aa r X i v : . [ ec on . T H ] J u l Equilibrium Refinement in Finite Evidence Games
Shaofei Jiang ∗ Evidence games study situations where a sender persuades a receiver by selectivelydisclosing hard evidence about an unknown state of the world. Evidence games oftenhave multiple equilibria. Hart et al. (2017) propose to focus on truth-leaning equilibria,i.e., perfect Bayesian equilibria where the sender prefers disclosing truthfully whenindifferent, and the receiver takes off-path disclosure at face value. They show thata truth-leaning equilibrium is an equilibrium of a perturbed game where the senderhas an infinitesimal reward for truth-telling. We show that, when the receiver’s actionspace is finite, truth-leaning equilibrium may fail to exist, and it is not equivalent toequilibrium of the perturbed game. To restore existence, we introduce a disturbedgame with a small uncertainty about the receiver’s payoff. A purifiable equilibrium isa truth-leaning equilibrium in an infinitesimally disturbed game. It exists and featuresa simple characterization. A truth-leaning equilibrium that is also purifiable is anequilibrium of the perturbed game.
Keywords:
Hard evidence, Verifiable disclosure, Equilibrium refinement
JEL Codes:
C72, D82, D83
1. Introduction.
In many real-life situations, communiation relies on hard evidence.For example, a jury’s verdict should be based on evidence presented in the court, ratherthan exchanges of empty claims. Evidence games study such situations. There is a sender(e.g., a prosecutor), and a receiver (e.g., a jury). The sender has some hard evidence aboutan unknown state of the world (e.g., whether a defendant is guilty) and wants to persuadethe receiver to take a certain action (e.g., to reach a guilty verdict) by selectively presentingevidence. Full revelation of evidence is often impossible in the presence of conflict of interestbetween the sender and the receiver–the receiver wants to learn the payoff relevant state andact accordingly, whereas the sender merely wants to induce her preferred receiver action.Therefore, the sender has an incentive to persuade the receiver that a certain state is morelikely by partially revealing evidence. ∗ Department of Economics, the University of Texas at Austin (email: [email protected]). I thankV Bhaskar, Maxwell Stinchcombe, Caroline Thomas, and seminar participants at UT Austin for helpfulcomments. distinguishing feature of evidence games (as opposed to signaling games and cheap-talkcommunication) is that the receiver’s private information (i.e., her evidence) is not payoffrelevant. Instead, it affects her ability to persuade the receiver by restricting her set offeasible actions (i.e., the set of evidence she can present to the receiver). That is, the sendercan selectively disclose evidence that she has but cannot fabricate evidence. For example,the prosecutor’s objective is to convict the defendant. This is not affected by what evidenceshe has.
In equilibrium , her chance of convicting the defendant may depend on the evidenceshe has, becuase when she has more evidence, there are more ways to present evidence inthe court, and thereby she may better persuade the jury.Evidence games often have multiple (Nash) equilibria. For example, there is a trivialequilibrim where, regardless of her evidence, the prosecutor presents no evidence, and thejury always acquits the defendant (this must be optimal on the equilibrium path for thejury if the presumption of innocence is practiced). Obviously, it is not a sensible predictionof what happens in courtrooms. However, this equilibrium is perfect (Selten, 1975) andsequential (Kreps and Wilson, 1982). Consider a perturbation to the prosecutor’s strategywhich puts higher probability on disclosing acquitting evidence than on disclosing convictingevidence, and a perturbation to the jury’s strategy such that the probability of convictingthe defendant after seeing any evidence is smaller than that after seeing no evidence. As bothperturbations converge to zero, this gives a sequence of ε -constrained equilibria in completelymixed strategies that converges to the trivial equilibrium. Hence, the trivial equilibrium isperfect. Moreover, given the perturbed sender’s strategy, it is consistent for the jury tobelieve that the actual evidence possessed by the prosecutor is more acquitting after seeingany disclosed evidence. Therefore, the trivial equilibrium is also a sequential equilibrium.Hart et al. (2017) (henceforth HKP) propose the following refinement to perfect Bayesianequilibrium in evidence games. A truth-leaning equilibrium is a perfect Bayesian equilibriumsuch that (Truth-leaning) Given the receiver’s strategy, the sender discloses her evidence truthfullyif doing so is optimal; (Off-path beliefs)
The receiver takes any off-path disclosure at face value (i.e., he believesthat the sender discloses truthfully).These conditions (especially the first one) are intuitive for evidence games. As is argued inHKP, these conditions follow the simple intuition that there is a “slight inherent advantage”for the sender to tell the whole truth, and “there must be good reasons for not telling it.” HKP defines truth-leaning equilibrium as a refinement to Nash equilibrium. However, we note thatany truth-leaning equilibrium is a perfect Bayesian equilibrium and sequential equilibrium. We regard allsolution concepts in the current paper as refinements to perfect Bayesian equilibrium. To address this problem, we propose the following solution conceptby introducing a small uncertainty (i.e., disturbance) to the receiver’s payoff `a la Harsanyi(1973). Suppose that the receiver receives a random private payoff shock associated witheach of his actions. In the disturbed game, the sender has a strict incentive to persuadethe receiver, and a truth-leaning equilibrium exists in the disturbed game. We define a purifiable equilibrium as the limit of a sequence of truth-leaning equilibria in the disturbedgames as the disturbances converge to zero. That is, a purifiable equilibrium is a truth-leaning equilibrium of an infinitesimally disturbed game. A purifiable equilibrium alwaysexists.Another problem of truth-leaning equilibrium in finite evidence games is that it may notfollow the intuition that the sender is slightly more advantageous if she discloses truthfully. In both HKP and Jiang (2019), if a piece of evidence e ′ is inherently better than the sender’s evidence e (i.e., the receiver’s optimal action knowing that the sender’s evidence is e ′ is strictly higher than his optimalaction knowing that the sender’s evidence is e ) and the sender can feasibly disclose e ′ , then the sender’spayoff from any randomization between disclosing e ′ and e is strictly higher than her payoff from disclosingonly e , given any Bayesian consistent system of beliefs of the receiver and any sequentially rational receiverstrategy. This is not the case when the receiver’s action is finite. For example, the prosecutor does not know how lenient the jury is (i.e., how convinced the jury hasto be in order to reach a conviction). However, she knows that after seeing more convicting evidence, theprobability that the jury will choose to convict the defendant is higher. Therefore, the prosecutor strictlyprefers presenting all convicting evidence.
3o formalize this intuition, we revisit the perturbed game in HKP, where the sender receivesa small reward if she discloses truthfully, and the sender must disclose truthfully with at leastsome small probability. We define a weakly truth-leaning equilibrium as the limit of a sequenceof perfect Bayesian equilibria of the perturbed games as the perturbations converge to zero.HKP shows that truth-leaning equilibrium is equivalent to weakly truth-leaning equilibrium.When the receiver’s action space is finite, however, the equivalence is no longer true. Itturns out that purifiability is the missing condition connecting truth-leaning and weaklytruth-leaning equilibrium–a purifiable truth-leaning equilibrium is weakly truth-leaning, anda purifiable weakly truth-leaning equilibrium is truth-leaning in “almost all” evidence games.
