evidence Games

AGENT wants the reward to be as high aspossible (same preference for all types )

↑Differs from signalling, screening,

Evidence Games: Truth

Agent reveals:

"the truth, nothing but the truth"

Agent reveals:

NOT necessarily "the whole truth"

Agent reveals:

all the evidence that the agent reveals mustbe true (it is verifiable)

Agent reveals:

the agent does not have to reveal all theevidence that he has

"truth-leaning"

revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability

Previous Work

UnravelingGrossman and O. Hart 1980Grossman 1981Milgrom 1981

Previous Work

Voluntary disclosureDye 1985Shin 2003, 2006, ...

Previous Work

MechanismGreen–Laffont 1986

Previous Work

MechanismGreen–Laffont 1986

Persuasion gamesGlazer and Rubinstein 2004, 2006

Evidence Games

EVIDENCE GAMES model very commonsetups

Evidence Games

In EVIDENCE GAMES there is equivalencebetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)

Evidence Games

In EVIDENCE GAMES there is equivalencebetween EQUILIBRIUM (without commitment)and OPTIMAL MECHANISM (with commitment)

The conditions of EVIDENCE GAMES areindispensable for this equivalence

Example 1

Professor wants salary as high as possible

Example 1

Dean wants salary to be as close as possibleto the Professor’s value

Example 1

Professor’s evidence (verifiable):

Example 1

Professor’s evidence (verifiable):[t0] 50%: no evidence[t+] 25%: positive evidence[t−] 25%: negative evidence

Example 1

Professor’s evidence (verifiable):[t0] 50%: no evidence → value = 60

[t+] 25%: positive evidence → value = 90

[t−] 25%: negative evidence → value = 30

Example 1 t+ : 25% 90

t0 : 50% 60

t− : 25% 30

Example 1: Equilibrium t+ : 25% 90

t0 : 50% 60

t− : 25% 30

t0 : 50% 60

t− : 25% 30

GAME: (G1) Professor provides evidence(G2) then Dean sets salary

t0 : 50% 60

t− : 25% 30

EQUILIBRIUM

t0 : 50% 60

t− : 25% 30

EQUILIBRIUMProfessor:

t+ provides positive evidencet0, t− provide no evidence

t0 : 50% 60

t− : 25% 30

Dean:to positive evidence gives salary = 90

to negative evidence gives salary = 30

t0 : 50% 60

t− : 25% 30

to no evidence gives salary = 50= (50% · 60 + 25% · 30)/(50% + 25%)

t0 : 50% 60

t− : 25% 30

unique sequential EQUILIBRIUMProfessor:

to no evidence gives salary = 50= (50% · 60 + 25% · 30)/(50% + 25%)

t0 : 50% 60

t− : 25% 30

t0 : 50% 60

t− : 25% 30

30 60 90value:

t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+

t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+

partial truth:

t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+

partial truth:

prof says:

t0 : 50% 60

t− : 25% 30

30 60 90value:

t− t0 t+

partial truth:

prof says:

dean pays: 30 50 90

Example 1: Mechanism t+ : 25% 90

t0 : 50% 60

t− : 25% 30

t0 : 50% 60

t− : 25% 30

MECHANISM: (M1) Dean commits to salary policy(M2) then Professor provides evidence

t0 : 50% 60

t− : 25% 30

OPTIMAL MECHANISM

t0 : 50% 60

t− : 25% 30

OPTIMAL MECHANISM

to no evidence gives salary = 50

to negative evidence gives salary ≤ 50

t0 : 50% 60

t− : 25% 30

OPTIMAL MECHANISM

Example 1: Explanation t+ : 25% 90

t0 : 50% 60

t− : 25% 30

t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0

t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0

the value of t0 is higher than the value of t−

t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0

in MECHANISM:

t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0

in MECHANISM:

the only way to separate t− from t0is to pay t− strictly more than to t0

t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0

in MECHANISM:

this is not optimal

t0 : 50% 60

t− : 25% 30

in EQUILIBRIUM :

t− says t0

in MECHANISM:

this is not optimal

OPTIMAL MECHANISM does notseparate more than EQUILIBRIUM

t0 : 50% 60

t− : 25% 30

t0 : 50% 60

t− : 25% 30

t− t0

partial truth:

