Correlated Equilibria in Sender-Receiver Games

Transcription

1 Correlated Equilibria in Sender-Receiver Games Andreas Blume Department of Economics University of Pittsburgh Pittsburgh, PA May, 2010 Abstract It is shown that the efficiency bound for communication equilibria identified by Goltsman, Hörner, Pavlov and Squintani [13] in the leading example of the Crawford-Sobel model can be obtained with strategy-correlated equilibria. Thus, unlike in earlier implementations of this bound, there is no need for communication to a mediator, for the sender s message to the receiver to be garbled, or for repeated message exchanges between sender and receiver. The occasional mismatch between the encoding and decoding rules used by sender and receiver in a correlated equilibrium can be interpreted as uncertainty about language use. I am grateful to Maria Goltsman and Joel Sobel for comments.

2 1 Introduction The amount of information that can be credibly transmitted from one strategic agent to another is constrained by how closely their preferences are aligned. In their seminal paper on sender-receiver games Crawford and Sobel (CS) [6] study this constraint in a model in which a privately informed sender sends a message to a receiver who then takes an action. Payoffs depend on the sender s private information, referred to as her type, and on the receiver s action but not on the message sent, which is therefore cheap talk. CS find that equilibria have a simple structure. With a type space that is a compact interval, as long as the sender prefers a strictly higher action than the receiver for any realization of the sender s type, in any equilibrium only a finite number of actions is taken and the set of types that induce any given action is a non-degenerate interval. In particular, with fixed preferences equilibria are bounded away from perfect information transmission where the receiver would take a distinct action for every types of the sender. The special case in which the sender s type is uniformly distributed, both sender and receiver have quadratic loss functions and the sender s and receiver s preferred actions differ by a constant bias has received considerable attention. In this uniform-quadratic model equilibria can be ranked in ex-ante efficiency terms and an appealing comparative statics result holds. The equilibrium with the largest number of equilibrium actions is ex-ante efficient and the number of actions in this equilibrium is inversely related to the sender s bias. There are a number of well-known mechanisms for improving efficiency in sender-receiver environments. Myerson [19] gives an example that illustrates the benefits of communicating through a noisy channel; Forges [11] shows that similar effects can be achieved with repeated simultaneous message exchange; Aumann and Hart [2] fully characterize the set of equilibrium payoffs in a class of finite games that are preceded by an infinity of opportunities for simultaneous message exchange; Krishna and Morgan [16] show that two communication rounds suffice to improve on the most efficient equilibrium in the uniform-quadratic CS game for almost all biases, including a range of biases too large to admit any communication in CS; Blume, Board and Kawamura [3] show that for almost any bias in the uniform-quadratic CS game there is a noise level in a noisy-channel game and an equilibrium that strictly improves 1

3 on the best CS equilibrium, again including biases for which communication is impossible in CS; and, Ivanov [14] shows that the same effect can be achieved by replacing the noisy channel with a strategic mediator. It follows from the revelation principle (Myerson [17]) that the set of all equilibrium payoffs that can be attained by one of the above schemes is a subset of the set of payoffs that can be achieved as equilibria of a direct mechanism in which the sender truthfully reports her type to a nonstrategic mediator and the receiver obediently executes the actions recommended to him by the mediator. In general, whenever a Bayesian game is extended by adding a nonstrategic mediator who privately receives messages from the players and privately recommends actions to them, the truthful and obedient equilibria of the extended game are referred to as communication equilibria of the original game (see Forges [9] and Myerson [18]). Recently, Goltsman, Hörner, Pavlov and Squintani (GHPS) [13] derived an efficiency bound for communication equilibria in Crawford-Sobel games with a uniform type distribution, quadratic utility functions and constant bias. The bound can be implemented in Krishna and Morgan s model for a range of biases and in the models of Blume, Board and Kawamura and of Ivanov for all biases. This note shows that the bound can also be implemented without the sender communicating to a nonstrategic mediator, without repeated communication by sender and receiver, without a noisy channel, and without a strategic mediator. Instead, it suffices that there is a correlation device that sends messages to both players before the sender sends her message to the receiver, which is equivalent to saying that the bound can be attained as a (strategy-) correlated equilibrium. Importantly, neither player needs to send messages to the device. The concept of a correlated equilibrium was introduced by Aumann[1] for finite strategic form games. He pointed out that they correspond to the Nash equilibria of games that are obtained by adding a correlation device to the original (base) game that privately sends messages to players before the base game is played. There are multiple ways of extending the correlated equilibrium idea to Bayesian games because of the added possibility that players may send messages to the correlation device before the device sends messages to the players. The most permissive adaptation corresponds to the above mentioned communication 2