Outline of the paper.
Section 2 presents an example where truth-leaning equilibriumdoes not exist and illustrates the constructions of purifiable equilibrium and weakly truth-leaning equilibrium. Section 3 presents the model. Section 4 formally defines purifiableequilibrium and shows the relationship between purifiable equilibrium, truth-leaning equi-librium, and weakly truth-leaning equilibrium. The last section concludes. Proofs are in theAppendix.
2. A Simple Example and Discussion.
Every new aircraft design has to be certifiedby the Federal Aviation Administration (FAA) before any aircraft built according to thisdesign can enter service (in the U.S.). Like other innovations, altering the design of anaircraft often entails high level of risks. The FAA often has to rely on information andtest results provided by airplane manufacturers, yet airplane manufactueres’ disclosure is farfrom complete. Consider an airplane manufacturer (the sender) seeking to get a new aircraft designcertified by the FAA (the receiver). The design can be good or bad with equal likelihood.If the design is bad, the aircraft manufacturer has some bad evidence (e.g., machanicalfailures during test flights) with probability . Otherwise, the aircraft manufacturer hasno evidence. The FAA does not know the quality of the design and chooses to Approve or Reject the aircraft design based on evidence disclosed by the sender. The disclosure ofbad evidence is voluntary and verifiable (i.e., the sender can disclose bad evidence or noevidence if it has bad evidence, and it can only disclose no evidence if it has no evidence). For example, a design deficiency of the batery system on board Boeing’s 787 Dreamliners had caused sev-eral aircraft fires in 2013, which led to the grounding of all 50 aircrafts at the time and a redesign of the batterysystem (see ).More recently, MCAS, a new fligh control software embedded into Boeing’s 737 MAX aircrafts, caused twodeadly crashes wihtin two years of the airliner’s first commercial operation. The entire fleet has since thenbeen grounded (see ). In the 737 MAX incident, Boeing failed to inform the FAA of a design change of MCAS (see ). , its optimal action is Approve ; if its posterior belief is lessthan , its optimal action is Reject ; if its posterior belief is exactly , either action as wellas any randomization between the two actions is optimal.A strategy of the sender describes how it discloses bad evidence. Let p be the probabilitythat the sender discloses no evidence if it has bad evidence. Since bad evidence fully revealsthat the design is bad, the receiver always chooses Reject (thus the sender gets 0) afterseeing bad evidence. Let q be the probability that the receiver chooses Approve after seeingno evidence. Let µ be the receiver’s posterior belief that the design is good after seeing noevidence. Since no evidence is always disclosed with postive probability, Bayes’ rule requiresthat µ = p . It is easy to verify that the game has a continuum ofperfect Bayesian equilibria, where p ≥ , q = 0, and µ = p ≤ . That is, the senderwith bad evidence discloses no evidence with at least probability , and the receiver alwaysrejects the design.However, there is no truth-leaning equilibrium. Given the receiver’s strategy, the senderwith bad evidence is indifferent between disclosing no evidence and disclosing truthfully(since both actions yield payoff 0). Truth-leaning therefore requires the sender to disclose badevidence truthfully (i.e., p = 0), which is not satisfied by any perfect Bayesian equilibrium. Suppose that the receiver receives a payoff shock ζ for choos-ing Approve , where ζ is normally distributed according to N (0 , ε ) and is private informationof the receiver (hence the receiver’s type). That is, the receiver’s payoff from approving agood design is 1+ ζ , and that from approving a bad design is ζ −
2. Hence, the receiver almostalways has a unique optimal action after seeing no evidence, which is
Approve if µ > − ζ (equivalently, ζ > − µ ) and Reject if µ < − ζ (equivalently, ζ < − µ ). This implies thatthe design is approved with probability Φ (cid:0) µ − ε (cid:1) > q ( ζ ) be the probability that the type ζ receiver approves the design5fter observing no evidence. The disturbed game has a continuum of perfect Bayesianequilibria, where p = 1, µ = , and q ( ζ ) = 0 if ζ < , q ( ζ ) ∈ [0 ,
1] if ζ = , q ( ζ ) = 1 if ζ > . Since the sender strictly prefers disclosing no evidence, all perfect Bayesian equilibriaof any disturbed game are truth-leaning. Moreover, in all equilibria, the receiver chooses Approve with probability Φ( − ε ) after observing no evidence. That is, the disturbed gamehas a unique truth-leaning equilibrium outcome, where the sender discloses no evidence, andafter seeing no evidence, the receiver chooses Approve with probability Φ( − ε ) and believesthat the design is good with probability.As the disturbance diminishes (i.e., as ε ↓ p = 1, q = 0, µ = . Let us consider the following perturbed game.Let ε and ε be small positive reals that are common knowledge to the sender and thereceiver. The sender receives a reward ε if it discloses (bad evidence) truthfully, and thesender must disclose truthfully with at least probability ε .If its posterior belief µ > , then the receiver has a unique optimal action Approve afterobserving no evidence. Then, for ε <
1, the sender strictly prefers disclosing no evidence,so the Bayesian consistent belief is µ = < . If µ < , the receiver’s unique optimalaction is Reject after observing no evidence. With the reward for truth-telling, the senderstrictly prefers disclosing truthfully, so the Bayesian consistent belief is µ = > . Hence,the receiver’s posterior belief µ = in any perfect Bayesian equilibrium of the perturbedgame. Indeed, for ε < ε ≤ , the perturbed game has a unique perfect Bayesianequilibrium, where p = , q = ε , µ = .As ε , ε ↓
0, the perfect Bayesian equilibrium of the perturbed game converges to aperfect Bayesian equilibrium of the original game, where p = , q = 0, µ = . Figure 1 summarizes the equilibria of the game. There is a continuumof perfect Bayesian equilibria which differ only on the sender’s strategy. Among them,the weakly truth-leaning equilibrium maximizes the probability that the sender disclosestruthfully. The purifiable equilibrium maximizes the receiver’s posterior belief on the gooddesign.The fact that this simple game does not possess a truth-leaning equilibrium suggests thattruth-leaning equilibrium may not be a proper solution concept for finite evidence games.A more fundamental problem of truth-leaning equilibrium is the discrepency between therefinement and the intuition behind it. The requirement that the sender weakly prefersdisclosing truthfully seemingly arises from the sender having infinitesimal reward for truth-6 1/4 1Weaklytruth-leaning PurifiablePBETruth-leaning(non-equilibrium)
Figure 1:
The probability that the sender discloses no evidence when having bad evidence telling, but in the example, the weakly truth-leaning equilibrium constructed by adding aninfinitesimal reward for truth-telling is not the same as imposing truth-leaning refinementon perfect Bayesian equilibria. Weakly truth-leaning equilibrium also has a few shortcomings. First, it may not be ro-bust to incomplete receiver payoff information. In our example, the sender strictly prefersdisclosing no evidence once we introduce a small uncertainty to the receiver’s payoff. There-fore, the weakly truth-leaning equilibrium where the sender having bad evidence disclosesno evidence with probability is not robust to incomplete receiver payoff information. Indefense of weakly truth-leaning equilibrium, the perfect Bayesian equilibrium in every per-turbed game where the sender receives a small reward for truth-telling (viz., p = , q = ε , µ = ) is robust to incomplete receiver payoff information in our example, but this is not ageneric result. In general, a weakly truth-leaning equilibrium may fail to be the limit pointof a sequence of equilibria of perturbed games that are robust to incomplete receiver payoffinformation.Second, different sequences of perturbations may select different weakly truth-leaningequilibria, and