Example 2

[t±] 5%: both evidences → value = 42

Example 2 t+ : 20% 102

t0 : 50% 60

t± : 5% 42

t− : 25% 30

[t±] 5%: both evidences → value = 42

t0 : 50% 60

t± : 5% 42

t− : 25% 30

t0 : 50% 60

t± : 5% 42

t− : 25% 30EQUILIBRIUM

t0 : 50% 60

t± : 5% 42

Professor:t+, t± provide positive evidencet0, t− provide no evidence

t0 : 50% 60

t± : 5% 42

Professor:t+, t± provide positive evidencet0, t− provide no evidence

Dean:to positive evidence gives salary = 90= (20% · 102 + 5% · 42)/25%

to no evidence gives salary = 50= (50% · 60 + 25% · 30)/75%

t0 : 50% 60

t± : 5% 42

t− : 25% 30

t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±

t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±

partial truth:

t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:

t0 : 50% 60

t± : 5% 42

t− : 25% 30

30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:

dean pays: 30 5042 90

t0 : 50% 60

t± : 5% 42

t− : 25% 30

t0 : 50% 60

t± : 5% 42

t− : 25% 30OPTIMAL MECHANISM

t0 : 50% 60

t± : 5% 42

to both evidences gives salary ≤ 90

t0 : 50% 60

t± : 5% 42

to both evidences gives salary ≤ 90

OPTIMAL MECHANISM = EQUILIBRIUM

t0 : 50% 60

t± : 5% 42

t− : 25% 30

t0 : 50% 60

t± : 5% 42

t− : 25% 30Another equilibrium

t0 : 50% 60

t± : 5% 42

Professor:always provides no evidence

t0 : 50% 60

t± : 5% 42

Dean:ignores all evidence and gives salary = 60= 50%·60+25%·30+20%·102+5%·42

t0 : 50% 60

t± : 5% 42

supported by the belief of the Dean whenreceiving the out-of-equilibrium positive evidencethat it mostly comes from t± rather than from t+

t0 : 50% 60

t± : 5% 42

t− : 25% 30Another equilibrium ("babbling" )

t0 : 50% 60

t± : 5% 42

t− : 25% 30Another equilibrium ("babbling" )SATISFIES ALL STANDARD REFINEMENTS

t0 : 50% 60

t± : 5% 42

t0 : 50% 60

t± : 5% 42

30 42 60 102value:

t− t0 t+t±

partial truth:

t0 : 50% 60

t± : 5% 42

30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:

t0 : 50% 60

t± : 5% 42

30 42 60 102value:

t− t0 t+t±

partial truth:

prof says:

dean pays: 30 6042 < 60

Example 3

Dean wants salary close to Professor’s value

Example 3

Professor t0 wants salary high

Example 3

Professor t− wants salary close to 50

Example 3

EQUILIBRIUM :

separation

Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30

Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)

Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)⇒ contradiction

Example 3

EQUILIBRIUM :

separation ⇒ x0 = 60, x− = 30⇒ t− does not reveal (prefers 60 to 30)⇒ contradiction

⇒ no separation (babbling), x = 45

Example 3

EQUILIBRIUM : no separation , x = 45

Example 3

MECHANISM:

Example 3

MECHANISM:

Example 3

MECHANISM:

to no evidence gives salary = 71(separating: t− prefers 30 to 71)

Example 3

MECHANISM:

REQUIRES COMMITMENT :

after no evidence Dean prefers salary = 60

Example 3

MECHANISM:

MECHANISM is strictly better for the Dean thanEQUILIBRIUM

Example 3: Commitment Helps

MECHANISM:

MECHANISM is strictly better for the Dean thanEQUILIBRIUM

↑NOT "evidence game"

Examples: Summary

Example 1 :equivalence (our result)

Examples: Summary

Example 2 :EQUILIBRIUM different fromOPTIMAL MECHANISM

⇒ "truth-leaning"

Examples: Summary

Example 2 :EQUILIBRIUM different fromOPTIMAL MECHANISM

⇒ "truth-leaning"

Example 3 :assumptions do not holdresult fails: commitment helps

AGENT (A)

