Perturbed Communication Games with Honest Senders and Naive Receivers∗ Ying Chen† Department of Economics, Arizona State University January, 2010
Abstract This paper studies communication games in which the sender is possibly honest (tells the truth) and the receiver is possibly naive (follows messages as if truthful). By characterizing message-monotone equilibria in the perturbed games, the paper provides an explanation for several important aspects of strategic communication not explained in the canonical model, including sender exaggeration, receiver skepticism and the clustering of messages. The paper also derives a surprising result that the strategic receiver may respond to more aggressive claims with more moderate actions. In the limit as the probabilities of the non-strategic players approach zero, (i) the limit equilibrium outcome corresponds to a most-informative equilibrium outcome of the limit (Crawford-Sobel) game; (ii) only the messages at the top of the message space are sent with positive probability. The paper also establishes equilibrium existence when the message space is finite and shows why existence may fail when the message space is continuous. Keywords: Communication, Honest Senders, Naive Receivers, Sender Exaggeration, Receiver Skepticism, Clustering of Messages, Non-montone Receiver Reaction, Finite Message Space J.E.L. Classification: C72, D82, D83 ∗
First version: December 2004. An earlier version of this paper was circulated under the same title. A portion of that earlier version has been taken out and incorporated into "Selecting Cheap-Talk Equilibria" (Chen, Kartik and Sobel, 2008). This is a drastically revised version of the earlier paper. I am indebted to my dissertation advisors David Pearce, Stephen Morris and Dino Geradi for their advice and support. I thank Vincent Crawford, Ezra Friedman, Navin Kartik, Alvin Klevorick, Edward Schlee, Joel Sobel and audience at various seminars and conferences for helpful comments and suggestions. † Department of Economics, Arizona State University, P.O. Box 873806, Tempe, AZ 85287-3806. Email:
[email protected].
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Introduction
Cheap-talk models following the seminal work by Crawford and Sobel (1982) and Green and Stokey (2007, earlier version 1981) have had tremendous success in explaining how information is strategically transmitted when agents have only partially aligned interests.1 Several important aspects of strategic communication, however, cannot be explained by the canonical model. To illustrate, consider a security analyst making stock recommendations to an investor. Analysts face a well-known conflict of interest — although they want to provide reliable advice to attract customers, they are often under pressure to bias their recommendations upwards, especially when affiliated with the underwriter. Empirical studies2 have found that analysts’ recommendations are often inflated, many investors adjust their trading responses downward and “buy” and “hold” recommendations are issued much more frequently than “sell” recommendations. The canonical model cannot explain at least three aspects of this example: sender exaggeration, receiver skepticism, and the relative frequency of different recommendations. It has long been understood, going back to Crawford and Sobel (1982), that the canonical model does not predict how messages are used or interpreted in equilibrium because the players do not care about messages per se and therefore the literal meanings of messages do not affect equilibrium behavior. Another element of the canonical Crawford and Sobel (1982) (henceforth “CS”) model is that all players are fully strategic. A main goal of this paper is to show, through a tractable model, that these aspects of strategic communication that cannot be explained in the canonical model arise naturally when the players are not all fully strategic.3 In the CS model, a sender privately observes the state of the world and then sends a costless message to the uninformed receiver; upon receiving the message, the receiver chooses an action that affects both players’ payoffs. My model departs from CS by assuming that with positive probability, the sender is honest (truthfully reports his observation) and the receiver is naive (follows whatever message is sent to her as if they were truth1
Examples from different areas abound, including, for example, Matthews (1989) in political economy and Morgan and Stocken (2003) in financial economics. 2 See, for example, Michaely and Womack (2005) and Malmendier and Shanthikumar (2007). 3 There is a long tradition, going back to at least Kreps and Wilson (1982) and Migrom and Roberts (1982), of introducing behavioral types to explain phenomena that cannot be explained by fully strategic players. That the players may be non-strategic is consistent with experimental and other empirical evidence. Both experimental studies by Forsythe, Lundholm and Rietz’s (1999) and Cai and Wang (2006) find that in cheap-talk games, some sender subjects displayed a tendency to reveal the true state and some receiver subjects showed a certain amount of gullibility even when their opponents have a clear incentive to lie. Malmendier and Shanthikumar’s (2007) empirical study finds that although large traders take into consideration the incentives of stock analysts, small traders follow stock analysts’ recommendations literally.
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ful).4 Otherwise the players act fully strategically to maximize their expected payoffs. (I call these players “dishonest” senders and “sophisticated” receivers, or strategic players in general.) Whether a player is strategic or not is private information. A sender’s “honesty” and a receiver’s “naivety” are relative to the literal meanings of the messages. In my model, there are a finite number of messages available. The message has a commonly understood literal meaning “my observation of the state is in the set ,” and the collection of ’s forms a partition of the state space. This captures the characteristics of the message spaces typically used in many communication situations. (For example, graduate admission offices often ask recommenders to describe a student as belonging to one of the following categories: extraordinary (top 2%), exceptional (top 5%), outstanding (top 10%), superior (top 15%), above average (top 25%), average (top 50%), and below average (bottom 50%).) My model assumes that an honest sender sends the message if and only if his observation of the state is in the set and when a naive receiver receives the message , she responds as if the state is in the set . Although the honest sender’s and the naive receiver’s behavior are fixed, the strategic players’ use and interpretation of messages arise endogenously in equilibrium, and they are the focus of my analysis. In section 3, I provide a characterization of the players’ strategies in the class of message-monotone equilibrium (i.e., equilibrium in which the sender’s strategy is nondecreasing in the state) when the probabilities of the non-strategic players are positive. Specifically, suppose the dishonest sender has an upward bias. Then, in a messagemonotone equilibrium, the dishonest sender pools with the honest sender who has on average a higher observation, i.e., the dishonest sender sends a message whose literal meaning implies a higher observation of the state: sender exaggeration arises in equilibrium. Of course, the strategic receiver cannot be fooled systematically. Since she does not observe whether the sender is honest or dishonest, she discounts any message potentially sent by the dishonest sender. For messages at the low end of the message space that are sent only by the honest sender, they are “credible” and even the sophisticated receiver believe them. However, every message higher than (0) (i.e., the equilibrium message sent by the dishonest sender with the lowest observation) is sent by a positive measure of the dishonest sender in equilibrium. Hence, for messages sufficiently high (higher than (0)), we have receiver skepticism: the sophisticated receiver discounts the face value of the claim and her response is strictly lower than the naive receiver’s. 4
These types correspond to the L0 types ("truster" and "believer") in Cai and Wang (2006). They find evidence of other behavior types of higher levels of sophistication (according to their classification) that are anchored on the L0 types. For tractability, my model incorporates only the L0 types.