4 equilibria. The direct adaptation of Aumann s definition to Bayesian game is the set of strategy-correlated equilibria as defined by Cotter [5]. The set of strategy-correlated equilibria is a (sometimes strict) subset of the set of communication equilibria. Note that since no message is sent to a third party (the correlation device, or a mediator), strategy-correlated equilibria preserve privacy vis-à-vis the third party, as long as the third party cannot observe sender messages and receiver actions. We show that GHPS s efficiency bound for communication equilibria in CS games can be achieved with strategy-correlated equilibria. The proof is constructive. We design a device that probabilistically and privately sends an encoding rule to the sender and a decoding rule to the receiver. The two rules agree (in the sense that the sender can predict the receiver s response, given her message) some but not all of the time. The probability of disagreement is the same as the probability of a transmission error in the noisy channel implementation of the GHPS bound by Blume, Board and Kawamura. One interpretation is that sender and receiver are uncertain about each other s language use. 2 The CS Environment We will be concerned with correlated equilibria in the leading example of Crawford and Sobel [6] (CS). In the uniform-quadratic example of CS a privately informed sender, S, communicates with a receiver, R. Their payoffs depend on the receiver s action, a R, the sender s type t T = [0, 1], and a parameter b > 0 that measures the sender s bias relative to the receiver. The sender s type t is drawn from a uniform distribution on T. When the sender s type is t and the receiver takes action a, the sender s payoff equals U S (a, t, b) = (t + b a) 2 and the receiver s payoff equals U R (a, t) = (t a) 2. CS study the game in which after observing her type t the sender sends a message m M to the receiver, who then takes an action in response to the sender s message. Messages do not directly affect payoffs. We will assume that the message space M equals the unit interval. 1 1 In the CS analysis of their model no assumption about the message space is needed, except that its cardinality be at least as large as the maximal number of equilibrium actions. Our proof makes use of the fact that the message space contains a compact non-degenerate interval. Our assumption that M = [0, 1] meets this requirement. Since we are in essence concerned with designing a communication mechanism, this 3

5 All of this is common knowledge among the players. We will refer to the above payoff structure, type set and type distribution as the (uniform-quadratic) CS environment and once the message and action stages are included as the (uniform-quadratic) CS game. In general communication need not be restricted to the sender sending a single message to the receiver as in the CS game. As discussed earlier, the literature has considered communication through noisy channels, repeated message exchanges and the use of strategic as well as non-strategic mediators, who both receive messages from and send messages to the players. Our focus will be on games in which initially both players privately receive a signal from a correlation device and then the CS game is played, i.e. after both players have received their signals from the device and the sender has learned her type, the sender sends a single message to the receiver who then takes an action. 3 Correlated Equilibria in the CS Game In Bayesian games there are multiple ways of defining correlated equilibria, which vary according to the degree to which correlated signals are allowed to depend on the players types. Strategy-correlated equilibrium, as defined by Cotter [5], can be implemented through a device that sends private signals to players that are independent of the players types. An alternative extension of correlated equilibrium to Bayesian games, that of a communication equilibrium, defined by Forges [9] and Myerson [18], permits players to send messages to the correlation device in addition to receiving instructions from the device. Evidently, the set of communication equilibria is a superset of the set of strategy-correlated equilibria. We will show in this section that in CS games the efficiency bound for communication equilibria can be attained through strategy-correlated equilibria. In CS games a behavior strategy for the sender is a function σ : T (M) and for the receiver it is a function ρ : M R, where we make use of the fact that for any belief the receiver has a unique best action. We write W S (σ, ρ) for the sender s expected payoff and W R (σ, ρ) for the receiver s expected payoff from the strategy profile (σ, ρ). Denote the set of behavior strategies for the sender by Σ and for the receiver by P. A strategy correlated does not appear to be restrictive. 4