PRINCIPAL (P ) (= "market")

AGENT (A)

(finite) set of TYPES: T

the type t ∈ T is chosen according to aprobability distribution p ∈ ∆(T )

AGENT (A)

the type t ∈ T is revealed to Agent and not toPrincipal

AGENT (A)

Agent’s MESSAGE: s ∈ T

AGENT (A)

Agent’s MESSAGE: s ∈ T

Principal’s DECISION: REWARD x ∈ R

Payoffs / Utilities

UA and UP do not depend on the message s

Payoffs / Utilities

UA does not depend on the type t

UA(s, x; t) = x

Payoffs / Utilities

UA(s, x; t) = x

UP : "Canonical" example

ht(x) := UP (s, x; t) = −(x − v(t))2

v(t) = the "value" of type t (to PRINCIPAL )

Payoffs / Utilities

UA(s, x; t) = x

UP : "Canonical" example

ht(x) := UP (s, x; t) = −(x − v(t))2

v(t) = the "value" of type t (to PRINCIPAL )

General assumption:(SP) UP is single-peaked w.r.t. UA

Single Peakedness (SP)

For every distribution of types (belief) q ∈ ∆(T )

the principal’s expected utilityhq(x) =

∑t∈T qt ht(x)

is a single-peaked function of the reward x

For every distribution of types (belief) q ∈ ∆(T )

the principal’s expected utilityhq(x) =

∑t∈T qt ht(x)

is a single-peaked function of the reward x

⇔ There exists v(q) such thath′

q(x) > 0 for x < v(q)

h′q(x) = 0 for x = v(q)

h′q(x) < 0 for x > v(q)

Canonical example:ht(x) = −(x − v(t))2

v(q) = Eq[v(t)] =∑

t qt v(t)

v(q) = Eq[v(t)] =∑

t qt v(t)

More general:ht(x) is a differentiable strictly concavefunction of x, for each t

v(q) = Eq[v(t)] =∑

t qt v(t)

More general:ht(x) is a differentiable strictly concavefunction of x, for each t

(SP) is more general than concavity

Evidence and Truth

Agent reveals:

Evidence and Truth

Agent reveals:

Evidence and Truth

Agent reveals:

Evidence and Truth

Agent reveals:

Evidence and Truth

Agent reveals:

⇒ Agent can pretend to be a type thathas less evidence

Evidence and Truth

L(t) = the set of types that t can pretend to be= the set of possible messages of type t

Evidence and Truth

E = set of (verifiable) pieces of evidence

Evidence and Truth

Et ⊆ E = set of pieces of evidence that t has(and can provide)

Evidence and Truth

L(t) = {s ∈ T : Es ⊆ Et}

Evidence and Truth

L(t) = {s ∈ T : Es ⊆ Et}

(L1) Reflexivity : t ∈ L(t)

Evidence and Truth

L(t) = {s ∈ T : Es ⊆ Et}

(L2) Transitivity : If s ∈ L(t) and r ∈ L(s)then r ∈ L(t)

Evidence and Truth

L(t) = {s ∈ T : Es ⊆ Et}

(L2) Transitivity : If s ∈ L(t) and r ∈ L(s)then r ∈ L(t)

Type (summary)

A type t has two characteristics:

Type (summary)

Value to the Principal(expressed by h(t) and its peak v(t))

Type (summary)

Evidence that he can provide(expressed by L(t))

Type (summary)

Evidence that he can provide(expressed by L(t))

No relation is assumedbetween Value and Evidence

(G1) Agent sends message s ∈ L(t) to Principal

(G2) Then Principal sets reward x ∈ R

STRATEGIES

(Agent) σ(s|t) = probability that type t sendsmessage s in L(t)

STRATEGIES

(Agent) σ(s|t) = probability that type t sendsmessage s in L(t)

(Principal) ρ(s) ∈ R = reward to message s

Equilibrium

(σ, ρ) is a NASH EQUILIBRIUM if

Equilibrium

(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)

Equilibrium

(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)

(P) σ̄(s) > 0 ⇒ ρ(s) = v(q(s))

whereσ̄(s) = total probability of message s

q(s) ∈ ∆(T ) = the posterior distributionon types conditional on message s

Equilibrium

(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)

