Coordinated Sequential Bayesian Persuasion

Transcription

1 Coordinated Sequential Bayesian Persuasion Wenhao Wu December 3, 017 Abstract his paper studies an extended Bayesian Persuasion model in which multiple senders persuade one Receiver sequentially and the subsequent players can always observe previous signaling rules and their realizations. In the spirit of Harris (1985), I develop the recursive concavification method to characterize the Subgame Perfect Equilibrium, which summarizes the multiplicity of possible strategic interactions among senders and identifies the ranges of their equilibrium payoffs. Furthermore, I prove the existence of a special type of equilibrium, called the Silent Equilibrium, where at most one sender designs a nontrivial signaling rule. Also, I show that in zero-sum games, the truth-telling information structure is always supported in equilibrium. Finally, I make comparisons with a simultaneous multi-sender Bayesian persuasion model (Gentzkow and Kamenica 017b) and examine the impact of the order of persuasion. A geometric version of Blackwell s order is adopted for examination of informativeness. Keywords: Bayesian Persuasion, Multiple Senders, Subgame Perfect Equilibrium, Markov Perfect Equilibrium, Communication, Infinite Game. JEL Classification Codes: D8, D83 I am indebeted to Andreas Blume for his encouragement and comments. I would also like to thank Martin Dufwenberg, Mark Walker, Asaf Plan, Peter Norman and participants at the U of Arizona theory lunch, the 8th International Conference on Game heory at Stony Brook, and the Fall 017 Midwest Economic heory meeting, for their helpful comments. All errors are my own. University of Arizona. wuw@ .arizona.edu 1

2 1 Introduction Bayesian persuasion, introduced by Kamenica and Gentzkow (011), describes a situation in which a sender, with full commitment to a signaling rule, can effectively induce a specific action of a Receiver by influencing her belief. When the only sender monopolizes information revelation, he is able to manipulate the Receiver s belief and exploit the rent of persuasion. he Receiver would still like to follow the sender s suggestions, even though she realizes that the information is generated from a biased source. When multiple senders get involved in persuasion, however, not only does the Receiver need to integrate information from different sources, but also the senders have to consider what information others disclose. Strategic interactions and conflicts of interest confound senders behavior and require in-depth analysis. Gentzkow and Kamenica (017b) study this class of Bayesian persuasion games with multiple senders who move simultaneously; in contrast, this paper is devoted to its counterpart in which senders move sequentially. Assuming public observance of previously designed signaling rules and their realizations, subsequent senders can correlate the probabilities of their signals with previous senders. 1 his is a coordinated information structure (Li and Norman 017a, 017b). From the perspective of informativeness, this setting implies that subsequent senders can select any weakly more transparent information structure than the existing one (Gentzkow and Kamenica 017b). I approach this problem by a recursive method that decomposes this multisender Bayesian persuasion game into separate single-sender games. Considering each sender s decision making, it is rational for him to update his belief based on previous information and evaluate the consequences of his persuasion as affected by reactions of subsequent players. hen, he confronts an updated prior and a current value referring to future equilibrium payoffs as if he is the only sender. herefore, the equilibrium of the entire game is a combinating result of these decentralized analyses. Next, I present an example to illustrate the above idea. Example 1: Judge, Prosecutor and Attorney his example is an extension of the judge-prosecutor example in Kamenica and Gentzkow (011) by adding an attorney who defends the suspect. he game proceeds as follows: the prosecutor moves first and picks a singaling rule that sends signal I or G randomly conditional on the underlying states innocent or guilty. After observing the prosecutor s rule and signal, the attorney designs his own signaling rule that yields another signal. Finally, the judge reviews reports from both sides and make a decision. No one knows the truth, while the common prior of the person s being guilty is 30% (µ 0 = 0.3). Suppose the judge wants to convict guilty people but acquit innocent people. She obtains 1 unit for doing so, 0 unit if she fails. he prosecutor (attorney) only wants to have the person convicted (acquited), whose payoff is 1 for conviction (acquitment) but 0 otherwise. Apparently, this is a Bayesian persuasion game with the state space {innocent, guilty}, the signal space {I, G}, and the action space {acquit, convict}. Let µ 1 and µ denote posteriors after period 1 and, 1 his is the same assumption on the space of signaling rules in Gentzkow and Kamenica (017b). he equilibrium outcome is independent of the order in this example.

3 v v µ µ Figure 1: Judge-prosecutor-attorney persuasion game respectively. In this section, I denote by V, V 1, V 1, V 1 1, V 1 0, V 0 senders values on posteriors of all periods, where the superscript indexes for the player and the subscript for the period. A complete explanation of notation is in Section 3.1. First, the judge s optimal strategy, for all possible belief, µ, she may hold at the end, can be described as: if µ < 0.5, she acquits; if µ > 0.5, she convicts; if µ = 0.5, she randomizes arbitrarily over convict and acquit. Endogenizing the judge s responses, both senders values can be defined upon µ, as below: V 1 (µ ) = V (µ ) = 0 if µ [0, 0.5), [0, 1] if µ = 0.5, 1 if µ (0.5, 1]. 1 if µ [0, 0.5), [0, 1] if µ = 0.5, 0 if µ (0.5, 1]. When the attorney is about to take an action at the second period, he must have observed the prosecutor s persuasion and formed a new belief, µ 1. Due to the flexibility of the prosecutor s persuasion, µ 1 could be any point in the 3