not all sequences of perturbed games have a sequence of perfect Bayesianequilibria that converges as the perturbation goes to zero.Consider a slight variant to our example, where the sender’s bad evidence is either type1 or type 2 (think about software failures and hardware failures). If the design is bad, the Recall that HKP shows the equivalence of truth-leaning equilibrium and weakly truth-leaning equilibriumin evidence games where the recevier continuously chooses an action, and its payoff function is single-peakedgiven any belief. In the current example, suppose that the receiver chooses an action a ∈ R , and the receiver’spayoff is quadratic, i.e., − ( a − x ) , where x is a random variable that equals 0 if the design is bad and 1 if thedesign is good. The unique truth-leaning equilibrium is as follows. The sender always discloses no evidence,the receiver’s belief and action are after seeing no evidence and 0 after seeing bad evidence. This is alsothe unique weakly truth-leaning equilibrium. To see this, consider a disturbed game where: (i) the sender receives ε if it discloses truthfully; (ii)the sender must disclose truthfully with at least probability ε ; (iii) the receiver receives a payoff shock ζ distributed according to N (0 , ε ) for choosing Approve , which is its private information. For ε < , ε < , and ε < − ε − ε · − Φ − ( ε ) , the disturbed game has a continuum of perfect Bayesian equilibria, where p = ε Φ − ( ε ) − µ = ε Φ − ( ε )3 , q ( ζ ) = 0 if ζ < − ε Φ − ( ε ), q ( ζ ) ∈ [0 ,
1] if ζ = − ε Φ − ( ε ), and q ( ζ ) = 1if ζ > − ε Φ − ( ε ). In any perfect Bayesian equilibrium, the design is approved with probability ε after thereceiver observes no evidence. As ε ↓
0, this equilibrium outcome converges to p = , q = ε , µ = . probability;if the design is good, the sender has no evidence. The sender with a certain type of badevidence can disclose truthfully or no evidence but cannot disclose the other type of badevidence. Let p i be the probability that the sender with type i bad evidence discloses noevidence, q the probability that the receiver chooses Approve after seeing no evidence, and µ the receiver’s belief that the design is good after seeing no evidence. The game has acontinuum of perfect Bayesian equilibria, where p + p ≥ , q = 0, and µ = p + p .Now, let us consider the following perturbed game. Given small positive reals ε , ε < ε | , ε | ≤ , the sender receives a reward ε i if it truthfully dicloses type i bad evidence,and the sender with type i bad evidence must disclose truthfully with at least probability ε i | i . If ε i < ε j , the unique perfect Bayesian equilibrium is p i = , p j = 0, q = ε i , µ = . If ε = ε , there is a continuum of perfect Bayesian equilibria, where p + p = , q = ε = ε , µ = . Hence, as ( ε , ε | , ε , ε | ) →
0, whether there exits a convergent sequence of perfectBayesian equilibria depends on the rates of convergence of ε and ε . If ε = ε almost always,then any perfect Bayesian equilibrium of the unperturbed game such that p + p = is thelimit point of a sequence of perfect Bayesian equilibria of the perturbed game. If ε i ≤ ε j almost always, and ε i < ε j infinitely often, then the unique weakly truth-leaning equilibriumis such that p i = , p j = 0, q = 0, and µ = . If neither case happens, there is noconvergent sequence of perfect Bayesian equilibria of the perturbed game. In conclusion, theunperturbed game has a continuum of weakly truth-leaning equilibria, where p + p = , q = 0, µ = , and different weakly truth-leaning equilibria may be selected by different setsof infinitesimal perturbations.Purifiable equilibrium is spared from similar problems. For “almost all” evidence games,any purifiable equilibrium is infinitesimally close to a truth-leaning equilibrium of any in-finitesimally perturbed game. That is, purifiability does not depend on the selection ofdisturbances. The normality of the receiver’s payoff shock in our example is dispensable.Moreover, the set of purifiable equilibria has a simple structure, and we give a characteriza-tion of all purifiable equilibria in any evidence game.
3. The Evidence Game.
There are two stages. Two players, a sender (she) and areceiver (he), move sequentially. At the outset of the game, a state of the world ω ∈ { G, B } is realized with probability π ∈ (0 ,
1) on ω = G . Neither player observes the realized state ω , and the prior π is common knowledge. In the first stage, the sender observes a piece ofhard evidence e ∈ E and discloses m ∈ E to the receiver, where E is a finite set of evidence.In the second stage, the receiver observes the disclosed evidence m and chooses an action Since the sender’s payoff is independent of the realized state, so it does not affect our analysis if therealized state is known to the sender. ∈ A , where A = { a < a < · · · < a K } is a finite subset of the real line. Let F G and F B be two distributions over the set of evidence E . The sender’s evidence e is a random draw from either F G or F B , depending on the realizedstate. If ω = G , e is drawn from distribution F G ; if ω = B , it is drawn from distribution F B .Disclosure is verifiable. That is, the set of evidence that the sender can feasibly disclosedepends on the evidence she has (in contrast, in a signaling game, the sender, regardless ofher type, chooses from the same set of signals). Throughout the paper, We maintain thefollowing assumptions that are standard in the literature: (Reflexivity) The sender can always truthfully diclose her evidence e ; (Transitivity) If the sender can disclose e ′ when she has evidence e , and she can disclose e ′′ when she has evidence e ′ , then she can disclose e ′′ if she has evidence e .Under these assumptions, we can represent the “disclosure rule” as a preorder - on E . Disclosing m is feasible given evidence e if and only if m - e , and the feasible set of disclosuregiven a piece of evidence e is its lower contour set LC ( e ) = { m ∈ E : m - e } . The receiver’s payoff u R ( a, ω ) depends on both his action and the realizedstate of the world (but not the true evidence or the disclosed evidence), and the receivermaximizes his expected payoff. We assume that the receiver’s payoff function satisfies thefollowing assumption: (Increasing differences) u R ( a, G ) − u R ( a, B ) is strictly increasing in a .Under this assumption, the receiver wants to match the sate of the world. That is, his optimalaction is weakly increasing in his posterior belief that the state is good. More precisely, given µ ∈ [0 , φ ( µ ) = argmax a ∈ A µu R ( a, G ) + (1 − µ ) u R ( a, B )is upper hemicontinuous and weakly increasing in µ . The sender’s payoff equals the action of the receiver, i.e., u S ( a, ω ) = a . Given theassumption on the receiver’s payoff, the sender has a weak incentive to persuade the receiverthat the state is good. Notice that the evidence e , the disclosed evidence m , and the realizedstate ω are payoff irrelevant to the sender.An evidence game is summarized by a tuple G = h π , ( E, - ) , F G , F B , A, u R i . A preorder - is a binary relation satisfying reflexivity ( e - e for all e ) and transitivity ( e ′′ - e ′ - e ⇒ e ′′ - e ). Throughout the paper, we say a correspondence φ : [0 , ⇒ A is weakly increasing if a i ≤ a j for all µ i < µ j , a i ∈ φ ( µ i ), and a j ∈ φ ( µ j ). .3. Strategies and perfect Bayesian equilibrium. A (behavioral) strategy of the senderis σ : E → ∆( E ) such that supp ( σ ( ·| e )) ⊂ LC ( e ), a (behavioral) strategy of the receiver is ρ : E → ∆( A ), and a system of beliefs of the receiver is µ : E → [0 , µ ( m ) denotesthe receiver’s posterior belief that the state is good after observing m .A perfect Bayesian equilibrium of G is a collection of the sender’s strategy, the receiver’sstrategy, and the receiver’s system of belief ( σ, ρ, µ ) such that: (Sender optimaltiy) Given ρ , supp ( σ ( ·| e )) ⊂ argmax m - e X a ∈ A a · ρ ( a | m )for all e ∈ E ; (Receiver optimality) Given µ , supp ( ρ ( ·| m )) ⊂ φ ( µ ( m ))for all m ∈ E ; (Bayesian consistency) For all on-path disclosure m ∈ S e ∈ E supp ( σ ( ·| e )), µ ( m ) = P e ∈ UC ( m ) σ ( m | e ) F G ( e ) π P e ∈ UC ( m ) σ ( m | e )[ F G ( e ) π + F B ( e )(1 − π )] .