(P) σ̄(s) > 0 ⇒ ρ(s) = v(q(s))

OUTCOME: πt = maxs∈L(t) ρ(s) = ρ(σ(·|t))

Equilibrium

(A) σ(r|t) > 0 ⇒ ρ(r) = maxs∈L(t) ρ(s)

(P) σ̄(s) > 0 ⇒ ρ(s) = v(q(s))

OUTCOME: πt = maxs∈L(t) ρ(s) = ρ(σ(·|t))

π = (πt)t∈T ∈ RT

Truth-Leaning

Revealing the whole truth gets a slight(= infinitesimal) boost in payoff andprobability

Truth-Leaning

(T1) Revealing the whole truth is preferablewhen the reward is the same

Truth-Leaning

(T1) Revealing the whole truth is preferablewhen the reward is the same(lexicographic preference)

Truth-Leaning

(T2) The whole truth is revealed with infinitesimalpositive probability

Truth-Leaning

(T2) The whole truth is revealed with infinitesimalpositive probability(by mistake, or because the agent may benon-strategic, or ... [UK])

Truth-Leaning

A Nash equilibrium is TRUTH-LEANING if itsatisfies:

Truth-Leaning

(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1

Truth-Leaning

(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1

(if message t is a best reply for type t then itis used for sure)

Truth-Leaning

(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1

(T2) σ̄(t) = 0 ⇒ ρ(t) = v(t)

Truth-Leaning

(T1) ρ(t) = maxs∈L(t) ρ(s) ⇒ σ(t|t) = 1

(T2) σ̄(t) = 0 ⇒ ρ(t) = v(t)

(if message t is not used then the rewardequals the value of type t; i.e., the belief isthat the [unexpected] message t comes fromtype t)

Truth-Leaning

Truth-Leaning equilibria:

Truth-Leaning

coincide with the equilibria selected in the"voluntary disclosure" literature

Truth-Leaning

satisfy all the refinement conditions in theliterature

Truth-Leaning

eliminate "unreasonable" equilibria (such as"babbling" in Example 2)

Truth-Leaning

eliminate "unreasonable" equilibria (such as"babbling" in Example 2)

Mechanism

MECHANISM:

Mechanism

MECHANISM: Reward scheme ρ : T → R

(ρ(s) = reward to message s)

Mechanism

Agent’s payoff when type is t:

πt = maxs∈L(t)

Mechanism

πt = maxs∈L(t)

Outcome : π = (πt)t∈T ∈ RT

Mechanism

πt = maxs∈L(t)

Outcome : π = (πt)t∈T ∈ RT

Principal’s payoff :

H(π) =∑

pt ht(πt)

Incentive Compatibility

Outcome π = (πt)t∈T ∈ RT is generated by a

mechanism ρ

if and only if

mechanism ρ

if and only if

πt ≥ πs

for all s, t ∈ T with s ∈ L(t)

mechanism ρ

if and only if

πt ≥ πs

Immediate because L satisfies reflexivity (L1)and transitivity (L2)

mechanism ρ

if and only if

πt ≥ πs

"direct" mechanism: ρ(t) = πt

mechanism ρ

if and only if

πt ≥ πs

"direct" mechanism: ρ(t) = πt

Optimal Mechanism

OPTIMAL MECHANISM :

Optimal Mechanism

OPTIMAL MECHANISM :

Maximize H(π) =∑

t∈T pt ht(πt)

Optimal Mechanism

OPTIMAL MECHANISM :

Maximize H(π) =∑

t∈T pt ht(πt)

subject to (IC):

πt ≥ πs

Optimal Mechanism

OPTIMAL MECHANISM :

Maximize H(π) =∑

t∈T pt ht(πt)

subject to (IC):

πt ≥ πs

OPTIMAL MECHANISM = Maximum underIncentive Constraints

Main Result

(i) There is a uniqueTRUTH-LEANING EQUILIBRIUM outcome.

Main Result

(ii) There is a uniqueOPTIMAL MECHANISM outcome.

Main Result

(iii) These two outcomes COINCIDE.