4 belief space. Nevertheless, no matter which µ 1 is realized, the attorney acts as if he is the only sender in a Bayesian persuasion game with this new prior µ 1 and the value V. Hence, the attorney s value dependent on the result of the prosecutor s persuasion, V1 (µ 1 ), is given by the concave closure as the lower red curve shows. In the equilibrium, the judge must break the tie by acquiting the person in favor of the attorney, otherwise, the attorney will deviate to a persuasion that randomizes over µ = 1 and 0.5 ɛ. ɛ is a sufficiently small number. 3 { V1 (µ 1 ) = 1 if µ 1 [0, 1 ], µ 1 if µ 1 ( 1, 1]. Furthermore, as the prosecutor and attorney play a zero sum game, the prosecutor s value defined on µ 1, V1 1, can be obtained by subtracting V1 from 1. { V1 1 (µ 1 ) = 0 if µ 1 [0, 1 ], 1 + µ 1 if µ 1 ( 1, 1]. With the knowledge of V1 1, the prosecutor completely understands the consequence of his persuasion. Like the attorney, he is equivalently playing a singlesender Bayesian persuasion game with the prior µ 0 and value V1 1. His initial value V0 1 is indicated as the concave closure of V1 1, shown as the blue line, which suggests that full revelation is his unique equilibrium strategy. Symmetrically, I can obtain the attorney s initial value, V0, as the opposite of V 1 0. V 1 0 (µ) = µ, µ [0, 1] V 0 (µ) = 1 µ, µ [0, 1] herefore, the equilibrium payoffs for both senders are V0 1 (µ 0 ) = 0.3 and V0 (µ 0 ) = 0.7. During this process, the equilibrium strategy is recovered by repeatedly imposing concave closures over relevant values. In equilibrium, the prosecutor chooses a truth-telling signaling rule such that it is optimal for the judge to follow the prosecutor s suggestion. As one main contribution of this paper, I formally develop this method as the recursive concavification method in Section 4. Besides the general technique, this paper solves for Subgame Perfect Equilibrium which is more general than the equilibrium concepts in existing literature, e.g. the Sender-preferred SPE and Markov Perfect Equilibrium (Kamenica and Gentzkow 011; Gentzkow and Kamenica 017b; Li and Norman 017b; Ely 017). Compared to those concepts, SPE has two advantages: 1) no behavioral restrictions and ) exhausting the strategic interactions among players. As will be shown in Sections 4.3 and 5.3, the equilibrium payoff sets of the Sender-preferred SPE and MPE are proper subsets of those of SPE. 3 Referring to heorem 1, Section 4. 4

5 Notice that it is an equilibrium when the prosecutor reveals the true state and the attorney sends a null signal. 4 In Section 6, I will show the pervasive existence of this type of equilibrium, called the Silent Equilibrium, where at most one sender reveals information. he existence of this equilibrium relies on the accessibility of an abundant set of signaling rules to the senders, which allows them to reveal beforehand what their followers would otherwise have revealed. he economic implication of this example is also significant. In contrast with the previous judge-prosecutor game (Kamenica and Gentzkow, 011), which only has partial revelation in equilibrium, introducing a sender with an opposite interest to the previous sender improves information transparency significantly. More strikingly, this result can be generalized to any pair of senders who have zero sum utilities (Section 7), which demonstrates that two competing senders can form a stable system that supports full revelation. More broadly, this example can be regarded as a justification for the adversarial system. Unlike in the simultaneous multi-sender Bayesian model (Gentzkow and Kamenica 017b), full revelation is not always an equilibrium. As discussed in Section 9, games could have unique partial revelation in equilibrium. he relationship between a sequential game and its corresponding simultaneous game is that when the sequential equilibrium is unique, 5 no simultaneous equilibrium is less informative than the sequential one (Li and Norman 017b). In this paper, the informativenesss is ordered by the Blackwell s order. For the purpose of comparing informativeness in an intuitive way, I adopt a geometric version of Blackwell s order in regard to posterior dispersion, as will be shown by heorem 5. he remainder of this paper is organized as follows. Section summarizes the existing literature in this area. Section 3 presents the basic model, explains the information environment, and introduces mathematical preliminaries. Section 4 explains the recursive concavification method and applies it to characterize SPE paths. Section 5 characterizes the MPE. Section 6 proves the existence of the Silent Equilibrium. Section 7 discusses the equilibrium in a zero sum game. Section 8 explores a geometric way of comparing informativeness under Blackwell s order. Section 9 discusses the relationship with the coordinated simultaneous multi-sender Bayesian persuasion model. Section 10 explores the effects of the order of persuasion. Section 11 concludes. Literature Review his paper lies in the domain of Bayesian Persuasion with multiple senders. Kamenica and Gentzkow (011) lay out a formal framework of the Bayesian persuasion game with a single sender and characterize the Sender-preferred SPE 6 by the concavification method (Aumann and Maschler 1995). hey also define a relationship, called Bayesian plausibility, between a signaling rule and its distribution of posteriors and suggest that one can work on the posterior belief space 4 A signal from a babbling signaling rule that leaves the belief unchanged. 5 he uniqueness of equilibrium means the distribution of actions conditional on the state is unique. 6 In this equilibrium, the receiver takes an action in favor of the sender when she is indifferent. 5

6 directly. Gentzkow and Kamenica (017a) (017b) analyze the case in which multiple senders move simultaneously and conclude that competition increases disclosure of information. Li and Norman (017a) attract attention to the Bayesian persuasion game with multiple senders who move sequentially by constructing a counter example to Kamenica and Gentzkow (017a) (017b) s general conclusion that competition improves transparency. hey also propose the Markov Perfect Equilibrium as an important equilibrium concept for the sequential Bayesian persuasion game, where belief serves as the state variable (Li and Norman 017a, 017b; Ely 017). Li and Norman (017b) independently address the same problem with this paper and conclude the same result that the Silent Equilibrium (the one-step equilibrium in their language) always exists. However, there are still many differences. First, they express coordinated signaling rules in the form of a partition framework (Li and Norman 017a, 017b; Kamenica and Gentzkow 017b), instead of the traditional definition set forth in this paper. Second, they assume a finite action space so that value functions can be divided into finite polytopes; however, this paper allows a compact action set and smooth value functions. hird, they frame the question as a linear programming problem and characterize the MPE, while this paper develops the recursive concavification method to characterize the SPE and MPE. Fourth, this paper discusses several related topics of zero-sum games, a geometric Blackwell s order, and the order of perusasion. Harris(1985), Harris, Reny and Robson (1995) and Hellwig, Leininger, Reny and Robson (1990) discuss the existence and characterization of SPE of infinite games, which class of games the Bayesian persuasion model belongs to. Following Harris (1985), I characterize the SPE paths by translating his method into a geometric way, i.e., the recursive concavification method. Ely (017) uses a similar method with the recursive concavification method to solve a dynamic persuasion mechanism, where one sender controls the receiver s belief over a stochastic process. o solve a Bellman equation with a concave closure on the right side, he approximates it by repeatedly concavifying a value function. In contrast, this paper concavifies a group of value correspondences 7 of all senders. his paper also broadly relates to literature in communication games with multiple senders or receivers. Krishna and Morgan (001), Battaglini (00) and Ambrus and akahashi (008) extend Crawford and Sobel (198) into multisender case and discuss the conditions under which full revealing equilibrium can be induced. Similar with Prop. 10, Krishna and Morgan (001) have a result that it is beneficial for the receiver to consult both senders with interests biased in opposite directions. Milgrom and Roberts (1986) and Forges and Koessler (008) study information disclosure model, introduced by Milgrom (1981) and Grossman (1981), with multiple senders. Bergemann and Morris (016) introduce a new equilibrium concept, Bayesian Correlated Equilibrium, which can be interpreted as the equilibrium of a persuasion model with one sender and multiple receivers. 7 As will be shown by heorem 1, concavifying correspondences means imposing concave closure over the minimum of the correspondences, which becomes a criterion for selecting SPE paths. 6