4. Refinements of Perfect Bayesian Equilibrium.
Sections 4.1 through 4.3 giveformal definitions to truth-leaning equilibrium, purifiable equilibrium, and weakly truth-leaning equilibrium. Section 4.2 also characterizes purifiable equilibrium. Section 4.4 showsthe relationship between the three refinements. A truth-leaning equilibrium of G is a perfect Bayesianequilibrium ( σ, ρ, µ ) such that: (Truth-leaning) Given ρ , e ∈ argmax m - e X a ∈ A a · ρ ( a | m ) ⇒ σ ( e | e ) = 1; (Off-path beliefs) For all off-path disclosure m , µ ( m ) = ν ( m ), where ν ( m ) = F G ( m ) π F G ( m ) π + F B ( m )(1 − π ) . As is shown in Section 2, a truth-leaning equilibrium of G may not exist.10 .2. Disturbed games and purifiable equilibrium. A disturbed game is where the receiverhas a private payoff shock (i.e., type) ζ : A → R . The receiver has type dependent payoff v R ( a, ω | ζ ) = u R ( a, ω ) + ζ ( a ). We identify the set of the receiver’s types with R K , where K = | A | is the number of available receiver actions. Let η be a distribution over R K thathas full support and is absolutely continuous with respect to the Lebesgue measure. Denoteby G R ( η ) the disturbed game where the receiver’s type is distributed according to η .In the disturbed game, a strategy of the sender is σ : E → ∆( E ) such that supp ( σ ( ·| e )) ⊂ LC ( e ), a strategy of the receiver in G R ( η ) is r : E × R K → ∆( A ), and a system of beliefsof the receiver is µ : E → [0 , µ ( m ) is the receiver’s posterior belief that the stateis good after observing m . Given any strategy of the receiver r , let ρ : E → ∆( A ) be theinduced distributions over the receiver’s actions. That is, ρ ( a | m ) = Z r ( a | m, ζ ) η ( dζ )is the probability that the receiver takes action a after m is disclosed. We shall also use theshorthand notation and write this as ρ = h r, η i .A truth-leaning equilibrium of G R ( η ) is a tuple ( σ, r, µ ) such that: (Receiver optimality in disturbed games) Given µ , supp ( r ( ·| m, ζ )) ⊂ τ ( µ ( m ) , ζ )for all m ∈ E and ζ ∈ R K , where τ (¯ µ, ζ ) ⊂ A is the solution to the type ζ receiver’sproblem given posterior belief ¯ µ ∈ [0 ,
1] on the good state, i.e., τ (¯ µ, ζ ) = argmax a ∈ A ¯ µu R ( a, G ) + (1 − ¯ µ ) u R ( a, B ) + ζ ( a ); (Sender optimality) , (Bayesian consistency) , (Truth-leaning) , and (Off-path be-liefs) , as are defined above for the original game G .If ( σ, r, µ ) is a truth-leaning equilibrium, we say ( σ, ρ, µ ) is a truth-leaning equilibrium out-come of G R ( η ).In any disturbed game, the sender has a strict incentive to persuade the receiver–from thesender’s perspective, the expected value of the receiver’s optimal action is strictly increasingin his posterior belief. Therefore, a truth-leaning equilibrium exists in any disturbed game. The assumption that the receiver’s belief is independent of his type is without loss for finding truth-leaning equilibrium, as on-path beliefs are determined by Bayes’ rule, and off-path beliefs are determined bythe refinement. all truth-leaning equilibria of all disturbed games. In fact, the set of truth-leaning equilibria is thesame for all disturbed games.
Lemma . A truth-leaning equilibrium exists in all disturbed games. Moreover, thereexist a closed set Σ ⋆ ⊂ ∆( E ) E and a system of beliefs of the receiver µ ⋆ such that for alldisturbed games G R ( η ) , ( σ, r, µ ) is a truth-leaning equilibrium of G R ( η ) if and only if σ ∈ Σ ⋆ , µ = µ ⋆ , and supp ( r ( ·| m, ζ )) ⊂ τ ( µ ( m ) , ζ ) for all m ∈ E and ζ ∈ R K . A purifiable equilibrium is the limit point of a sequence of truth-leaning equilibria ofdisturbed games as the payoff uncertainty goes to zero. Formally, a purifiable equilibrium of G is a tuple ( σ, ρ, µ ) such that there exists a sequence of disturbances { η n } ∞ n =1 and foreach η n , a truth-leaning equilibrium outcome ( σ n , ρ n , µ n ) of G R ( η n ) such that η n w −→ δ and( σ n , ρ n , µ n ) → ( σ, ρ, µ ).By Lemma 1, it is easy to see that a purifiable equilibrium exists, and in any purifiableequilibrium, σ ∈ Σ ⋆ and µ = µ ⋆ . Since the receiver’s problem in any disturbed game dependsonly on his type and his poterior belief, the receiver’s action after seeing a disclosed evidencein any purifiable equilibrium should depend only on his posterior belief. That is, if two piecesof evidence m and m ′ are such that µ ⋆ ( m ) = µ ⋆ ( m ′ ), then ρ ( ·| m ) = ρ ( ·| m ′ ) in any purifiableequilibrium. It turns out that conversely, any perfect Bayesian equilibrium satisfying theseconditions is a purifiable equilibrium.