6 equilibrium (SCE) in the CS model is a probability distribution λ (Σ P ) that satisfies for all measurable functions δ : Σ Σ and γ : P P : W S (σ, ρ)dλ(σ, ρ) Σ P W R (σ, ρ)dλ(σ, ρ) Σ P W S (δ(σ), ρ)dλ(σ, ρ), and (1) Σ P W R (σ, γ(ρ))dλ(σ, ρ) (2) Σ P An SCE in the CS model can be interpreted as a correlation device that privately sends an encoding rule, σ Σ, to the sender and a decoding rule, ρ P, to the receiver, with the property that both sender and receiver prefer to adhere to their suggested rules. Inequality (1) is the incentive compatibility constraint of sender. It states that the sender must prefer always to use the strategy σ recommended by the device to any deviation rule δ that determines which strategy to follow as a function of the recommendation. Inequality (2) is the analogous constraint for the receiver. Using the revelation principle, Myerson [17], one can characterize the set of communication equilibria in the uniform-quadratic CS model as corresponding to a family of conditional distributions on R, {p( t)} t T, that satisfies: [ ] t = arg max (t + b a) 2 dp(a t ), t T, and (3) t T R ap(a t)dt = tp(a t)dt, a R. (4) T T The family of conditional distributions can be interpreted as a mediator who recommends action a with probability p(a t) to the receiver when the sender reports a type t to the mediator. The first equation (3) is the incentive constraint for the sender. It says that it must be optimal for the sender to report her type truthfully to the mediator. The second equation (4) is the incentive constraint for the receiver and requires that conditional on the action a being recommended, this action is equal to the receiver s expectation of the sender s type conditional on that recommendation; recall that with a quadratic loss function the receiver s optimal action is equal to her expectation of the sender s type. Goltsman, Hörner, 5

7 Pavlov and Squintani (GHPS) [13] use this characterization to show that the receiver s ex ante payoff in any communication equilibrium of the CS model is bounded above by 1b(1 b). Since the ex ante payoffs of the receiver, V 3 R and the sender, V S, are related through V R = V S + b 2, this is also the efficiency bound for communication equilibria CS model. Blume, Board and Kawamura [3] (henceforth BBK) provide a mechanism that attains the efficiency bound for communication equilibria in the CS model. In BBK players communicate through a noisy channel that lets the message pass through with probability ɛ and otherwise transmits a random draw from a distribution G on the message space M = [0, 1] that has an everywhere positive density. 2 BBK show that for any b there exists a noise level ɛ(b) and an equilibrium of the corresponding ɛ(b)-noise game, Γ(ɛ(b)), that achieves the GHPS bound. We will show that this bound can also be attained in a strategy correlated equilibrium of the CS game and therefore requires no communication of the players to a mediator or through a noisy channel. Our proof strategy is to show that for any front-loading equilibrium of the game Γ(ɛ(b)) with noise level ɛ(b) that achieves the GHPS efficiency bound in BBK there exists an outcome-equivalent (and therefore payoff equivalent) strategy-correlated equilibrium. As background it is useful briefly to recall the key elements of the construction of front loading equilibria in BBK. In a front-loading equilibrium the type set, [0, 1], is partitioned into a finite collection of intervals, for any partition element that is not the leftmost interval there is a single message that is sent by types in that partition element and types in the leftmost partition element randomize over all the remaining messages, using the distribution G. The benefit from having the lowest interval send nearly all messages is that upon receiving such messages the receiver assigns relatively high probability to the error event, which biases his action upward toward the mean of the type distribution, which in turn relaxes incentive constraints for the sender. Indeed, using this construction one can show that a sufficiently small amount of noise in message transmission is almost always welfare improving. Moreover, for any b > 0 there exists a noise-level ɛ(b) and a corresponding front-loading equilibrium that achieves the GHPS bound. 2 While the exact nature of the distribution G is irrelevant, it may help to think of it as being the uniform distribution. 6