7 N ω 1 ω S1 S1 π 1 π 1 S π,1 s 1 s s 1 s S π, R s 1 s s 1 s s 1 s s 1 s R a 1 a a 1 a a 1 a a 1 a a 1 a a 1 a a 1 a a 1 a Figure : he game tree of a path in a Bayesian persuasion model with senders 3 Model 3.1 Basic Setup his section lays out a formal framework of a multi-sender coordinated sequential Bayesian persuasion game. he states of the world are Ω = {ω 1,..., ω l }. here are senders and 1 Receiver. Each sender has access to a set of costless signaling rules, Π. A signaling rule is a mapping from the state space to distributions over the signal space π : Ω (S), where the finite signal space is S = {s 1,..., s m }. 8 Equivalently, Π = l ω=1 (S). Specifically, S t denotes Sender t s signal space and s t the signal realized in period t. A common prior µ 0 (Ω) is shared by all players. µ t represents the updated posterior belief after period t, for any t = 1,...,. he Receiver takes an action a A, A is compact. Utility functions of all players depend on the state of the world and the action taken by the Receiver. Additionally, I impose continuity on the senders utility functions, v t (a, ω), t = 1,...,, and the Receiver s utility function, u(a, ω). he timing of the game is as follows: Date 0 Nature picks a state ω according to the probability of µ 0. ω is unknown to all players. Date 1 Sender 1 chooses π 1 Π which generates s 1 S. Date After observing {π 1, s 1 }, Sender chooses π s S. Π which generates... Date After observing {π i, s i } i=1, Sender chooses π Π which generates s S. 8 he assumption of finite signal space is made WLG for expositional simplification. As Prop. 7 will show, the equilibrium values are invariant in the cases of finite and infinite signal spaces. 7

8 Date + 1 After observing {π i, s i } i=1, the Receiver makes a decision a A. he extensive form, as in Figure 1, consists of two types of information sets, one at which a sender has chosen the signaling rule but has not sent the signal, and the other at which both the signaling rule and the signal have been sent (denoted by nodes labeled S1, S and R ). he former leads to fixed simple probability distributions over the second type of information sets without any strategic interaction. So it suffices to emphasize on the second type. Combining a pair of a signaling rule and its signal into a period of history, a history up until the end of date t is denoted by e t = (π 1, s 1,..., π t, s t ). E t is the set of histories including e t. For t = 1,...,, Sender t s strategy is a mapping σ t : E t 1 Π and the Receiver s strategy is ρ : E A. he set of Sender t s strategies is Σ t, for t, and the Receiver s strategy set is Σ +1. In this paper, I focus on pure-strategy equilibrium. However, the result equivalently applies to the mixed strategy equilibrium with finite actions. Because for senders, there is no difference between using mixed strategies and pure strategies; 9 for the Receiver, the mixed strategy set with finite actions is compact. Next, let me clarify the meaning of subgame in this paper. A subgame is defined to begin at an information set for each history e t E t, t = 1,...,, instead of at one decision node as SPE usually requires. he rationale depends on that in each history and its corresponding information set, players observe, equivalently, a sequence of experiments and their realizations, so that it is possible to prescribe beliefs over all information sets by Bayes rule, no matter whether they are on or off the equilibrium path. With such a fixed belief system, the sequential rationality is satisfied by the Receiver s maximizing her expected payoffs in all subgames originating from those information sets. hroughout this paper, I refer to SPE as with subgames in this sense. 10 I denote the subgame starting at history e t by Γ(e t ), and denote the unique belief over the information set associated with e t by µ t (e t ). ne SPE of Γ(e t ) is represented as γ(e t ) which consists of a strategy profile of subsequent players {σ t+1,..., σ, ρ}. ne SPE of the whole game is then γ(e 0 ). he set of SPE paths of Γ(e t ) is denoted by Γ(e t ), and the equilibrium path of γ(e t ) is written as γ(e t ). With these preliminaries at hand, I can give the formal definition of SPE: Definition 1. A Subgame Perfect Equilibrium of this game, γ(e 0 ), is a strategy profile {σ 1,..., σ, ρ} such that for any t = 0,...,, e t E t and σ h Σ h, h {t + 1,..., }, it satisfies: E µt(e t)[v h (σ h, σ h )] E µt(e t)[v h (σ h, σ h )] And for all e E and a A, E µ (e )[u(ρ(e ), ω)] E µ (e )[u(a, ω)] 9 Kamenica and Gentzkow 011, footnote his variation of SPE agrees with Li and Norman (017a) (017b). hey propose SPE as a proper concept because there is no private information and players don t have to update beliefs about others types. 8