Proposition . A purifiable equilibrium exists and is a perfect Bayesian equilibrium.Moreover, ( σ, ρ, µ ) is a purifiable equilibrium if and only if σ ∈ Σ ⋆ , µ = µ ⋆ , supp ( ρ ( ·| m )) ⊂ φ ( µ ( m )) for all m ∈ E , and µ ( m ) = µ ( m ′ ) ⇒ ρ ( ·| m ) = ρ ( m ′ , · ) . As is noted in Appendix A.1, the equilibrium system of beliefs µ ⋆ and the set of thesender’s equilibrium strategies Σ ⋆ are determined by the evidence space ( E, - ) and thedistributions F G and F B . They are independent of the receiver’s payoff function u R .It is also worth noting that, since the sender’s action (i.e., the disclosed evidence) is notpayoff relevant, evidence games are not generic in the sense of Harsanyi (1973). However,all purifiable equilibria in “almost all” evidence games can be approached using an arbitrary That is, η n converges weakly to the point mass at 0, i.e., R f dη n → f (0) for all bounded continuousfunctions f : R K → R . By contrast, a perfect Bayesian equilibrium only requires that supp ( ρ ( ·| m )) = supp ( ρ ( ·| m ′ )). All Nash equilibria of an evidence game are irregular under the definition of van Damme (1996), sincegiven the receiver’s strategy, the sender is indifferent between all of her strategies. φ ( µ ⋆ ( m )) is a singleton for all m ∈ E , then the receiver’s purifiable equilibrium strategy is unique and is a pure strategy(viz. ρ ( a | m ) = a ∈ φ ( µ ⋆ ( m )) ). We show in Appendix A.2 that, in this case, any purifiable equi-librium can be approached using arbitrary disturbances. That is, for all purifiable equilibria( σ, ρ, µ ) and all sequences of disturbances η n w −→ δ , there exists a sequence of truth-leaningequilibrium outcomes of the disturbed games ( σ n , ρ n , µ n ) that converges to ( σ, ρ, µ ). In Ap-pendix A.5, we show that φ ( µ ⋆ ( m )) is a singleton for all m ∈ E in all but a measure zeroset of evidence games.If φ ( µ ⋆ ( m )) is not a singleton for some m ∈ E , there exists a continuum of the receiver’spurifiable equilibrium strategies. A given purifiable equilibrium may be the limit point oftruth-leaning equilibrium outcomes only for some sequences of disturbed games, and notall sequences of disturbed games have a convergent sequence of truth-leaning equilibriumoutcomes. For example, consider a slight variant of the example in Section 2 where thereceiver’s payoff from approving a bad design is -1 (instead of -2). As a result, the receiver’sbelief threshold is . There exists a continuum of purifiable equilibria, where p = 1, q ∈ [0 , µ = . Specifically, there exists a purifiable equilibrium in which the receiver chooses Approve and
Reject with equal probability after seeing no evidence (i.e., q = ). But in orderto approach this equilibrium using truth-leaning equilibria of disturbed games, the sequenceof disturbances { η n } ∞ n =1 must be such that η n ( { ζ ( Approve ) > ζ ( Reject ) } ) → . That is,along the sequence of disturbances, the probability that the receiver has a strict incentiveto choose Approve at belief must converge to , equating the probability that the receiverchooses Approve in the intended purifiable equilibrium.
Let ε = { ε e , ε e | e } e ∈ E be acollection of positive real numbers. The perturbed game G S ( ε ), as is defined in HKP, isan evidence game where the sender who has evidence e receives an extra payoff ε e if shediscloses truthfully, and she must disclose truthfully with at least probability ε e | e . That is,the sender’s payoff is v S ( a, e, m ) = a + ε e e = m , and a strategy of the sender is σ : E → ∆( E )such that supp ( ρ ( ·| e )) ⊂ LC ( e ), and σ ( e | e ) ≥ ε e | e for all e .A perfect Bayesian equilibrium of G S ( ε ) is a collection of the sender’s strategy, the re-ceiver’s strategy, and the receiver’s system of beliefs ( σ, ρ, µ ) such that: (Sender optimality) Given ρ , σ ( m | e ) > ⇒ m ∈ argmax m - e X a ∈ A v S ( a, e, m ) · ρ ( a | m )for all e and m = e ; 13 Receiver optimality) and (Bayesian consistency) , as are defined for G .A weakly truth-leaning equilibrium of G is a tuple ( σ, ρ, µ ) such that there exists a sequenceof perturbations { ε n } ∞ n =1 and for each ε n , a PBE ( σ n , ρ n , µ n ) of G S ( ε n ) such that ε n →
0, and( σ n , ρ n , µ n ) → ( σ, ρ, µ ). A weakly truth-leaning equilibrium exists. As is shown in Section2, diferent sequences of perturbations may select different weakly truth-leaning equilibria. Proposition . A weakly truth-leaning equilibrium exists andis a perfect Bayesian equilibrium.4.4. Relationship between truth-leaning, weakly truth-leaning, and purifiable equilibrium.
The example in Section 2 indicates that purifiable equilibrium and weakly truth-leaningequilibrium do not imply each other, and neither implies truth-leaning equilibrium.In a slight variant of the example in Section 2, we shall see that an equilibrium that isboth truth-leaning and weakly truth-leaning can fail to be purifiable. Suppose that we alterthe distribution of the sender’s evidence when the design is good such that the sender hasbad evidence and no evidence with equal probability. The distribution when the design isbad remains unchanged. The game has a unique truth-leaning equilibrium, where p = 0, q = 0, µ = . That is, the sender discloses truthfully, the receiver always rejects the design,and the receiver’s belief on the good design is after seeing no evidence. Notice that this isalso the unique weakly truth-leaning equilibrium of the game. However, it is not a purifiableequilibrium. In the unique purifiable equilibrium of the game, the sender always disclosesno evidence, the receiver always rejects the project, and his belief on the good design is after seeing no evidence (i.e., p = 1, q = 0, µ = ).HKP shows that truth-leaning equilibrium and weakly truth-leaning equilibrium areequivalent in a setting where the receiver continuously chooses an action on the real line.This is not the case in finite evidence games. It turns out that purifiability is the missingcondition. On the one hand, if a weakly truth-leaning equilibrium is also purifiable, thenit is a truth-leaning equilibrium. On the other hand, for “almost all” evidence games, atruth-leaning equilibrium that is also purifiable is a weakly truth-leaning equilibrium. Proposition . If a truth-leaning equilibrium is purifiable, then it is also a weaklytruth-leaning equilibrium.