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Another novel aspect of the perturbed game is the clustering of messages, i.e., only those messages at the top of the message space are sent by a “large” measure of the dishonest sender. This is illustrated in figure 1 and it provides an explanation for the relative frequency of messages used: for example, professors’ recommendations of students tend to concentrate in the highest categories, and “strong buy” and “buy” are issued much more frequently by financial analysts than “sell” recommendations. Perhaps the most surprising result is that the sophisticated receiver’s equilibrium actions are not necessarily increasing in the messages she receives, even if the sender follows an increasing strategy. That is, the sophisticated receiver may react to a more aggressive claim with a more moderate action. How can this happen? Roughly speaking, a higher message may lead to a lower action if the receiver believes that with high probability, the message is sent by the dishonest sender and therefore discounts the claim heavily. As the probabilities of the non-strategic players go to zero, the perturbed game converges to the canonical Crawford-Sobel game. Another goal of this paper is to investigate what equilibrium outcomes survive in the limit and how messages are strategically used and interpreted when the perturbation is vanishingly small. The main result, Proposition 2 in section 4.1, shows that as the probabilities of the non-strategic players go to zero, the limit equilibrium outcome corresponds to a “most informative” equilibrium outcome (i.e., one that has the maximal number of steps) in the limit game. This complements the findings in Chen, Kartik and Sobel (2008). They introduce a selection criterion called “no incentive to separate” (NITS) and shows that it uniquely selects the most informative equilibrium in the CS game under some regularity conditions. My analysis provides a foundation of this selection criterion by showing that NITS must be satisfied in a limit equilibrium outcome of the perturbed games. Additionally, Proposition 2 says more than what equilibrium outcome survives in the limit — it also shows that the sender uses only the messages at the top of the message space to convey information in the limit, resulting in the starkest form of message clustering. These results show that we have sharp predictions about how players use and interpret messages strategically when they have a preexisting common language, and with (arbitrarily small) positive probability, some players are “literal minded.” The perturbation approach taken in this paper complements various other ways to understand the role of a preexisting common language in strategic communication. For example, Farrell (1993) and Matthews, Okuno-Fujiwara and Postelwaite (1991) focus on how messages’ literal meanings restrict players’ beliefs about unexpected announcements and how that affects equilibrium selection. More recent work by Lo (2006) takes a somewhat different approach and looks at the implications of the restrictions a common language may have on
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players’ strategies in cheap-talk games. These papers, including mine, that incorporate a preexisting common language have rather distinct goals from another strand of literature (e.g., Blume, Kim and Sobel (1993) and Warneryd (1993)) that use evolutionary arguments to explain the emergence of the meaning of words when a common language does not exist. One modeling difference between my paper and related papers is that my paper assumes a finite message space whereas other papers typically assume a continuous message space. Besides the advantage of fitting many communication situations well, a finite message space also has interesting theoretical implications — while a message-monotone equilibrium always exists when the message space is finite, as shown by Proposition 5, existence may fail when the message space is continuous. In section 5, I use an example to illustrate this possible failure of existence. Applying results in Manelli (1996), I also show that adding “cheap-talk” extension restores existence in the game with a continuous message space. Other Related Literature A number of recent and independent papers have introduced non-strategic players and other perturbations into sender-receiver games. Let me briefly discuss the connection between these papers and mine. In Kartik, Ottaviani and Squintani (2006), the receiver is possibly naive, or the sender has a cost of lying. Under the crucial assumption that the state and message spaces are unbounded, they show that a fully revealing equilibrium exists in which the sender follows a strictly monotone (hence invertible) strategy. Ottaviani and Squintani (2006) also allow the possibility that the receiver is naive but assume that the state and message spaces are bounded.5 They construct an equilibrium in which the sender’s strategy is fully revealing in a low range of the state space and partitional in the top range. While this is somewhat analogous to the characterization in my model,6 their construction does not derive how messages are used. Instead, it assumes that the sender uses messages close to one another for the partitions in the top range, so it does not explain message clustering. Moreover, Ottaviani and Squintani (2006) find that all equilibrium outcomes in the CS model are robust to their perturbation of the naive receivers and hence do not provide 5
So there are two main modeling differences between my paper and theirs. One is that my paper allows the possibility of nonstrategic types on both sides whereas their paper only allows the possibility that the receiver is naive. The other is that the message space is finite in my paper whereas it is continuous in theirs. 6 It is only somewhat analogous because in my model the strategic sender’s strategy is never fully revealing, even at the low range of the state space — the strategic sender always pools with the honest sender. Additionally, because the message space is finite, the strategic sender of different observations also pool on the same message.