8 The following result shows that there exists a strategy-correlated equilibrium in the CS game that achieves the GHPS bound. The proof proceeds by constructing a correlation device and an equilibrium for the game that is augmented by the correlation device that translates the BBK-front-loading equilibria of ɛ(b)-noise games into strategy-correlated equilibria. Proposition The CS game with quadratic preferences and a uniform type distribution has a strategy correlated equilibrium that achieves the efficiency bound for communication equilibria. Proof: We will show that any outcome induced by a front-loading equilibrium of an ɛ- noise game can be replicated with a strategy-correlated equilibrium in the game without noise. The result then follows by applying this observation to the case where ɛ = ɛ(b) and the corresponding optimal front-loading equilibrium. Consider any k-step front-loading equilibrium in the ɛ-noise game, Γ(ɛ). In such an equilibrium there is a partition {Θ 1,..., Θ k } = {[0, θ 1 ], (θ 1, θ 2 ],..., (0, θ k ]} of the type space T = [0, 1] such that types in the partition element Θ 1 randomize uniformly over the entire message space M = [0, 1], types in any other partition element Θ i send a distinct message m i M, m i m j i j, i, j > 1, the receiver responds to m i with the action θ i+θ i 1 2 i > 1 and to any other message m m i i > 1 with the action (1 ɛ)θ 1 θ 1 2 +ɛ 1 2 (1 ɛ)θ 1 +ɛ. Note that the latter action is a weighted average of the action 1 2, which would be optimal if the receiver was sure that the message m was generated by noise and of the action θ 1 2, which would be optimal if the receiver was sure that the message m was sent intentionally (by types in the set Θ 1 ); the weights are the posterior probability that message m was received as a result of noise, was received because it was sent intentionally, ɛ (1 ɛ)θ 1, and the posterior probability that it +ɛ (1 ɛ)θ 1 (1 ɛ)θ 1 +ɛ. We will construct a strategy correlated equilibrium in the zero-noise game, Γ(0) that induces the same outcome (joint distribution over types of actions). To this end consider the following correlation device D : The device first draws a pair of potential signals (p 1, p 2 ) from independent uniform distributions on [0, 1] and then privately sends actual signals ξ 1 to the sender and ξ 2 to the receiver. The sender s signal ξ 1 is always equal to p 1. With probability 1 ɛ the receiver s signal ξ 2 equals p 1 as well; otherwise, it equals p 2. Denote the zero-noise 7

9 game augmented by the device D by Γ(0, D). Consider the following strategies in the game Γ(0, D). Sender types in Θ 1 randomize uniformly over M, regardless of the signal ξ 1 that they receive from the correlation device. Types in the interval Θ j, j = 2... k send the message ξ 1 + j 2 ξ k 1 + j 2 k (where x denotes the largest integer less than or equal to x). For j = 2,..., k the receiver responds to message ξ 2 + j 2 ξ k 2 + j 2 θ k with the action j +θ j 1. The receiver s response to any other 2 message is (1 ɛ)θ 1 θ 1 2 +ɛ 1 2 (1 ɛ)θ 1 +ɛ. That these strategies are mutual best replies in Γ(0, D) can be seen as follows: The event that ξ 1 and ξ 2 coincide in Γ(0, D) matches the event that the sender s messages goes through in Γ(ɛ) and has the same probability. If in this event the receiver gets one of the messages of the form ξ 2 + j 2 ξ k 2 + j 2 k, he can with probability one identify the set of types who sent that message, just like in the k-step front-loading equilibrium of Γ(ɛ). If instead in this event he receives any of the other messages, he does not know whether this is because of randomization by a type in the lowest partition element or because the signals ξ 1 and ξ 2 differ. Therefore, in this case his best reply will be a weighted average that assigns probability (1 ɛ)θ 1 (1 ɛ)θ 1 +ɛ Θ 1, and probability to the event that the message was sent by a type in the lowest interval, ɛ (1 ɛ)θ 1 +ɛ to the event that it was sent by one of the other types, again just like in the k-step front-loading equilibrium of Γ(ɛ). In the event that ξ 1 and ξ 2 do not coincide the receivers s response is the same weighted average and agrees with his action in the k-step front-loading equilibrium of Γ(ɛ). Finally, the incentives of the sender are the same in the games Γ(ɛ) and Γ(0, D). In one game, Γ(0, D), the sender can only affect the receiver s action with positive probability in the event that the signals ξ 1 and ξ 2 coincide, and in the other, Γ(ɛ), she can only affect the receiver s action with positive probability in the no noise event, in which the message is transmitted as sent. Thus in both cases the arbitrage condition that makes the boundary type θ i indifferent between the action that is induced by types in Θ i and the one induced by types in Θ i+1 is the same as in the CS model for all i = 1,..., k. 8