Proposition . Fix π , ( E, - ) , F G , F B , and A . Let G be the set of all evidence games Although bad evidence is not fully revealing of the state, the receiver knows that the sender has badevidence after seeing bad evidence. Therefore, the receiver’s posterior belief on the good design is , and thereceiver chooses Reject after seeing bad evidence in any equilibrium. Hence, we can describe an equilibriumof the game using p , q , µ , as are defined in Section 2. ith prior π , evidence space ( E, - ) , distributions of evidence F G and F B , and receiver actionspace A . Identify G with a subset of R K by the bijection h π , ( E, - ) , F G , F B , A, u R i 7→ { u R ( a, G ) , u R ( a, B ) } a ∈ A . Let N ⊂ G be the set of evidence games that have a purifiable truth-leaning equilibrium thatis not a weakly truth-leaning equilibrium. N has Lebesgue measure zero. For nongeneric games, a purifiable truth-leaning equilibrium need not be weakly truth-leaning. Cnosider again the example presented after Proposition 2 where the receiver’s beliefthreshold is . There exists a continuum of truth-leaning equilibria, where p = 1, q >
0, and µ = . As we have seen, all truth-leaning equilibria are purifiable in this game. However,there is a unique weakly truth-leaning equilibrium in which p = 0, q = 0, and µ = . Allother truth-leaning equilibria are not weakly truth-leaning. This example is not generic,since the receiver is indifferent between Approve and
Reject after seeing no evidence intruth-leaning equilibria.
5. Conclusion.
HKP propose truth-leaning equilibrium as a solution concept in ev-idence games. The intuition is that the sender may find it slightly more advantageous todisclose evidence truthfully when indifferent. This paper points out two problems of apply-ing this solution concept to finite evidence games. First, it may fail to exist. Second, it maynot agree with the intuition that the sender receives an infinitesimal reward for truth-telling.That is, truth-leaning equilibrium is not equivalent to weakly truth-leaning equilibrium infinite evidence games.We propose a simple solution to restore existence by adding a small payoff uncertainty tothe receiver. In the disturbed game, the sender is as if she faces a single receiver whom she hasstrict incentive to persuade, and therefore, a truth-leaning equilibrium exists. A purifiableequilibrium is the limit point of a sequence of truth-leaning equilibria of disturbed games.That is, a purifiable equilibrium is a truth-leaning quilibrium in an infinitesimally disturbedgame. We show that a purifiable equilibrium always exists and has a simple characterization.Purifiability also solves the second problem. If a weakly truth-leaning equilibrium isalso purifiable, then it is a truth-leaning equilibrium. Conversely, in almost all finite ev-idence games, a truth-leaning equilibrium that is also purifiable is a weakly truth-leaningequilibrium.
Appendix A. Proofs
A.1. Proof of Lemma 1. roof. Given any posterior belief µ ∈ [0 , a i and a j are both optimal fortype ζ receiver only if ζ ( a j ) − ζ ( a i ) = µ [ u R ( a i , G ) − u R ( a j , G )]+(1 − µ )[ u R ( a i , B ) − u R ( a j , B )].By assumption, this is true only for an η -null set of ζ . Hence, τ ( µ, · ) is η -a.e. a singletonset. This allows us to define ϕ ( µ ) = Z sup τ ( µ, ζ ) η ( dζ ) = Z inf τ ( µ, ζ ) η ( dζ ) . In any equilibrium ( σ, r, µ ) of the disturbed game, ϕ ( µ ( m )) is the sender’s expected payoffif she discloses m .Moreover, τ ( · , ζ ) is weakly increasing for all ζ ∈ R K . Let µ i < µ j , a i ∈ τ ( µ i , ζ ), and a j ∈ τ ( µ j , ζ ). Then µ i u R ( a i , G ) + (1 − µ i ) u R ( a i , B ) + ζ ( a i ) ≥ µ i u R ( a j , G ) + (1 − µ i ) u R ( a j , B ) + ζ ( a j ) ,µ j u R ( a j , G ) + (1 − µ j ) u R ( a j , B ) + ζ ( a j ) ≥ µ j u R ( a i , G ) + (1 − µ j ) u R ( a i , B ) + ζ ( a i ) . Hence,(A.1) ( µ j − µ i )[ u R ( a j , G ) − u R ( a j , B )] ≥ ( µ j − µ i )[ u R ( a i , G ) − u R ( a i , B )] . Since u R ( a, G ) − u R ( a, B ) is strictly increasing in a , (A.1) implies that a j ≥ a i .Therefore, ϕ : [0 , → R is strictly increasing. Suppose that, contrary to the claim, thereexist µ i < µ j such that ϕ ( µ i ) = ϕ ( µ j ). Then, for an η -a.e. set of ζ , τ ( µ i , ζ ) = τ ( µ j , ζ ).This is true only if u R ( a, G ) − u R ( a, B ) is constant across all a ∈ A , which contradicts theassumption of increasing differences.Now consider an auxiliary evidence game G ( ϕ ) without receiver type, where the receiverchooses an action in R , and given any posterior belief µ ∈ [0 , ϕ ( µ ). This is the standard setup in Jiang (2019). We are to establish a dualitybetween truth-leaning equilibria of G R ( η ) and truth-leaning equilibria of G ( ϕ ).Let (ˆ σ, ˆ a , ˆ µ ) be a truth-leaning equilibrium of G ( ϕ ). Let r : E × R K → ∆( A ) be suchthat supp ( r ( ·| m, ζ )) ⊂ τ (ˆ µ ( m ) , ζ ) for all m ∈ E and ζ ∈ R K . We are to show that (ˆ σ, r, ˆ µ ) is atruth-leaning equilibrium of G R ( η ). By construction, it satisfies receiver optimality, Bayesianconsistency, and the condition on off-path beliefs. We only need to verify sender optimalityand truth-leaning. Since τ (ˆ µ ( m ) , · ) is η -a.e. a singleton for all m , r ( a | m, ζ ) = a ∈ τ (ˆ µ ( m ) ,ζ ) for all m, a , and almost all ζ . Hence, with slight abuse of notation, P a ∈ A a · r ( a | m, ζ ) = τ (ˆ µ ( m ) , ζ ) for all m and almost all ζ . Integrating over ζ on both sides, P a ∈ A a · ρ ( a | m ) = ˆ a : E → R is a pure strategy of the receiver. Since given any posterior belief µ , the receiver has a uniqueoptimal action ϕ ( µ ). Thus, he uses a pure strategy such that ˆ a = ϕ ◦ ˆ µ in any equilibrium of G ( ϕ ). (ˆ µ ( m )) = ˆ a ( m ). That is, the sender’s problem given ˆ a in G ( ϕ ) is the same as the sender’sproblem given r in G R ( η ). Since (ˆ σ, ˆ ρ, ˆ µ ) is sender optimal and truth-leaning, (ˆ σ, r, ˆ µ ) istherefore also sender optimal and truth-leaning.Conversely, let (ˆ σ, ˆ r, ˆ µ ) be a truth-leaning equilibrium of G R ( η ), and define a = ϕ ◦ ˆ µ . Itis easy to see that (ˆ σ, a , ˆ µ ) is a truth-leaning equilibrium of G ( ϕ ).Jiang (2019) characterizes the truth-leaning equilibria of the auxiliary disclosure game.Fixing a finite evidence space ( E, - ) and distributions F G and F B , a truth-leaning equilib-rium exists in G ( ϕ ) for all strictly increasing ϕ : [0 , → R . Moreover, there exists a systemof beliefs µ ⋆ : E → [0 ,
1] such that for all strictly increasing ϕ , ( σ, a , µ ) is a truth-leaningequilibrium of G ( ϕ ) if and only if µ = µ ⋆ , a = ϕ ◦ µ , σ ( e | e ) = µ ( e ) ≤ ν ( e ) , and(A.2) µ ( m ) = min (cid:26) ν ( m ) , P e ∈ E σ ( m | e ) F G ( e ) π P e ∈ E σ ( m | e )[ F G ( e ) π + F B ( e )(1 − π )] (cid:27) for all m ∈ E , where ν ( · ) = F G ( · ) π F G ( · ) π + F B ( · )(1 − π ) . Notice that the right hand side of (A.2) isa continuous function of σ ∈ ∆( E ) E . Therefore, there exists a closed subset Σ ⋆ of ∆( E ) E such that ( σ, a , µ ) is a truth-leaning equilibrium of G ( ϕ ) if and only if σ ∈ Σ ⋆ , µ = µ ⋆ , and a = ϕ ◦ µ . By the above duality, for all disturbances η , ( σ, r, µ ) is a truth-leaning equilibriumof the disturbed game G R ( η ) if and only if σ ∈ Σ ⋆ , µ = µ ⋆ , and supp ( r ( ·| m, ζ )) ⊂ τ ( µ ( m ) , ζ )for all m ∈ E and ζ ∈ R K . A.2. Proof of Proposition 2.