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any argument for equilibrium selection in the CS model.7 Kartik (2008)8 introduces another kind of perturbation: the sender incurs a convex cost of lying. His equilibrium characterization has similar properties to those in Ottaviani and Squintani (2006): low types separate while high types pool and the receiver’s action is increasing. While this explains sender exaggeration, it does not explain message clustering.9 Moreover, Kartik’s results are derived under the refinement of monotonic D1 criterion that restricts beliefs off the equilibrium path whereas refinement is not needed in my paper because the existence of the honest sender implies that Bayes’ rule always applies. In all of the related papers discussed, the receiver’s strategy is increasing in the message received. So the interesting phenomenon of more moderate reactions to more aggressive claims never arises in these models. Another paper that incorporates boundedly rational players into communication is Crawford (2003). He looks at a binary, asymmetric, zero-sum game in which one player can costlessly signal his intention of play. By introducing “mortal types,” Crawford (2003) finds that misrepresentation of intentions can be successful sometimes. My paper is also related to a number of studies of strategic communication in which incomplete information about the players’ preferences plays an important role. Morgan and Stocken (2003) analyze how uncertainty about stock analysts’ incentives affects stock recommendations. Sobel (1985) models the dynamics of a sender’s “credibility” in a long term relationship when he can be either a “friend” or an “enemy” to the receiver. Morris (2001) explains that an advisor whose preferences are identical with the decision maker’s may have a reputational incentive to lie and be “politically correct” because he does not want to be perceived as being biased.
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The Model
The benchmark is the classic model of strategic information transmission introduced by Crawford and Sobel (1982). There are two players, a sender () and a receiver (). At the beginning of the game, privately observes the state of the world, and sends a 7 In addition to considering two-sided perturbations, my paper also explores what happens if the perturbation is only on one side. In contrast to the one-sided perturbation of the naive receivers, my paper finds that only the most informative equilibrium outcome in the CS model is robust to the perturbation of the honest sender. This highlights the role of the discipline on the receiver’s belief in ruling out less informative equilibria. 8 An earlier version of Kartik (2008) was circulated under the title “Information Transmission with Almost-Cheap Talk.” 9 In Kartik’s characterization, types separate below a cutoff, and above this cutoff, types pool on messages that have the lowest lying cost for the highest type and they may further separate by using different messages that have the same lying cost.
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costless message, , to . Upon receiving , chooses an action, , which affects both players’ payoffs. Suppose the von Neumann-Morgenstern utility functions for and are: ( ) = − ( − − )2 and ( ) = − ( − )2 . The “bias” of the sender, , parameterizes the divergence of interest between the two parties. Without loss of generality, I assume that ≥ 0. So for any , the sender’s ideal point is always higher than the receiver’s. The state space is Ω = [0 1] and the action space is = R. The players’ common prior on has distribution , with distribution function (·) ∈ C 1 and density (·). Assume 0 () ∞ for ∈ [0 1]. If () = 1 for all ∈ [0 1], then we have the leading “uniform-quadratic” case of CS. My model departs from CS by introducing two non-strategic types. On the sender’s side, there is an “honest” type who reports truthfully. On the receiver’s side, there is a “naive” type who follows the messages as if truthful. A player’s “honesty” or “naivety” is relative to the literal meanings of messages, and they are described as follows. Fix a partition of the state space Ω : I = {1 2 } where each element of the partition has a positive measure and sup{ } = inf{+1 } for = 1 − 1. Let the message space be = {1 2 −1 }.10 One can interpret ∈ as having the literal meaning: “My observation of is in .” The honest sender reports truthfully in the sense that he reports if and only if ∈ .11 Since the naive receiver follows the messages as if they were truthful, it follows that when receiving , she believes that ∈ . Given the quadratic payoff function, she chooses an action equal to the expectation of for conditional on ∈ . To simplify notation, let = (| ∈ ) = ∈ (∈ ) 12 = 1 . So the naive receiver chooses an action = when receiving . One can think of the non-strategic players as having different preferences from the 10
There are other ways to model the message space. For example, one can define the message space as a finite set = { }=1 where ∈ [0 1]. Assume that the honest sender sends the message in that is closest to her observation and the naive receiver believes that is sent by those types that find closest. The equilibrium properties found in Proposition 1 still hold under this assumption. 11 This paper assumes that the honest sender observes . An alternative assumption is that the honest sender has a noisy signal that tells him the element of the partition I . All the results go through under this assumption. 12 The broad idea behind the notion of honest sender and naive receiver is the following: there is an exogenous, commonly-known mapping between the sender’s type and messages. The honest sender is faithful to this mapping in her report and the naive receiver responds to the messages as if the sender was faithful to this mapping. The exact form of this exogenous mapping will of course affect the results, but sometimes there may be particularly natural ways to define it. In the current model, it is natural to define honesty as reporting if and only if ∈ . Moreover, because there is a one-to-one correspondence between a message and the naive receiver’s best response in my model, one can without loss of generality assume that is equal to the naive receiver’s best response. This particular assumption was chosen only to simplify notation so that the naive receiver’s response to is equal to . There are other ways to define messages without changing the model conceptually. For example, we can just let = and the results will go through.
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strategic ones. For example,13 assume that the honest sender’s payoff function is ( ) = and the naive receiver’s payoff function is 0 if ∈ and ( ) = 0 if ∈ 2 ( ) = − ( − ) . Then it is a dominant strategy for the honest sender to report when ∈ and for the naive receiver to choose = when receiving . I will call those players who have the same preference as in the original CS model the “dishonest sender” and the “sophisticated receiver”, or the “strategic players” in general. Denote by the probability that is honest and by the probability that is naive. ’s type space is = Ω × where = { }.14 ’s type space is = { }. Assume that the two elements of ’s type have independent distributions and are also independent of ’s type distribution. Let denote the players’ common prior on . The game described above is not a standard signaling game because both the sender and the receiver have private information. But the only significant role the receiver’s private information plays is changing the payoff function of the sender. So for the remainder of the paper, I will study the following signaling game Γ[( ) I ], a reformulation of the game described above. In game Γ, player first observes his type = ( ) from and then sends a signal from (as defined on page 7, is a function of I and ). Player receives , infers ’s probable type and selects an action from . The game ends and each player receives payoff ( = ), which is defined as follows. Let 1(·) be the indicator function such that 1() = 1 if = , 1() = 0 ¡ ¢ if = . Define ( ) = − ( − − )2 − (1 − ) ( − − )2 1() − ( ) (1 − 1()) and ( ) = −( − )2 . Everything in the game except is common knowledge. For notational convenience, sometimes I will write Γ for the game with the message space . Another useful piece of notation is ( () ) = − ( − − )2 − (1 − ) ( () − − )2 , the dishonest sender’s interim payoff function.