The equilibrium strategies need not be unique(happens only when the Agent is indifferent—andthen the Principal is also indifferent)

All the conditions are indispensable :

Truth structure: reflexivity

Truth structure: transitivity

Truth Leaning: whole truth slightly better

Truth Leaning: whole truth slightly possible

Agent’s utility: independent of type

Principal’s utility: single-peaked with respectto Agent’s utility

EQUILIBRIUM (without commitment)yields the same as COMMITMENT

EQUILIBRIUM yields OPTIMAL SEPARATION(for the principal / "market")

EQUILIBRIUM yields PARETO EFFICIENCY(in the canonical setup)

UNDER INCENTIVE CONSTRAINTS

EQUILIBRIUM yields PARETO EFFICIENCY(in the canonical setup)

Applications

Applications: Finance

Disclosure by public firms

Disclosing false information is a criminal act

Withholding information is allowed in somecases

Withholding information is allowed in somecases, and is difficult (if not impossible) todetect

Impacts asset prices (e.g.: quarterly reports)

Our result:

Our result:Market’s behavior in EQUILIBRIUM yields theOPTIMAL SEPARATION between "good" and "bad"firms (even if mechanisms and commitmentswere possible)

The equilibria considered in this literature turnout to be the truth-leaning equilibria

Applications: Law

Commitments: constitutions, laws, legaldoctrine, precedents, ...

Applications: Law

Affect evidence provided in court

Applications: Law

Our result:The power of these commitments may not gobeyond selecting the truth-leaning equilibria

Applications: Law

Our result:The power of these commitments may not gobeyond selecting the truth-leaning equilibria– which are most natural here

Law: Right to Remain Silent

U.S.: "You have the right to remain silent.Anything you say can and will be usedagainst you in a court of law ..."(Miranda Warning, following the 1966Miranda v. Arizona Supreme Court decision)

U.K.: "You do not have to say anything. But itmay harm your defence if you do not mentionwhen questioned something which you laterrely on in court. Anything you do say ..."(Criminal Justice and Public Order Act 1994)

(TRUTH-LEANING )

EQUILIBRIA entail (Bayesian) inferences

Our equivalence result implies that the sameinferences apply to OPTIMAL MECHANISMS

Therefore: Committing to not making adverseinferences ("RIGHT TO REMAIN SILENT") isNOT OPTIMAL

Instead: TRUTH-LEANING , which is OPTIMAL

Instead: TRUTH-LEANING , which is OPTIMAL(reinforce the advantages of truth-telling)

Medical Over-Treatment

Doctors and hospitals prefer to over-treat asthey are paid more when doing so

Give doctors incentives to provide evidence

Equivalence Theorem: Intuition

In a TRUTH-LEANING EQUILIBRIUMif t pretends to be s (6= t) then:

s reveals his type (i.e., says s)

Else: s has a strictly better message (by (T1)),and then so does t (by transitivity)

v(t) < v(s)

No one pretends to be worth less than they are

v(t) < v(s)

v(t) < πt = πs ≤ v(s)

v(t) < v(s)

NOTE: These conclusions need not hold forequilibria that are not truth-leaning

v(t) < v(s)

⇒ t and s cannot be separated in any OPTIMALMECHANISM

v(t) < v(s)

- To separate: ρ(t) > ρ(s) (else t says s)

v(t) < v(s)

- To separate: ρ(t) > ρ(s) (else t says s)- Not optimal: decreasing ρ(t) or increasing ρ(s)brings rewards closer to values

v(t) < v(s)

CONCLUSION:OPTIMAL MECHANISM cannot separate

more than TRUTH-LEANING EQUILIBRIUM

Equivalence Theorem: Proof

0. Preliminaries

1. Every TL-EQUILIBRIUM outcome equalsthe unique OPTIMAL MECHANISM outcome

0. Preliminaries

1. Every TL-EQUILIBRIUM outcome equalsthe unique OPTIMAL MECHANISM outcome

2. A TL-EQUILIBRIUM exists

Proof: 0. Preliminaries

"In betweenness" : v(t1) ≦ v(t2) implies

v(t1) ≦ v({t1, t2}) ≦ v(t2)