9 f special importance is a cluster of values {Vt k (µ)} t, k, t {0,..., }. k {1,..., }. Vt k : (Ω) R is defined as Vt k (µ) = {v R v is the payoff for Sender k in some γ t (e t ) where µ t (e t ) = µ, e t E t }, for t, k, µ. hey represent the sets of equilibrium payoffs for sender k after period t at a certain posterior belief µ. For example, if v Vt k (µ), it means there is a SPE in which Sender k holds a belief µ after period t and his expected payoff is v. In Section 4, I use V and to represent the maximum and minimum selections of a sender s V value. I also use V k : Γ(e t ) R, for t, to denote Sender k s expected payoff on SPE paths. 3. Coordinated Information Structure he information structure is determined by signaling rules designed by senders. he signaling rule is equivalent to the experiment defined in Blackwell (1951) (1953) which provides the Receiver with a more informative environment. If senders impose signaling rules individually, the probabilities of signals are independent with each other. However, in this model senders move sequentially and subsequent senders can vary their signaling rules over previous realizations, so each strategy profile specifies a sequence of signaling rules as {π 1, {π,s } s S1,..., {π,s } s S1 S }. he coordinated information structure is where senders are able to correlate the probabilities of their own signals with that of previous senders. Denote a coordinated signaling rule by π c. Definition. {π1, c..., π c } are coordinated signaling rules if for t, {πc 1,..., πt 1} c and their realizations {s 1,..., s t 1 }, πt c is a mapping: Ω ( t i=1 S i) consistent with the probability distribution ( t 1 i=1 S i) prescribed by {π1, c..., πt 1}, c i.e., m P (s 1,..., s t 1, s j t ω) = P (s 1,..., s t 1 ω), for ω, {s 1,..., s t 1 } S 1 S t 1. With a coordinated signaling rule a sender can unilaterally deviate to any information structure weakly more informative than the existing one (Gentzkow and Kamenica 017b), which plays an important role in the proof of heorem 4 and Prop. 1. Prop. 1 reveals the equivalence relation between information structures generated by {π 1, {π,s } s S1,..., {π,s } s S } and {π1, c..., π c }. It also clarifies the coordinated information structure embedded in this game and estabilishes connections with other work in multi-sender persuasion games (Li and Norman 017a, 017b; Gentzkow and Kamenica 017b). It shows that the analysis in this paper also applies to an alternative game in which senders move sequentially and design coordinated signaling rules. Proposition 1. he information structure given by independent signaling rules {π 1, {π,s } s S1,..., {π,s } s S1 S } is equal to that generated by some coordinated signaling rules {π1, c..., π c }. he converse is true. Proof. See appendix. 3.3 Preliminaries his part mentions a series of knowledge about Bayesian plausibility, some facts in convex analysis, and Blackwell s order, that form the mathematical basis of 9

10 following sections. In addition, I introduce notation for expositional simplification Signaling rules and posteriors Each signal s j from a signaling rule π leads to a posterior φ j by Bayes Law. hese posteriors, {φ j } m, and the probabilities of their associated signals, {p(φ j )} m, construct a distribution of distributions that summarizes all information content of a signaling rule. he relationship between signaling rules and their posteriors is called Bayesian plausibility. 11 Definition 3. (Bayesian plausibility) For any π Π and its posteriors {φ j } m, µ = m p(φ j)φ j, p(φ j ) [0, 1], for j. Conversely, any Bayesian plausible distributed posteriors are outcomes of some signaling rule. his equivalence relation allows me to define a signaling rule indirectly as its posteriors. Specifically, I refer φ 1,..., φ m µ to a signaling rule that generates a set of Bayesian plausible distributed posteriors {φ 1,..., φ m } whose expectation is µ, such that µ = m p(φ j)φ j, p(φ j ) [0, 1], for j. { φ 1,..., φ m µ, γ 1,..., γ m } represents an SPE path on which a sender designs a signaling rule φ 1,..., φ m µ, and his followers strategies form another SPE path, γ j receiving his signal s j, for j. Furthermore, I denote by φ n 1,..., φ n m µ φ 1,..., φ m µ the convergence of a sequence of signaling rules, where posteriors φ n j φ j and coefficients p(φ n j ) p(φ j), for j. I should point out that Bayesian plausibility is preserved in the limit, i.e., φ 1,..., φ m µ is still a signaling rule. Notice that p(φ j ) could be zero that makes the corresponding signal redundant, but it is a convenient expression in Section Convex Analysis Most proofs draw on useful results from convex analysis, so it is necessary to point them out at this stage. he first is a key concept that has been widely used in economics to identify the potential rents of persuasion for senders, that is, the concave closure. Mathematically, there are two equivalent definitions. 1 he most well known definition describes it as the lowest concave function that is not lower than a specific function. Another definition, however, is given by Eq. (1). hough complicated, it perfectly fits the context of a persuasion game, and is adopted as the main definition in this paper. { m m } cl(f(x)) = sup α j f(x j ) α m, α j x j = x, m = 1,,... Each component of the equation contains a corresponding economic meaning. he objective function f refers to a value function of a sender, the set {x j } m a distribution of posteriors of a singaling rule, {α j} m 1 the probabilities of these signals. In addition, the restriction α m, m α jx j = x implies the satisfication of Bayesian plausibility. his definition is central to the recursive concavification method that drives the recursion in Sections 4 and Referring to Kamenica and Gentzkow (011), Prop Hiriart-Urruty, Jean-Baptiste, and Claude Lemaréchal, Fundamentals of Convex Analysis, pp.99, Proposition.5.1. (1) 10

11 n a belief space (Ω), one posterior belief is a point and the convex hull of a set of posteriors {φ 1,..., φ m } is denoted by conv{φ 1,..., φ m }. A regularity in regard to convex hulls, which determines uniqueness in Section 8, is the Carathéodory s heorem. 13 Carathéodory s heorem For any set S R n and x convs, x can be represented as a convex combination of n + 1 elements of S Blackwell s rder Adopted as the measure of informativeness, Blackwell s order is a partial order on the space of signaling rules. A more informative signaling rule creates a better environment for a decision maker to achieve higher expected payoffs. Specifically, the meaning of Blackwell s order is given by Definition 4 and Definition 5, both of which are equivalent based on Blackwell s theorem. Suppose there are two signaling rules π = λ 1,..., λ m1 µ0 and π = µ 1,..., µ m µ0, i.e. µ 0 = p 1 λ 1 + +p m1 λ m1 and µ 0 = q 1 µ 1 + +q m µ m. he signal generating mechanisms can be described as two matrices A m1 l and B m l, where the (i, j)th entry represents the probability of generating the i-th signal at state j, i.e. a ij = P (s i ω j ) and b ij = P (s i ω j ) where s i induces λ i and s i induces µ i. Given the Receiver s utility function u, her value for a posterior µ is defined as V u : (Ω) R, V u (µ) = max a A E µ [u(a, ω)]. Her value for a signaling rule π = λ 1,..., λ m1 µ0 is V u : Π R, Vu (π) = p 1 V u (λ 1 ) + + p m1 V u (λ m1 ). An action rule is a mapping from the signal space to the action space, f : S A, and the set of action rules is F. Defining the expected payoff of an action rule f at state i as v i (f, π) = m π(s j ω i )u(f(s j ), ω i ), i = 1,..., l, the range of the payoff vector is denoted by D π = {(v 1 (f, π),..., v l (f, π)) f F}. Before introducing Blackwell s theorem, I define sufficiency and informativeness as two relations among signaling rules. Definition 4. (Sufficiency) If π is sufficient for π, written π π, then there exists a matrix C m1 m, c ij 0, i c ij = 1, i, j, such that A = CB. Definition 5. (Informativeness) If π is more informative than π, written π π, then D π D π, for any utility function u and any compact action set A. Blackwell (1951)(1953) demonstrates the equivalence between the statistical relation (sufficiency) and the Bayesian decision relation (informativeness), regardless of the common prior, the action set and the payoff structure. his model, however, is endowed with a common prior that facilitates a geometric expression of Blackwell s order, as discussed in Section 8. Blackwell s heorem π π iff π π. 13 Ewald, Günter, Combinatorial convexity and algebraic geometry, Section.3, pp