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Message-monotone Equilibria in Γ
Since the strategies of the honest sender and the naive receiver are given exogenously, I only need to find the the strategic players’ equilibrium strategies. Let () : Ω → be the dishonest sender’s (pure) reporting strategy and () : → be the sophisticated 13
Of course, there are many other payoff functions with which reporting truthfully and following the messages as if truthful are dominant strategies. Different functional form will not change the results. 14
For convenience, I will also refer to the sender of type ( ) as the type- dishonest sender.
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receiver’s (pure) action strategy. (The strict concavity of the sophisticated receiver’s utility function implies that she never plays a mixed strategy in equilibrium.) My analysis will focus on the following class of equilibrium, the message-monotone equilibrium. Definition 1 A message-monotone equilibrium in Γ is a sequential equilibrium15 in which () is weakly increasing in So a message-monotone equilibrium requires that the dishonest sender’s strategy be pure and increasing in the state. Since both the sender and the receiver have monotone preferences — they prefer higher actions for higher states, and that a fraction of receivers follow the messages literally, this is a natural restriction to make. Note that no restriction is made on the receiver’s strategy. Let me first use an example to illustrate certain properties of a message-monotone equilibrium and then provide a general characterization and interpretation.
3.1
An Example
Suppose is uniformly distributed on [0 1], = = 001 and = 005. Let = 10 and = [01( − 1) 01) for = 1 9 and = [09 1]. So = {005 015 095}. 0.95 0.85 0.75 0.65
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Figure 1: ()
15 Kreps and Wilson (1982) define sequential equilibrium only for finite games. Manelli (1996) adapts their definition to infinite signaling games and I use Manelli’s definition in this paper. The definition requires that the sender selects a best reponse for any type realization and that the receiver selects a best response to any message on and off the equilibrium path. Since the existence of the honest sender implies that every message is sent on the equilibrium path when 0, this definition of sequential equilibrium coincides with Bayesian Nash Equilibrium with the requirement of interim optimality.
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0.8 0.7 0.6
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0.4 0.3 0.2 0.1 0
0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 m
Figure 2: ()
Figure 1 shows that (0) = 1 = 005 and all the messages in are sent by a positive measure of the dishonest sender. The dishonest sender exaggerates his claims: he always sends a message higher than his observtion of (except when 095 since the highest message = 095). Figure 2 shows that the sophisticated receiver discounts the sender’s claims — she responds to a message with an action lower than the message. Moreover, as figure 1 shows, equilibrium messages cluster at the top. In particular, a threshold ¯ (= 065) exists such that a “large” measure of the dishonest sender pools at each of the three messages higher than ¯ whereas lower messages are sent by only “small” measures of the dishonest sender. Figure 2 shows that the sophisticated receiver’s responses to the top three messages are far apart, reflecting the pooling of large measures of the dishonest sender, but the responses to the lower messages are relatively flat.
3.2
Characterization of Message-monotone Equilibria in Γ
Fix 0 and ∈ (0 1). A message-monotone equilibrium always exists, as will be shown in section 5. In this section, I characterize the properties of players’ equilibrium strategies. Since Ω is a continuum and is finite and has potential messages, the sender’s increasing strategy () is a step function that can be represented by a vector x ∈ [0 1]+1 . The components of x are ( = 0 ) where 0 = 0 = 1 and ≤ +1 . Each component of x is a “jump point” of (·), i.e., the value of at which the sender jumps from one message to the next higher one. Formally, x is defined as follows: for ∈ such that (| () = ) 6= 0, let −1 = inf{ : () = } and = sup{ : () = }; for ∈ such that (| () = ) = 0, if 10
(0) (1), then = inf{ : () ≥ }, if ≤ (0), = 0, if ≥ (1), = 1. Suppose () and () are strategies in a message-monotone equilibrium. Let x ∈ [0 1]+1 represent (). The existence of the honest sender implies that every message is sent with positive probability ex ante and therefore the receiver’s posterior can always be pinned down by Bayes’ rule.16 Given the sophisticated receiver’s quadratic payoff function, her best response to is equal to the conditional expectation of . If 1 ≤ (0), then the sophisticated receiver believes that was sent by the honest sender with probability one and hence ( ) = . If ≥ (0), then the sophisticated receiver believes that was sent by the honest sender with probability ( ( ∈ )) ( ( ∈ ) + (1 − ) (| () = )) and by the dishonest sender with probability³ ((1 − ) (| () ´ = )) ( ( ∈ ) + (1 − ) (| () = )). Let R ( ) = {:()= } ( (| () = )). (So ( ) is the sophisticated receiver’s best response if she believes that was sent by only the dishonest sender.) The best response of the sophisticated receiver is a weighted average of and ( ), i.e., ( ∈ ) + (1 − ) (| () = ) ( ) ( ∈ ) + (1 − ) (| () = ) R R ∈ + (1 − ) {:()= } = ( ∈ ) + (1 − ) (| () = )
( ) =
Clearly, if −1 = , then (| () = ) = 0 and ( ) = . Suppose −1 6= . So ( : () = ) 0. Suppose 1. Let = min{ : ( () = ) 0}, i.e., is the lowest message higher than sent by the dishonest sender with positive probability. Then at the “jump point” , the dishonest sender must be indifferent between sending and .17 Without loss of generality, assume that the dishonest sender of type sends the higher message, with probability one. Let = min{ ∈ : ( : () = ) 0}. Without loss of generality, assume that if the dishonest sender of type 0 is indifferent between sending and 0 ( ), then he sends with probability 1. Then, for any ∈ such that {| () = } 6= ∅, 16
This implies that the question of how players form their beliefs off the equilibrium path, which is a prominent issue in signaling and cheap-talk games (e.g., Banks and Sobel (1987), Cho and Kreps (1987), Farrell (1993), Matthews, Okuno-Fujiwara and Postlewaite (1991)), does not arise here. 17 Suppose the indifference of type does not hold. Then WLOG, assume there exists an 0 such that ( ( ) ) − ( ( ) ) . Because ( ( ) ) − ¡ ( ( ) ) is continuous in , there¢exists ¡ a ∈ (0 − −1 ) such that for all ∈ ( ¢ − ), | ( ( ) ) − ( ( ) ) − ( ( ) ) − ( ( ) ) | . So for the dishonest sender of type ∈ ( − ), sending is strictly better than sending , a contradiction.