Proof.
The first statement is implied by the second statement, since µ ⋆ is Bayesianconsistent with any sender’s strategy σ ∈ Σ ⋆ by Lemma 1, and φ is nonempty-valued.For the “only if ” part of the second statement, let ( σ, ρ, µ ) be a purifiable equilibrium.There exists a sequence of disturbances η n w −→ δ and for each η n , a truth-leaning equilibrium( σ n , r n , µ n ) of G R ( η n ) such that ( σ n , ρ n , µ n ) → ( σ, ρ, µ ), where ρ n = h r n , η n i . By Lemma 1, σ n ∈ Σ ⋆ for all n , and Σ ⋆ is closed. Therefore, σ ∈ Σ ⋆ . Additionally, µ n = µ ⋆ for all n , so µ = µ ⋆ . Fix any m ∈ E and a ∈ A such that a / ∈ φ ( µ ( m )) = τ ( µ ( m ) , τ is upperhemicontinuous in ζ , there exists a neighborhood U of 0 in R K such that a / ∈ τ ( µ ( m ) , ζ ) forall ζ ∈ U . By receiver optimality, r n ( a | m, ζ ) = 0 for all n and ζ ∈ U . Hence, as η n w −→ δ , ρ n ( a | m ) = R r n ( a | m, ζ ) η n ( dζ ) →
0. That is, a / ∈ supp ( ρ ( ·| m )). Lastly, let m, m ′ ∈ E be suchthat µ ( m ) = µ ( m ′ ). Since τ ( µ ( m ) , ζ ) = τ ( µ ( m ′ ) , ζ ) for all ζ , r n ( a | m, ζ ) = r n ( a | m ′ , ζ ) for all n , a , and almost all ζ . Therefore, ρ n ( ·| m ) = ρ n ( ·| m ′ ) for all n , so their limits also coincide,i.e., ρ ( ·| m ) = ρ ( ·| m ′ ).For the “if” part of the second statement, let ( σ, ρ, µ ) be such that σ ∈ Σ ⋆ , µ = µ ⋆ , supp ( ρ ( ·| m )) ⊂ φ ( µ ( m )) for all m ∈ E , and µ ( m ) = µ ( m ′ ) ⇒ ρ ( ·| m ) = ρ ( ·| m ′ ). We are17o show that it is a purifiable equilibrium. Let µ < µ < · · · < µ N be elements of µ ( E ),i.e., all possible posterior beliefs of the receiver. Since τ is upper hemicontinuous in ζ , thereexists r > τ ( µ i , ζ ) ⊂ φ ( µ i ) for all i and all ζ ∈ B r (0), where B r (0) denotesthe open ball of radius r around 0 in R K . For each α = ( α , α , . . . , α N ) ∈ × Ni =1 φ ( µ i ), let V α be the set of ζ ∈ R K such that τ ( µ i , ζ ) = { α i } . Notice that V α are pairwise disjoint, S α V α = B r (0), and λζ ∈ V α for all ζ ∈ V α and λ ∈ (0 , V α has positiveLebesgue measure. Let q n → ρ be a sequence such that supp ( q n ( ·| m )) = φ ( µ ( m )) for all m ∈ E , and µ ( m ) = µ ( m ′ ) ⇒ q n ( ·| m ) = q n ( ·| m ′ ). By abuse of notation, we write q n ( a | m ) as q n ( a, µ ( m )), and let x nα = Π Ni =1 q n ( α i , µ i ). x nα > α and all n . Therefore, for each n , wecan define a distribution η n over R K with full support and absolutely continuous with respectto the Lebesgue measure such that η n (cid:0) n V α (cid:1) = n − n x nα for all α , where n V α = { ζ : nζ ∈ V α } is a subset of V α . By construction η n w −→ δ . Let r be any receiver strategy in the disturbedgames such that supp ( r ( ·| m, ζ )) ⊂ τ ( µ ( m ) , ζ ) for all m ∈ E and ζ ∈ R K . By Lemma 1,( σ, r, µ ) is a truth-leaning equilibrium of G R ( η n ). Let ( σ, ρ n , µ ) be the associated equilibriumoutcome. Notice that ρ n ( a | m ) = R r ( a | m, ζ ) η n ( dζ ) is bounded from below by n − n q n ( a | m )and from above by n − n q n ( a | m ) + n , and recall that q n → ρ . Hence, ρ n → ρ , and ( σ, ρ, µ ) isa purifiable equilibrium. Remarks.
The above proof implies that, if φ ( µ ⋆ ( m )) is a singleton for all m ∈ E , thereexists a sequence of truth-leaning equilibrium outcomes of the disturbed games ( σ n , ρ n , µ n )that converges to ( σ, ρ, µ ) for all purifiable equilibia ( σ, ρ, µ ) and all disturbances η n w −→ δ .Let r be any receiver strategy in the disturbed games such that supp ( r ( ·| m, ζ )) ⊂ τ ( µ ⋆ ( m ) , ζ )for all m ∈ E and ζ ∈ R K . By Lemma 1, ( σ, r, µ ) is a truth-leaning equilibrium of alldisturbed games G R ( η n ). Since τ is upper hemicontinuous in ζ , and φ ( µ ⋆ ( m )) is a singletonfor all m ∈ E , r ( a | m, · ) is constant on a small neighborhood of 0 in R K for all a ∈ A and m ∈ E . Hence, ρ n ( a | m ) → r ( a | m,
0) = ρ ( a | m ) for all a ∈ A and m ∈ E . A.3. Proof of Proposition 3.