12 4 Subgame Perfect Equilibrium SPE identifies a process of strategic information revelation for each history, where the realized belief becomes the prior for the following subgame. he class of subgames starting at the same period associated with the same initial belief share the same set of SPE paths, i.e. for any e t, e t E t such that µ t (e t ) = µ t (e t), Γ(e t ) = Γ(e t). herefore, it is feasible to define senders equilibrium payoffs on belief spaces, which are given as Vt k (µ) in Section 3.1. However, the characterization of SPE is not straightforward, because this game is an infinite game due to the abundant strategy and history sets (Harris, Reny and Robson 1995; Hellwig, Leininger, Reny and Robson 1990). Fortunately, Harris (1985) provides a convenient criterion, which I translate geomatrically into a concave closure, for selecting SPE paths. In the remainder of this section, I will show how to characterize SPE paths with this criterion, and discuss related topics about potential ranges of SPE payoffs and multiplicity of equilibria. 4.1 Harris (1985) Implicit in any SPE is a notion of credible punishment on all potential deviations, which is reflected in Harris criterion. From this perspective, Harris criterion can be understood as the supremum of payoffs a sender can achieve subject to subsequent players most severe credible punishment. o make the punishment effective, a sender s action followed by an equilibrium path is justified as an SPE path if and only if the sender obtains payoff no less than Harris criterion. he construction of Harris criterion includes two steps. First, For any t {1,..., }, suppose Γ t (e t ) are known for e t E t, calculate the worst equilibrium payoff for Sender t when he takes a signaling rule π t := φ 1,..., φ m µt 1(e t 1), denoted by b(π t ). { m } b(π t ) =min p(φ j )v j v j Vt t (µ t (e t 1, π t, s j )), j m = p(φ j ) min{vt t (µ t (e t 1, π t, s j ))} m t = p(φ j (µ t (e t 1, π t, s )V j )) () t his is well defined because all value correspondences have closed graphs by Prop. and Prop. 5, and thus the minimum can be achieved. he second step is to find out the supremum of these worst payoffs, sup{b(π ) π Π}, which is the definition of Harris criterion. Formally, suppose γ j is an SPE path in Γ((e t 1, π t, s j )) for j, then {π t, { γ 1,..., γ m }} is one of the SPE paths in Γ t 1 (e t 1 ) if and only if Sender t s expected payoff from it is no less than sup{b(π ) π Π}. Because of the definition of the concave closure in Eq. (1) and the equivalence between signaling rules and Bayesian plausible distributed posteriors, Harris criterion takes another geometric form. 1

13 sup{b(π ) π Π} { m } = sup p(φ t (µ t (e t 1, π t, s j)v j )) φ 1,..., φ m µt 1(e t 1) = cl(v t t t )(µ t 1 (e t 1 )) (3) Eq. (3) establishes a connection between the criterion and the concavification method, which will be justified as a necessary and sufficient condition in Prop. 3 and heorem 1. With this criterion, one can pin down the sequence of values {Vt k } t, k as well as the set of SPE paths recursively: {V t } t=1 ) {V cl(v t 1} t=1 cl(v )... {V t 0 } t=1 4. Recursion 4..1 Period + 1 At the beginning of period +1, the Receiver has observed all signaling rules and signals from senders and formed a new belief µ, based on which she optimizes her expected utility by choosing an action. he optimal action set is A (µ ) = {a A a arg max a A E µ (u(a, ω))} as a correspondence A : (Ω) A. Because of continuity of u and compactness of the domain, by the Maximum heorem, A is non-empty valued and has a closed graph. he next proposition establishes a more significant result that V t is non-empty valued and has a closed graph for t = 1,...,. Proposition. For each t = 1,...,, V t(µ ) = {E µ (v t (a, ω)) a A (µ )}. V t is non-empty valued and has a closed graph. Proof. For {µ n } µ and {v n v n V t (µn )} v, {a n } A s.t. a n A (µ n ), E µ n(v t (a n, ω)) = v n. Because A is compact and A has a closed graph, {a n k } {a n } a A (µ). herefore E µ (v t (a, ω)) V t (µ). Because {a n k } a and v t continuous, v = lim n vn = lim k E µ n k [v t (a n k, ω)] = E µ [v t (a, ω)] V t (µ) Prop. shows that the condition for the application of Harris criterion is satisfied. he set of SPE paths, Γ (e ), for e E, is easily shown as Γ (e ) = A (µ (e )) (4) 13