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it must be that ( : () = ) 0. That is, if a message is sent by some type of the dishonest sender, it must be sent by a positive measure of the dishonest sender. Let = min { ∈ : ≥ } if and = if ≥ . Let ¯ = max=1 { ( ∈ )}, i.e., ¯ is the maximum measure of the honest sender who sends a particular message. So a smaller ¯ implies a finer message space. The following proposition characterizes the properties of a message-monotone equilibrium when the message space is sufficiently fine. Proposition 1 (Properties of equilibrium strategies) Fix 0 and ∈ (0 1). Suppose () and () are strategies in a message-monotone equilibrium in Γ . There exists a 0 such that if ¯ ,18 the strategies satisfy the following properties: 1. (The sender’s on-path messages) If (0) ≤ ≤ , then −1 6= ; also, (0) ≤ . 2. (Sender exaggeration and receiver skepticism) If (0) ≤ ≤ , then ( ) ( ) and ( ) + . Furthermore, if (0) ≤ , then . 3. (Clustering of messages at the top) There exists an ¯ ∈ such that for ∈ (a) if , ¯ then ( ) ≥ −1 + and ( ) ; (b) if (0) ≤ ≤ , ¯ then ( ) −1 + . 4. (Non-monotonicity of the receiver’s actions) ¯ If (+1 − − )2 ( − − )2 , then (+1 ) ( ); Suppose +1 ≤ . If (+1 − − )2 ( − − )2 , then (+1 ) ( ). The proof is in the appendix. Part (1) says that any message higher than (0) is sent by a positive measure of the dishonest sender. It also says that the message sent by the lowest type of the dishonest sender is no higher than , which implies that any message higher than is sent by the dishonest sender in equilibrium. Part (2) says that the dishonest sender inflates his claims: the dishonest sender who sends has on average a lower observation than the honest sender who sends . In fact, a stronger result holds: except for those types who necessarily find every message lower 18
may depend on and
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than his observation (i.e., when ), the dishonest sender’s equilibrium message is always higher than his type. As a result, the sophisticated receiver is skeptical of the “face value” of any message above (0). Her best response to any above (0) is lower than , but higher than if she knew that the sender was dishonest. Part (2) also points out that the sophisticated receiver’s best response to never exceeds the ideal point of the highest type of the dishonest sender who sends . Part (3) says that one can view the dishonest sender’s strategy as having two parts. ¯ For these messages, the dishonest One part is at the top of the message space ( ). sender’s reporting strategy is reminicent of what happens in the CS model: roughly, a “large” measure (in the sense that ( ) ≥ −1 + ) of the dishonest sender pools on a particular message and the sophisticated receiver’s best response to is higher than the ideal point of the lowest type of the dishonest sender who sends .19 As to the messages below , ¯ only a “small” measure pools on a particular message. This “clustering” result implies that one should expect the messages at the higher end of the message space being used more frequently than those at the lower end. Perhaps the most surprising result is part (4), which points out that the sophisticated receiver’s best response may be non-monotone in the message she receives.20 Note that when the dishonest sender of type makes a strategic choice of what message to send, he weighs the strategic response of the sophisticated receiver against the fixed response of the naive receiver. If (+1 − − )2 ( − − )2 , then sending the higher message +1 induces a worse response from the naive receiver. To make the dishonest sender indifferent, the higher message +1 must induce a better response from the sophisticated receiver. Hence (+1 ) ( ). If (+1 − − )2 ( − − )2 , however, sending the higher message +1 induces a better response from the naive receiver. To make the dishonest sender indifferent, the higher message +1 must induce a worse response from the sophisticated receiver. Hence (+1 ) ( ).21 It may seem somewhat counter-intuitive that the sophisticated receiver’s best response might be decreasing when both the senders use increasing reporting strategies. But recall that the sophisticated receiver’s best response to is a weighted average of and ( ). While it is true that conditional on the sender being honest or dishonest, a 19
Consider a CS equilibrium with more than one steps. Except in the first step, the receiver’s action in any step must be higher than the ideal point of the lowest type of the sender in that step. In the first step, however, depending on the parameters, the receiver’s action may or may not be higher than the ideal point of the lowest type. 20 Although it is possible that () is non-monotone, it is not necessarily so. For instance, () is weakly increasing in the example in section 3.1. But one can easily find examples in which () is non-monotone. 21 Of course, a worse response could also be a response that exceeds the sender’s ideal point and therefore is too high, but this is ruled out in equilibrium for . ¯
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higher implies a higher , a higher message may also imply a higher probability that the sender is dishonest — the message is “too good to be true.” Overall, the expectation of may be lower when is higher and therefore the sophisticated receiver can rationally respond to more aggressive claims with more moderate actions. The equilibrium characterization in Proposition 1 is consistent with empirical findings. For example, Malmendier and Shanthikumar (2007) study stock analysts’ recommendations to investors both large and small. They find that while large investors account for the incentives of the analysts to bias their recommendations upward, small investors seem to take the recommendations literally. Their analysis of the NYSE Trades and Quotations database shows that although all categories of recommendation (“strong sell”, “sell”, “hold”, “buy”, “strong buy”) are issued by analysts, there are very few “strong sell” or “sell” recommendations (458%). So over 95% of all recommendations are clustered at the three positive or neutral categories. Moreover, the recommendations influence the trading behavior of both large and small investors (this indicate that there is substantial common interest between analysts and investors). But large investors are skeptical of the face value of the recommendations: in general, they shift down their reaction to positive recommendations (e.g., they hold in response to buy recommendations); they react to negative or neutral recommendations by selling, but the differential reaction to hold, sell and strong sell is insignificant.