More generally: if q ∈ ∆(T ) is a weightedaverage of q1, q2, ..., qn ∈ ∆(T ) then

v(qi) ≦ v(q) ≦ maxi

v(t1) ≦ v({t1, t2}) ≦ v(t2)

More generally: if q ∈ ∆(T ) is a weightedaverage of q1, q2, ..., qn ∈ ∆(T ) then

v(qi) ≦ v(q) ≦ maxi

Proof : Follows from single-peakedness (SP)and differentiability

Let (σ, ρ) be a TL-EQUILIBRIUM with outcome π.Then:

message t is used in equilibrium:σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)

message t is not used in equilibrium:σ̄(t) = 0 ⇔ σ(t|t) = 0 ⇔ πt > v(t) = ρ(t)

Pf.If σ(t|t) = 0 then π(t) > ρ(t) = v(t) (by (T2)).If σ(t|t) > 0 then: σ(t|s) > 0 for s 6= t impliesπt = πs > v(s); but πt = v(q(t)) and soπt ≤ v(t) by in-betweeness. 2

Corollary.Let s 6= t. If σ(s|t) > 0 then v(s) > v(t).

Pf. v(s) ≥ ρ(s)

(s is used)

Pf. v(s) ≥ ρ(s) = πt

(s is optimal for t)

Pf. v(s) ≥ ρ(s) = πt > v(t)

(t is not used)

Pf. v(s) ≥ ρ(s) = πt > v(t) 2

Proof: 1. equilibrium → mechanism

Let (σ, ρ) be TL-EQUILIBRIUM , with outcome π.

Special Case :There is a single message s that is used(i.e., σ(s|t) = 1 for all t).

⇒ πt = ρ(s) = v(T ) for all t

⇒ v(t) < v(T ) ≤ v(s) for all t 6= s

⇒ π is the unique OPTIMAL MECHANISM

Pf. π is optimal even if we keep only the(IC) constraints πt ≥ πs for all t 6= s,

⇒ v(t) < v(T ) ≤ v(s) for all t 6= s

Pf. π is optimal even if we keep only the(IC) constraints πt ≥ πs for all t 6= s,because v(t) < v(T ) ≤ v(s) for all t 6= s

Special Case :

v(t) v(t′) v(s)

t t′ s

Special Case :

v(t) v(t′) v(s)

t t′ s

L:v(T )

Special Case :

v(t) v(t′) v(s)

t t′ s

L:v(T )

IC: πs ≤ πt

πs ≤ πt′

General Case.

General Case. For each message s that isused (i.e., σ̄(s) > 0)

General Case. For each message s that isused (i.e., σ̄(s) > 0) apply the Special Casewith:

set of types = Ts := {t : σ(s|t) > 0}(the types that use message s)

set of types = Ts := {t : σ(s|t) > 0}(the types that use message s)probability distribution = q(s)(the posterior given message s)

⇒ π restricted to Ts is the unique OPTIMALMECHANISM, for each s

Proof: 2. existence of TL-equilibrium

For every ε > 0, let Γε be the perturbation of theGAME Γ:

UA = x + ε1s=t

(revealing the whole truth increases agent’spayoff by ε)

UA = x + ε1s=t

(revealing the whole truth increases agent’spayoff by ε)

σ(t|t) ≥ ε(probability of revealing the whole truth is atleast ε)

Proposition. Γε has a Nash equilibrium.

Proof. Standard use of Kakutani’s Fixed PointTheorem.

Proposition. A limit point (σ, ρ) of Nashequilibria (σε, ρε) of Γε yields aTL-EQUILIBRIUM (σ′, ρ) of Γ.

Proof.

Proof.σ(t|t) < 1 ⇒

Proof.σ(t|t) < 1 ⇒ σ(t|r) = 0 for all r

⇒ q(t) = 1t

⇒ ρ(t) = v(t)

⇒ q(t) = 1t

⇒ ρ(t) = v(t)

Proof: 2’. mechanism → equilibrium

Let π be an OPTIMAL MECHANISM outcome.

We will construct a TL-EQUILIBRIUM (σ, ρ) withoutcome π.

Recall that in a TL-EQUILIBRIUM we have

σ̄(t) > 0 ⇔ σ(t|t) = 1 ⇔ v(t) ≥ πt = ρ(t)