14 4.. Period Subgames of period consist of the Receiver and Sender that resemble a single-sender persuasion game. Equivalently, one can solve them by the concavification method with Harris criterion. Proposition 3. For µ (Ω), v V (µ) if and only if θ 1,..., θ m µ s.t. v = m p(θ j)v j, v j V (θ j) and v )(µ). cl(v Proof. (Sufficient) Let Sender send a null signal and the Receiver take an action that support v in a SPE. For other signaling rules Sender designs, φ 1,..., φ m µ, let the Receiver take an optimal action that yields (φ j ) for V Sender upon receiving each φ j, j = 1,..., m. hen by the definition of )(µ), the value for Sender of playing any other signaling rule is dominated by )(µ), and by v as well. cl(v cl(v (Necessary) If v < )(µ), by the definition of ), φ 1,..., φ m µ cl(v cl(v s.t. m p(φ (φ j ) > v, i.e. the signaling rule φ 1,..., φ m µ dominates the j)v value v in any SPE following this strategy. As Harris criterion is necessary and sufficient, the SPE paths are simply action profiles satisfying it. Γ (e ) = { {π, a 1,..., a m } π := φ 1,..., φ m µ (e ), a j Γ (e, π, s j ), j, and m } p(φ j )V ({a j }) )(µ (e )) cl(v (5) Eq. (5) identifies Γ (e ), e E, which naturally yield the values of period 1 for senders. As mentioned before, all subgames at this period, starting with the same prior, share the same set of SPE paths, which allows me to group histories by beliefs and define value correspondences on the belief space. herefore, {V t 1 } t=1 incorporate all payoffs supported by Γ (e ), e E. { V t 1(µ) = V t ( γ) e E, µ (e ) = µ, γ Γ } (e ) (6) he existence of SPE is also guaranteed. In contrast with the minimum of the value for characterization, I use the maximum of the value to show the existence. Due to the closedness of V, the maximum of the value correspondence is USC. With this property, those signaling rules that support the concave closure of the maximum of the value are equilibrium strategies by Prop. 3, as they also generate payoffs no less than Harris criterion. Proposition 4. For µ (Ω), and any v = cl( V )(µ), v is an equilibrium payoff for Sender in a SPE γ(e ) s.t. µ (e ) = µ. 14

15 Proof. By the definition of cl( V ), { θn 1,..., θm n µ }, such that m k=1 pn V k (θn k ) v. Because (Ω) is compact, θ n k 1,..., θn k m µ θ1,..., θm µ. Because V has a closed graph, V is USC, i.e. V (θj ) lim V (θn k j ), for j. herefore, m p V j (θ j ) lim m pn k j V (θn k j ) = v = cl( V )(µ ) )(µ ). Also, as V cl(v has closed graph, V (θ j ) V (θ j ), so θ 1,..., θm µ is an equilibrium play of Sender at period Period t, t < his part extends the analysis to earlier stages by repeating the above argument. Similar results with Prop. 3 and 4 hold for {Vt t } t=1, for t. ake period 1 for instance. With the knowledge of V, Sender 1 can evaluate the consequence of a certain signaling rule as well as Sender does. hus, the identification of the set of SPE paths, Γ (e ), for e E, is subject to the same criterion except for a new value correspondence, V. he same logic applies to Sender t, for any t <. heorem 1. For t, µ (Ω), v Vt t+1 (µ) if and only if θ 1,..., θ m µ s.t. v = m p(θ j)v j, v j Vt+1 t+1 (θ t+1 j) and v )(µ). cl(v Proof. Similar with the proof of Prop. 3. t+1 heorem. For t, µ (Ω), and any v = cl( V t+1 )(µ), v is an equilibrium payoff for Sender t + 1 in a SPE γ t (e t ) s.t. µ t (e t ) = µ. Proof. Similar with the proof of Prop. 4. heorems 3 and 4 suggest that the set of SPE paths, Γ t 1 (e t 1 ), and its corresponding set of equilibrium payoffs, Vt 1, k for k, can be obtained by formulas similar with Eq. (5)(6). For e t 1 E t 1, t+1 Γ t 1 (e t 1 ) = { {π t, γ 1,..., γ m } π t := φ 1,..., φ m µk 1 (e k 1 ), γ j Γ t (e t 1, π t, s j ), j, and m } p(φ j )V t t ({ γ j }) )(µ t 1 (e t 1 )) cl(v { Vt 1(µ) k = V k ( γ) e t 1 E t 1, µ t 1 (e t 1 ) = µ, γ Γ } t 1 (e t 1 ) t (7) (8) Now I have given the set of SPE paths and defined a sequence of values for each sender at each period. Conversely, I refer to {τt k } t, k as the mappings from each value point in Vt k to its corresponding SPE paths. For any (µ, w) graph(vt k ), let τt k : (µ, w) Γ(e t ) be the set of all SPE paths, in which µ t (e t ) = µ and Sender k s expected payoff is w. As closedness underpins heorems 1 and and facilitates iterations of the concavification method, the remaining technical challenge is to show that {Vt k } t, k 15

16 v1 w 1 1 v w 1 v1 v1 V 1 V 1 1 A1 C A0 w 0 1 C0 C1 A θ1 θ θ µ φ1 φ φ µ1 v v V V 1 B1 B D B0 w 0 D0 D1 θ1 θ θ µ φ1 φ φ µ1 Figure 3: A Geometric Illustration of the Backward Induction V 1 0 V 0 µ0 µ0 16

17 are closed. his desirable mathematical property builds on many features of this game, including the continuous value functions, the compact action space, and the compact set of signaling rules, Π. In addition, finiteness is another important condition approached by focusing on equilibrium paths including only finite action profiles. In summary, all these properties such as continuity, compactness and finiteness, lead to the closedness of the value correspondences. In any Vt k, any converging sequence of value points has a limiting point supported by the limiting SPE path. his Upper hemi-continuity property is similar with the result in Hellwig, Leininger, Reny and Robson (1990), where they use SPE paths of nearby finite games to approximate the SPE path of a targeting infinite game. Proposition 5. V k t Proof. See Appendix. have closed graphs, for t, k. Finally, the assumption of finite signal space can be relaxed. Denote a persuasion game with infinite signal space by Γ, and another with sufficiently large finite signal space by Γ 0, such that S l. he equivalence between Γ and Γ 0 is presented below. Proposition 6. Any equilibrium values in Γ are also equilibrium values in Γ 0. Proof. See Appendix. 4.3 A Geometric Illustration Figure 3 provides a geometric illustration of a -sender case. he process starts from the left to the right. he left column depicts value correspondences {V i } i=1,. Sender s equilibrium strategy is determined by Harris criterion, the red curve in the lower-left figure. As Sender s persuasion causes a Bayesian plausible distribution over µ, the values in {V1 i } i=1, are convex combinations of points in {V i } i=1,. o specify, let θ be any point in (Ω), a 1 τ 1 (A 1 ) τ (B 1 ), and a τ 1 (A ) τ (B ). a 1 and a are the Receiver s best responses that support A 1, B 1 and A, B, respectively. hen, { θ 1, θ θ, a 1, a } is justified as an SPE path because Sender s payoff, w1, is above the criterion. herefore, A 0 and B 0 are contained in V1 1 and V1, respectively, and w1 1 V1 1 (θ ), w1 V1 (θ ). Because the set of SPE paths, Γ 1 (e 1 ), e 1 E 1, is well defined, one is able to derive {V1 i (µ)} i=1,, for µ. btaining {V1 i } i=1,, which are supposed to be closed by Prop. 5, Harris criterion for selecting Sender 1 s strategy is indicated by the blue curve. he transformation from {V1 i } i=1, to {V0 i } i=1, follows the same pattern. Suppose there are SPE paths γ 1 τ1 1 (C 1 ) τ1 (D 1 ) and γ τ1 1 (C ) τ1 (D ). Because C 0 lies above the red curve, { φ 1, φ φ, γ 1, γ } is an SPE path for any subgame γ 0 (e 0 ) such that µ 0 (e 0 ) = φ. So w0 1 V0 1 and w0 V0. he value correspondences, V0 1 and V0, are generated in this way. 4.4 Ranges of Values Figure 5 illustrates the limits of persuasion in three scenarios: a single-sender case, a multi-sender case and a case in which the player can not persuade. 17