4
Limit Equilibrium Outcomes
In this section, I explore the properties of the limit equilibrium outcomes as the perturbation of the non-strategic players vanishes. Section 4.1 analyzes the limit equilibrium outcomes when both and approach 0. Section 4.2 provides a discussion of the properties of limit equilibrium outcomes as either or approaches 0. Section 4.3 addresses the question of whether one-sided perturbation is sufficient to select the most informative equilibrium in the limit game.
4.1
Limit Equilibrium Outcomes as Both and Go to 0
Fix everything o in game Γ except for the parameters and . Consider a sequence of n games Γ with → 0 and → 0. The limit game Γ00 is a Crawford-Sobel game.22 (I also use Γ to refer to Γ00 ) 22
I use Manelli’s (1996) Definition 4 for the convergence of games. In my model, the requirement is that ( ) → 00 ( ) and ( ) → 00 ( ).
14
Denote by ( () ()) a message-monotone equilibrium strategy profile in the game Γ . Suppose the message space is sufficiently fine that the properties in Proposition 1 hold along the sequence.23 Together with the prior on the players’ types, ( () ()) generates an equilibrium outcome, a distribution on × × . Let ˆ denote a message-monotone equilibrium outcome of the game Γ . As → 0 and → 0, ˆ converges (in a subsequence) to a limit distribution ˆ. The following upper hemi-continuity result holds. Lemma 1 The limit distribution ˆ is a sequential equilibrium outcome of the limit game Γ00 . This result follows from Theorem 1 and Corollary 1 in Manelli (1996). His Theorem 1 implies that if ˆ can be generated by a strategy ˆ () of the sender, a strategy ˆ () of the receiver, and ˆ () is continuous, then the result holds. To show that these conditions are met, one can use the same argument as Manelli’s for his Corollary 1. First, it is always possible to find a strategy ˆ (·) to generate ˆ × 24 the sender’s share of the limit distribution. Second, because is finite, the receiver’s strategy is continuous ˆ (). The uniform convergence implies and () converges uniformly to a strategy that the limit distribution ˆ is generated by ˆ (). Hence ˆ is a sequential equilibrium outcome of the limit game Γ00 . Although the limit distribution is an equilibrium outcome of the limit game, the converse is not true, i.e., not all equilibrium outcomes in Γ are limit distributions of equilibrium outcomes in a sequence of converging games. As is well known, the CrawfordSobel game Γ has multiple equilibria. In particular, fix any sender bias , there is an upper bound, , on the size of an equilibrium (i.e., the number of subintervals in an equilibrium partition). Equilibria of each size from 1 through exist. Crawford and Sobel (1982) show that under a regularity condition, the condition () 25 the equilibrium that has the highest number of steps ex ante Pareto dominates the other equilibria with fewer steps and is therefore called the “most informative” equilibrium. In Proposition 2 below, we will see that (i) as the probabilities of the non-strategic types vanish, the limit equilibrium outcome corresponds to a most informative equilibrium outcome in the limit CS game, and (ii) in this limit equilibrium outcome, only the top messages are used. 23
As shown in the proof of Proposition 1, is bounded away from 0 as and approach zero. So such exists. 24 Note that although () may not converge, the vector that represents () does converge (in a subsequence). Suppose converges to a vector ˆ. Then ˆ represents the strategy ˆ (·) that generates ˆ × . 25 See Crawford and Sobel (1982) page 1444 for the definition of ( ). For example, the uniformquadratic case satisfies ( ).
15
Say an equilibrium outcome ˆ in Γ is a limit distribution of the equilibrium outcomes of the perturbed games if there exist sequences { } and { } with → 0 and → 0 such that the corresponding sequence of message-monotone equilibrium outcomes {ˆ } ¯ converges to ˆ. Recall that = max=1 { ( ∈ )}, the maximum measure of the honest sender who sends a particular message. Proposition 2 (Limit equilibrium outcome as and → 0) Suppose ˆ is a limit distribution of the equilibrium outcomes of the perturbed games. There exists a ˜ 0 such that if ¯ ˜, then (i) ˆ is a most informative equilibrium outcome in Γ and (ii) only the top messages in are sent with positive probability in ˆ. The proof is in the appendix. The following is a sketch. Sketch of proof. First, I show that ˆ is a most informative equilibrium outcome, using a result (their Proposition 3) in Chen, Kartik and Sobel (2008). They show that in a CS game, under (), only the most informative equilibrium outcome satisfies the following “no incentive to separate” (NITS) condition: the type-0 sender’s equilibrium payoff is at least as high as the payoff he gets if the receiver knew his type and responded optimally. In my model, NITS requires that type 0’s equilibrium payoff is at least − (0 − 0 − )2 . Proposition 1 implies that if the message space is sufficiently fine, NITS is satisfied along the sequence {ˆ }. This is easily seen because 1 ∈ (0 ) and 0 (1 ) ≤ 1 and hence the type-0 sender can guarantee a payoff of at least −2 by sending 1 . Because (·) converges uniformly, NITS must be satisfied in the limit equilibrium outcome ˆ. Hence ˆ is a most informative equilibrium outcome in the limit CS game. Then, I show that if a message is sent with positive probability in ˆ, then any message above it must be sent with positive probability as well. Hence, only the top messages are sent with positive probability in the limit equilibrium outcome.