10 4 Additional Comments on the Literature As noted earlier, one interpretation of a correlated equilibrium in the CS game is as describing situations where sender and receiver are uncertain about each other s language use. De Jaegher [7] has linked strategic vagueness to correlated equilibrium and illustrated the potential benefits of such vagueness by example. 3 Our result implements this idea in CS games. A number of authors have explored the relation between communication equilibria and correlated equilibria in Bayesian games. Forges [8] studies infinitely repeated finite senderreceiver games with limit of the average payoffs and finds that the sets of communication equilibrium payoffs of such games coincide with the sets of correlated equilibrium payoffs. Forges [12] shows that correlation followed by two rounds of interim communication can achieve all communication equilibrium payoffs in finite games with at least three players, provided the message space has the cardinality of the continuum. Vida [20] shows that the set of communication equilibrium payoffs of any finite Bayesian game can be realized via correlated equilibria of an extended game in which players have infinitely many opportunities to exchange messages from a countable message space before they play the base game. The present paper, in contrast, is concerned with comparing the efficiency frontiers of communication equilibria and correlated equilibria in one-shot CS games, i.e. with one round of interim communication. Most directly related to the present paper is Forges [10], which implies the equivalence of the sets of correlated and communication equilibrium payoffs in finite sender-receiver games. Because of the finiteness requirement, this result does not directly carry over to CS games with their continuous type and action spaces. In addition, finiteness is relied upon in the proof; e.g., the encryption of the sender s strategy involves a random draw from a uniform distribution over the set of all permutations of the outcome space (the set of type-action pairs). A natural open question is how to extend Forges [10] result to a class of infinite games that includes CS games. 3 In his paper, as in Blume and Board [4], the focus is on the strategic use of an existing vague language, as opposed to the question of why language is vague that is raised by Lipman [15]. 9

11 References [1] Aumann, Robert J. [1974], Subjectivity and Correlation in Randomized Strategies, Journal of Mathematical Economics 1, [2] Aumann, Robert J. and Sergiu Hart [2003], Long Cheap Talk, Econometrica 71, [3] Blume, Andreas, Oliver J. Board and Kohei Kawamura [2007]: Noisy Talk, Theoretical Economics 2, [4] Blume, Andreas and Oliver J. Board [2009], Intentional Vagueness, University of Pittsburgh working paper. [5] Cotter, Kevin D. [1991], Correlated Equilibrium in Games with Type-Dependent Strategies, Journal of Economic Theory 54, [6] Crawford, V.P.and J. Sobel [1982], Strategic Information Transmission, Econometrica 50, [7] De Jaegher, Kris [2003], A Game-Theoretic Rationale for Vagueness, Linguistics and Philosophy 26, [8] Forges, Françoise [1985], Correlated Equilibria in a Class of Repeated Games with Incomplete Information, International Journal of Game Theory 14, [9] Forges, Françoise [1986], An Approach to Communication Equilibria, Econometrica 54, [10] Forges, Françoise [1988], Can Sunspots Replace a Mediator, Journal of Mathematical Economics 17, [11] Forges, Françoise [1990], Equilibria with Communication in a Job Market Example, Quarterly Journal of Economics 105, [12] Forges, Françoise [1990], Universal Mechanisms, Econometrica 58,

12 [13] Goltsman, Maria, Johannes Hörner, Gregory Pavlov, and Francesco Squintani [2009], Mediation, Arbitration and Negotiation, Journal of Economic Theory 144, [14] Ivanov, Maxim [2009], Communication via a Strategic Mediator, Journal of Economic Theory, forthcoming. [15] Lipman, Barton L. [2009], Why is Language Vague?, Boston University working paper. [16] Krishna, Vijay and John Morgan [2004], The Art of Conversation: Eliciting Information from Experts Through Multi-Stage Communication, Journal of Economic Theory 117, [17] Myerson, Roger B. [1982], Optimal Coordination Mechanisms in Generalized Principal-Agent Problems, Journal of Mathematical Economics 10, [18] Myerson, Roger B. [1986], Multistage Games with Communication, Econometrica 54, [19] Myerson, Roger B. [1991], Game Theory: Analysis of Conflict, Harvard University Press, Cambridge, MA. [20] Vida, Péter [2007], From Communication Equilibria to Correlated Equilibria, University of Vienna working paper. 11