18 V A C B µ Figure 4: Ranges of values in 3 cases Suppose the sender, given the Receiver s optimal actions, has a value correspondence identified by the union of the two black curves. 14 In a single-sender case, SPE payoffs are required to be no less than cl(v ) by heorem 1 and can not be larger than cl( V ), as indicated by A. When the sender competes with others, his payoffs may drop down cl(v ), but not lower than the hyperplane going across three lowest values at vertices, denoted by h; this is his insured payoff by revealing the truth. he range of payoffs in the second case is A B. If the sender is not allowed to persuade, he may get a more detrimental result in C. he lower bound of C is the convex closure of from bottom, denoted V by b. he possible equilibrium payoffs are confined in A B C, the convex hull of the whole value correspondence. Proposition 7. For any Bayesian persuasion model with one Receiver and one sender: 1) the range of SPE payoffs in a single sender game is between cl( V ) and cl(v ); ) the range of potential SPE payoffs in a multi-sender sequential game is between cl( V ) and hyperplane h; the range of potential SPE payoffs when he cannot persuade is between cl( V ) and surface b. he inclusion relation between these sets sheds light on the impact of persuasion environment on senders welfare. When a sender monopolizes information disclosure, he is able to receive the highest range of payoffs. When he competes with other senders, his payoff may drop down, but is still guaranteed a basic rent from revealing the truth. If he cannot persuade, his equilibrium payoff could be even lower. 4.5 Multiplicity ne challenge is multiplicity of equilibria resulting form the abundant strategy sets of both senders and the Receiver. nce the Receiver has multiple opitmal actions, those value correspondences become thick, and Harris criterion is 14 By construction, the receiver has optimal actions at each µ. 18

19 V µ Figure 5: he judge charges a penalty slack in a sense that uncountably many action profiles satisfy it. hus, the explicit form of the set of SPE could be intractable, even in a simple example as below. Example : he Judge Charges a Penalty Suppose the judge should not only decide to acquit or convict, but also impose a penalty on the defendant once he gets convicted. he penalty could be any number between 0. and 1 (thousand dollars), constructing a compact action set {0} [0., 1] ( 0 refers to acquit ). he judge s target is still raising the judgement accuracy, and her optimal strategies can be described as: a = 0 if µ < 0.5; a {0} [0, 1] if µ = 0.5 and a [0., 1] if µ > 0.5. hough the judge is herself indifferent to the penalty, the prosecutor would like as high a fine to be charged as possible. Specifically, his payoff function is v(a, ω) = a and his value is represented as a correspondence in Figure 5. Harris criterion is represented as the red curve and implies that the prosecutor s expected payoff in equilibrium should be at least 0.1. Satisfying Harris criterion, any signaling rule in together with the Receiver s optimal response is an SPE path. For example, { 0, , 0, a}, a [0., 1] is a family of SPE paths that identifies the range of SPE payoffs for the prosecutor as [0.1, 0.6]. However, there are still uncountably many other SPE paths, which are impossible to explicit individually. 5 he Markov Perfect Equilibrium As shown above, the SPE might be intractable due to multiplicity, while the Markov Perfect Equilibrium can be a desirable refinement. Because in a persuasion game the only payoff relevant factor is belief, it undoubtedly becomes the state variable (Li and Norman 017a, 017b; Ely, 017). In MPE, players correspond to realized beliefs instead of histories, which implies that players only consider making full use of information regardless of how the information is revealed. his section discusses this concept from 3 respects. First of all, 19

20 I characterize the MPE with a simpler version of the recursive concavification method. Next, I present a general example of the MPE. Finally, I show that the MPE payoff set is properly included by the SPE payoff set. 5.1 Characterization Formally speaking, Sender t s Markovian stratege can be defined as a mapping σt M : (Ω) Π, for t, and the Receiver s Markovian strategy as ρ M : (Ω) A. he MPE shared by the class of subgames {Γ(e t ), µ t (e t ) = µ} is denoted by Γ M t (µ). In MPE, senders values reduce to functions and cl( V ) coincides with ). cl(v herefore, Harris criterion is satisfied if and only if the expected payoff takes on the value on the unique concave closure, and existence is preserved by substituting supremum in Eq. (1) with maximum. Proposition 8. For each sender, given his value function over posterior belief, f(x), { the optimal signaling rule exists for each prior if and only if: cl(f)(x) = m max α jf(x j ) α l, } m α jx j = x. When this condition is satisfied, the value function is called achievable. { m Proof. When cl(f)(x) = max α jf(x j ) α l, } m α jx j = x, for any prior µ, there exists {φ 1,..., φ m } such that m p(φ j)φ j = µ and m p(φ j)f(φ j ) = cl(f)(µ). bviously, φ 1,..., { φ m, µ is an optimal signaling rule. m If for some µ (Ω), max α jf(x j ) α l, } m α jx j = µ is not well defined, which means that there is a sequence of signaling rules φ n 1,..., φ n m µ whose values approximate cl(f)(µ) but do not reach the limit, which rejects the existence of an optimal signaling rule. he recursive concavification method resembles that for SPE, as described in Section 4. At period + 1, the Receiver takes an action ρ M (µ) A (µ), for µ, that captures Γ M (µ). For any µ, ΓM (µ) exists and determines senders values of the last period, {V t(µ)} t=1. V t (µ) = E µ [v t (ρ M (µ), ω)] (9) {V t} t=1 could be any selections of the value correspondences {V t} t without restriction, but Γ M (µ) exists for any µ if and only if V is achievable according to Prop. 8. Based on the concavification method, σ M := φ 1(µ),..., φ m (µ) µ is an optimal strategy if and only if for µ: cl(v )(µ) = m p(φ j (µ))v (φ j (µ)) (10) he satisfication of Eq.(10) directly implies that {σ M (µ), ρ} ΓM (µ) by Prop. 8. Furthermore, this definition disentangles the contingent strategies of Sender on all possible priors µ, and exhibits the influence of his persuasion on other passive senders. In parallel, {V t 1 } t can be transformed from {V t} t by a formula, 0