Proof.
The proof works similarly as in HKP despite different settings. First, observethat a perfect Bayesian equilibrium exists in every perturbed game G S ( ε ). The set of senderstrategies in the perturbed game Σ ⊂ ∆( E ) E and the set of receiver strategies ∆( A ) E areconvex and compact. Given a strategy of the receiver, the set of sender strategies that aresender optimal is closed and nonempty. This yields an upper hemicontinuous best responsecorrespondence of the sender Γ S : ∆( A ) E ⇒ Σ. Given a sender strategy σ , since all evidence Since τ is upper hemicontinuous in ζ , we only need to show that V α is nonempty for all α . Noticethat τ ( µ i , ζ ) = { α i } if and only if ζ ( α i ) > ζ ( a ′ ) for all a ′ ∈ φ ( µ i ), a ′ = α i . The assumption of increasingdifferences guarantees that all inequalities can be simultaneously satisfied for all i . Hence, V α is nonempty.
18s disclosed with positive probability, there is a unique Bayesian consistent system of beliefs µ σ , and the mapping σ µ σ is continuous. Since the solution to the receiver’s optimalityproblem φ is upper hemicontinuous, we have an upper hemicontinuous best response corre-spondence of the receiver Γ R : Σ ⇒ ∆( A ) E such that Γ R ( σ ) = × m ∈ E ∆( φ ( ρ σ ( m ))). Thenby the Kakutani fixed point-theorem, there exists σ, ρ such that σ ∈ Γ S ( ρ ) and ρ ∈ Γ R ( σ ).That is, the perturbed game has a Nash equilibrium. The Nash equilibrium paired with thesystem of beliefs µ σ consists of a perfect Bayesian equilibrium of the perturbed game.Since the set of sender’ strategies { σ : supp ( σ ( ·| e ) ⊂ LC ( e ) } ⊂ ∆( E ) E , ∆( A ) E , and [0 , E are compact, any sequence of perfect Bayesian equilibria of perturbed games { ( σ n , ρ n , µ n ) } ∞ n =1 has a convergent subsequence. Hence, a weakly truth-leaning equilibrium exists. It is easyto verify that any weakly truth-leaning equilibrium is a perfect Bayesian equilibrium. A.4. Proof of Proposition 4.
Proof.
Let ( σ, ρ, µ ) be a weakly truth-leaning equilibrium that is also purifiable. Weshow that (1) if σ ( e | e ) >
0, then σ ( e | e ) = 1, and (2) if σ ( e | e ) = 0, then e / ∈ argmax m - e P a ∈ A a · ρ ( a | m ), and µ ( e ) = ν ( e ).The first claim is due to purifiability. Let η n w −→ δ , and ( σ n , ρ n , µ n ) → ( σ, ρ, µ ) be suchthat ( σ n , ρ n , µ n ) is a truth-leaning outcome of G R ( η n ) for all n . If σ ( e | e ) >
0, then thereexists N such that σ n ( e | e ) > n ≥ N . However, ( σ n , ρ n , µ n ) is truth-leaning, so σ n ( e | e ) = 1 for all n ≥ N . Therefore, σ ( e | e ) = 1.The second claim is due to weakly truth-leaning. Let ε n →
0, and ( σ n , ρ n , µ n ) → ( σ, ρ, µ )be such that ( σ n , ρ n , µ n ) is a perfect Bayesian equilibrium of G S ( ε n ) for all n . If σ ( e | e ) = 0, e / ∈ argmax m - e P a ∈ A a · ρ ( a | m ). Otherwise, for all n , e is the unique maximizer to thesender’s problem in G S ( ε n ), so σ n ( e | e ) = 1, and σ n σ . Hence, for all e ′ ≻ e and all n , σ n ( e | e ′ ) = 0. By Bayes’ rule, µ n ( e ) = ν ( e ) for all n . Therefore, µ ( e ) = ν ( e ). A.5. Proof of Proposition 5.
Proof.
Notice that the receiver’s system of beliefs µ ⋆ is the same across all puri-fiable equilibria of all games in G . Moreover, given any two actions a i , a j and a be-lief µ , the receiver is indifferent between actions a i and a j at µ ∈ [0 ,
1] if and only if u ( a i , G ) , u ( a i , B ) , u ( a j , G ) , u ( a j , B ) are on a hyperplane in R . Therefore, the receiver isindifferent at some belief µ ⋆ ( m ) only on a Lebesgue null set of G . We are to show that, if φ ( µ ⋆ ( m )) is a singleton for all m ∈ E , then a truth-leaning equilibria that is also purifiableis weakly truth-leaning. This concludes that N has Lebesgue measure zero.Let G ∈ G be such that φ ( µ ⋆ ( m )) is a singleton for all m , and ( σ, ρ, µ ⋆ ) a truth-leaningequilibrium of G that is also purifiable. Given any perturbtion ε = { ε e , ε e | e } e ∈ E , we define19 σ ε , ρ ε , µ ε ) as follows:(1) σ ε ( e | e ) = 1 if σ ( e | e ) = 1;(2) σ ε ( e | e ) = ε e | e , and σ ε ( m | e ) = (1 − ε e | e ) σ ( m | e ) for all m = e if σ ( e | e ) = 0;(3) ρ ε = ρ ;(4) µ ε is by Bayes’ rule, i.e., µ ( m ) = P e ∈ UC ( m ) σ ε ( m | e ) F G ( e ) π P e ∈ UC ( m ) σ ε ( m | e )[ F G ( e ) π + F B ( e )(1 − π )] . For sufficiently small ε , ( σ ε , ρ ε , µ ε ) is a perfect Bayesian equilibrium of G ( ε ). Senderoptimality is satisfied if ε e < max m X a ∈ A a [ ρ ( a | m ) − ρ ( a | e )]for all e ∈ E such that σ ( e | e ) = 0. For all m ∈ E , since φ is upper hemicontinuous and µ ⋆ ( m ) is a singleton for all m , there exists δ > φ ( µ ) = φ ( µ ⋆ ( m )) for all m and all µ ∈ [0 ,
1] such that | µ − µ ⋆ ( m ) | < δ . Since µ ε → µ ⋆ , when ε is sufficiently small, ρ ε ( a | m ) = ρ ( a | m ) = a = φ ( µ ε ( m )) = a = φ ( µ ⋆ ( m )) for all m ∈ E . That is, receiver optimality issatisfied. By construction, it is also Bayesian consistent, and ( σ ε , ρ ε , µ ε ) → ( σ, ρ, µ ) for anysequence ε →
0. Therefore, ( σ, ρ, µ ) is a weakly truth-leaning equilibrium.
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