4.2
Limit Equilibrium Outcomes as Either or Goes to 0
It is also useful to understand what happens if the probability of only one kind of the nonstrategic players vanishes. Below I discuss properties of the limit equilibrium outcomes as either or goes to 0. (i) Fix everything in game Γ (including 0) except . Consider a sequence of ª © games Γ with → 0. That is, in the limit, the receiver is sophisticated with probability 1, but the probability that the sender is honest is positive. In the limit equilibrium, the sender’s strategy preserves the properties in Proposition 1. In particular, any message between (0) and is stil sent with positive probability and the messages 16
cluster at the top of the message space (i.e., above the threshold ). ¯ However, in the absence of the naive receiver, the sender’s indifference condition implies that the limit equilibrium strategy () is constant below . ¯ So the receiver’s limit action strategy differs from the characterization in Proposition 1. (ii) Fixn everything in game Γ (including 0) except . Consider a sequence o of games Γ with → 0. That is, in the limit, the sender is dishonest with probability 1, but the probability that the receiver is naive is positive. In the absence of the honest sender, the messages below ¯ are no longer sent by the dishonest sender and only those messages above ¯ are sent with positive probability in the limit equilibrium. Because the probability that the receiver is naive is positive even in the limit, the sophisticated receiver’s limit equilibrium strategy is not constant be¯ the limit equilibrium strategy () satisfies low . ¯ In particular, for , 2 ¯ − 0 − )2 −(1 − ) ( () ¯ − 0 − )2 . − ( − 0 − ) −(1 − ) ( ( ) − 0 − )2 = − ( That is, the receiver’s response makes the type-0 sender indifferent among all the mes¯ sages between and .
4.3
One-sided Perturbation
Proposition 2 shows that only the most informative equilibrium outcome in the CS game is robust to the two-sided perturbation of non-strategic players, providing a rationale for selecting the most informative equilibrium in the CS game. Can we still select the most informative equilibrium outcome if the perturbation is only on one side, i.e., either the receiver is possibly naive or the sender is possibly honest, but not both? Below, I address this question by analyzing the limit equilibrium outcome when perturbation is one-sided. 4.3.1
Perturbation only on the receiver’s side ( = 0 and 0)
Let Γ0 denote a game in which the probability that the sender is honest is 0 and the probability that the receiver is naive is . Fix the parameters in Γ0 . Proposition 3 Any equilibrium outcome in the limit CS game Γ00 is also an equilibrium outcome of the perturbed game Γ0 . This result follows from Proposition 2 in Ottaviani and Squintani (2006). I refer the reader to their paper for the proof, but discuss the main idea here. Let ( 0 1 ) denote the equilibrium partition of size in the CS game Γ00 . For the game Γ0 with 0, one can construct the following equilibrium: for = 0 − 1 and the interval ( +1 ), the sender sends the message ∈ that is closest to the strategic receiver’s 17
best response ¯ ( +1 ) = (| ∈ ( +1 )).26 Call this message . For an off-theequilibrium-path message ∈ ( +1 ) that is lower (or higher) than , the belief assigned to the sophisticated receiver leads to an action that is higher (or lower) than ¯ ( +1 ). Because of the strict concavity of the sender’s payoff in the receiver’s action, this construction prevents the sender from deviating. It also generates the equilibrium partition (0 1 ). So all equilibrium outcomes in the limit CS game are robust to the perturbation only on the receiver’s side. Note that the construction of the equilibrium described in the preceding paragraph involves implausible beliefs off the equilibrium path that would have been ruled out if the sender were possibly honest. 4.3.2
Perturbation only on the sender’s side ( = 0 and 0)
In this case, one can show that only the most informative equilibrium outcome in the limit CS game is robust to the perturbation only on the sender’s side. That is, we have the following result. Proposition 4 There exists a ˆ 0 such that if ¯ ˆ, then only the most informative equilibrium outcome in the limit CS game Γ00 is a limit distribution of message-monotone equilibrium outcomes in a converging sequence of perturbed games {Γ 0 } with → 0. The proof is in the appendix. The crucial observation is that in any message-monotone equilibrium in a perturbed game Γ0 with 0, the “no incentive to separate” condition is satisfied. This is due to the discipline imposed on the receiver’s beliefs by the existence of the honest sender. Although perturbation on the sender’s side is sufficient to select the most informative equilibrium outcome in the CS game, it is worth noting that certain equilibrium properties away from the limit as characterized in Proposition 1 depend on being positive. In particular, the interesting non-monotonicity of the receiver’s actions arises only when 0.
5
Existence
One important difference between my model and related models is that my model assumes a finite message space whereas others typically assume a continuous message space. In addition to being a more realistic assumption in many communication situations, a finite 26
In Ottaviani and Squintani (2006), the message space is continuous and hence the sender sends a message that is equal to the sophisticated receiver’s optimal response, but the argument is the same.