21 { m } V t 1(µ) = p(φ j (µ))v t (φ j (µ)) σ M (µ) = φ 1 (µ),..., φ m (µ) µ (11) In the same pattern, σ M 1 (µ), for µ, is an MPE strategy if and only if V is achievable. Similarly, {σ M 1 (µ), σm, ρ} ΓM (µ) if and only if Sender s payoff reaches cl(v )(µ), so on and so forth. As long as {σm 1,..., σ M, ρm } is a MPE, this recursive process will go through until it yields the sequence of value functions {Vk t} t, k. heorem 3. {σ M 1,..., σ M, ρm } forms a MPE if and only if, for µ (Ω), k = 1,...,, t = 1,..., and σ M t (µ) = φ t 1(µ),..., φ t m(µ) µ : ρ M (µ) A (µ) (1) V k (µ) = E µ [v k (ρ M (µ), ω)] (13) cl(v t t )(µ) = m p(φ t j(µ))v t t (φ t j(µ)) (14) V k t 1(µ) = m p(φ t j(µ))v k t (φ t j(µ)) (15) 5. A General Example In the MPE, the recursive concavification method takes a more transparent form. I present such an example in Figure 6. he Receiver s Markovian strategy yields senders value functions on final beliefs. Following the analysis in Section 4., the persuasion rents for both senders are concave closures over V1 1 and V. 15 When Sender 1 (Sender ) reveals information, he designs signaling rules that randomizes posteriors over the vertices of the domain of the blue (red) line the prior lies in. Also, their persuasion causes parallel randomization for the other sender that leads to transformation from V 1, V to V1 1, V1, and from V1 1, V1 to V0 1, V0, respectively. Geometrically, these transformations are as easy as connecting two value points. Finally, the equilibrium payoffs are V0 1 (µ 0 ) and V0 (µ 0 ), and the equilibrium paths are recovered from V1 1 and V. If µ 0 (0, µ ), the equilibrium path is that Sender 1 sends 0, µ µ0, Sender designs a null signaling rule; if µ 0 (µ, µ), nobody reveals any information; if µ 0 ( µ, 1), Sender 1 sends µ, 1 µ0, while Sender does not reveal any information. 1

22 v1 v V 1 V µ µ v1 v V 1 1 V 1 Figure 6: A General Example of MPE µ1 µ1 v1 v3 V 1 0 µ µ V 0 µ0 µ0

23 v µ µ v µ µ Figure 7: ne SPE that is not an MPE 5.3 MPE vs. SPE his part differentiates SPE from MPE by presenting SPE payoffs that are not in MPE. In this example, there are two states {ω 0, ω 1 }, µ = P (ω 1 ), and µ 0 = 1. Denote the SPE payoffs by (v 1 s, v s) and the MPE payoffs by (v 1 m, v m). Suppose the Receiver s action set is [4, 6] [0, ] and her optimal reactions to final beliefs are represented by A (µ). A (µ) = { [4, 6] if µ [0, 1 ], [0, ] if µ [ 1, 1]. here are two senders, whose utilities are given as below. v 1 (a, ω 0 ) = { 10 a if a [4, 6], a if a [0, ]. 15 Here the concave closures of the minimum and the maximum of value correspondences coincide, Harris criterion is satisfied by signaling rules that achieve the concave closure. 3

24 v 1 (a, ω 1 ) = 6 a v (a, ω 0 ) = a v (a, ω 1 ) = { a 4 if a [4, 6], a + 4 if a [0, ]. Under this payoff structure, V 1 and V are depicted as the grey areas in Figure 11. ne feature of this payoff structure is that v 1 and v are increasing and decreasing functions in a, respectively, which means they have opposite interests in the Receiver s actions. he Receiver s punishment on one sender can automatically become a reward to the other. For any MPE in which vm = 4, the Receiver must take 4 at µ = 0 and 0 at µ = 1, generating the lowest payoffs Sender could receive in equilibrium. therwise, he can deviate to the truth-telling signaling rule that brings about a payoff higher than 4. But the Receiver s behavior against Sender is beneficial for Sender 1, in a sense that as long as he reveals the state, his payoff is 6, the highest possible payoff for him in this game. herefore, (vm, 1 vm) = (6, 4) is the unique pair of MPE payoffs in which vm = 4. In SPE, however, the Receiver has more lattitude in punishing senders for deviations from certain information structures, and consequently there will be more abundant equilibria. Focusing on a class of equilibria where Sender keeps silent and the Receiver takes actions satisfying Harris criterion of the second period, a subset of V1 1 is shown as the deeper grey area in the upper figure. 16 It is sufficient to obtain Harris criterion of the first period from this subset of V1 1, as shown by the blue line. 17 hen, I have one SPE path: Sender 1 designs 1 4, 3 4 1, Sender sends null signals and the Receiver takes 5 at µ = 1 4 and 1 at µ = 3 4. Here, (v1 s, vs) = (4, 4). 6 he Silent Equilibrium ne observation is that when the equilibrium information structure is unique, in either Example 1 or Figure 6, it is an equilibrium that only Sender 1 sends a nontrivial signaling rule, while Sender babbles. his is called the Silent Equilibrium. Definition 6. (he Silent Equilibrium) An SPE in which at most one sender reveals information on the equilibrium path. 16 Because my goal is to find a specific SPE, I do not need to characterize the whole set of V Harris criterion cannot be higher than the concave closure of the minimum of this subset, and it cannot be lower than the hyperplane going across the value points at the verticies, according to Prop. 7. 4