18
message space has another advantage: it guarantees the existence of a message-monotone equilibrium, as Proposition 5 below shows, whereas a message-monotone equilibrium may fail to exist when the message space is continuous. Proposition 5 (Equilibrium existence with a finite message space) Fix 0, and ∈ (0 1), ∈ (0 1). A message-monotone equilibrium exists in Γ . The proof is in the appendix. The following is a sketch. Sketch of Proof. The proof has two main steps. First, the sophisticated receiver is restricted to choose her strategy () from the set in which + (1 − ) () is weakly increasing.27 Under this restriction, the dishonest sender’s best response is weakly increasing because of the single crossing property. Application of Kakutani’s fixed point theorem shows that the “restricted” best response correspondence has a fixed point. Second, I show that for a fixed point found in the first step, the sophisticated receiver’s strategy that corresponds to the fixed point is still a best response even without the constraint that + (1 − ) () is weakly increasing, thus establishing the existence of message-monotone equilibrium. When the message space is continuous, however, a message-monotone equilibrium may fail to exist. The example below illustrates this. }. Example: Suppose () = 1, ∀ ∈ [0 1], = 005 and = { 1 2 −1 For simplicity, let’s look at the limit message-monotone equilibrium outcome ˆ as and go to 0. The most informative CS equilibrium partition consists of 3 subintervals: 2 2 7 7 ) [ 15 15 ) [ 15 1]. Proposition 2 in section 4 implies that in ˆ, only the top three [0 15 2 ) = messages are sent with positive probability. In particular, in ˆ, { ∈ [0 15 −3 1 2 2 7 3 7 = 15 } = 15 , { ∈ [ 15 15 ) = −2 = 10 } = 13 and { ∈ [ 15 1] = −1 11 8 = 15 } = 15 . As converges to [0 1],28 the equilibrium outcomes will converge weakly to the 2 1 2 2 7 3 ) = 1 = 15 } = 15 , { ∈ [ 15 15 ) = 1 = 10 }= distribution where { ∈ [0 15 1 7 11 8 and { ∈ [ 15 1) = 1 = 15 } = 15 . But no strategy pair in the limit game Γ 3 (same as Γ except the message space in Γ is [0 1]) can generate this distribution. To generate this distribution, the sender must send = 1 for all ∈ [0 1] and the receiver 1 2 3 2 7 when ∈ [0 15 ), with = 10 when ∈ [ 15 15 ) and must respond to = 1 with = 15 11 7 with = 15 when ∈ [ 15 1], which is obviously impossible. In any finite game Γ , the top three messages in the message space are used to convey different information. But the infinite game Γ has no top three messages and equilibrium 27
The restriction is that ( + (1 − ) ()) is increasing, not that () is increasing. Indeed, as will be shown in section 3.2, () is not necessarily increasing in a message-monotone equilibrium. 28 Convergence is in the Hausdorff metric.
19
breaks down in the limit because the sender is unable to convey his private information effectively to the receiver. The nonexistence problem in infinite signaling games has been investigated in the literature. Manelli (1996) provides an ingenious way to solve the problem. The idea is to extend the sender’s strategy space and allow him to make a costless, nonbinding suggestion of action to the receiver, in addition to sending a message . Manelli (1996) terms this new game the canonical cheap-talk extension of the original game. His Theorem 2 shows that the limit distribution of sequential equilibrium outcomes in a sequence of converging games is a sequential equilibrium outcome of the canonical cheap-talk extension of the limit game. Since my Proposition 5 has shown that a message-monotone equilibrium exists whenever the message space is finite, one can apply Manelli’s Theorem 2 to obtain the following result. Result: A message-monotone equilibrium exists in the canonical cheap-talk extension of Γ, the game with a continuous message space [0 1]. In particular, going back to the example above, one can see that the canonical cheaptalk extension of Γ has a message-monotone equilibrium in which the dishonest sender 1 2 3 2 7 when ∈ [0 15 ), = 10 when ∈ [ 15 15 ) and = 11 sends = 1 and suggests = 15 15 7 when ∈ [ 15 1], and the sophisticated receiver follows the suggestions.
6
Conclusion
In this paper, I enrich the standard communication game between parties of partial common interest by incorporating non-strategic players — the honest sender and the naive receiver — into the model. Introducing the possibility that the players may be non-strategic fundamentally changes the way the game is played. The communication is no longer “cheap talk” because the naive receiver responds to messages in a literal-minded way. The existence of the honest sender implies that every message is sent with positive probability ex ante and it disciplines the receiver’s posterior belief. One contribution of the paper is that it provides an explanation for several important and interesting aspects of strategic communication that the canonical model fails to explain. For example, empirical studies on stock recommendations have found that analysts typically exaggerate their claims upwards, many investors discount the analysts’ “buy” recommendations, and positive and neutral categories are issued much more frequently than negative ones. While these aspects are absent in the predictions of the canonical model, my paper shows that they arise naturally in the perturbed model. A surprising 20
result also arises in the perturbed model: the strategic receiver’s equilibrium strategy may be non-monotone, that is, she may respond to more aggressive claims with more moderate actions. Another contribution of the paper is that it generates sharper predictions than the canonical model does even in the limit. In particular, as the probabilities of the nonstrategic players go to zero, the equilibrium outcomes in the perturbed games converge to a most informative equilibrium outcome in the limit game where the sender partially separates by using the messages at the top of the message space.
Appendix Proof of Proposition 1: Suppose −1 6= , 1 and = min{ : ( () = ) 0}. Let 4 ( ) = ( ( ) ) − ( ( ) ) = − ( − ) ( + − 2 − 2)−(1 − ) ( ( ) − ( )) ( ( ) + ( ) − 2 − 2). Monotonic4( ) = 2 ( − ) + ity of () requires that 4 ( ) 0. Hence 2 (1 − ) ( ( ) − ( )) 0. Type ’s indifference requires that 4 ( ) = 0. Combining the two conditions, we have ( ( )+ ( )− − ) ( ( ) + ( ) − 2 − 2) ≥ 0. Call this condition (∗). Condition (∗) implies that if ( )+ ( )−2 −2 (≥) ≤ 0, then ( ) + ( ) (≥) ≤ + . Recall that = min { ∈ : ≥ } if and = if ≥ . Lemma 2 For all ≥ , −1 6= . Proof. By contradiction. Suppose and there exists an ≥ such that −1 = . Then ( ) = ≥ ≥ . So the dishonest sender of type 0 = ( − ) gets his highest possible payoff by sending = and therefore he has a strict incentive to deviate to sending , a contradiction. Suppose = and −1 = . Then ( ) = ≤ . Since (0) and ( (0)) , the dishonest sender of type 0 has a strict incentive to deviate to , a contradiction. Next, I show that in a message-monotone equilibrium, the sophisticated receiver’s best response to cannot be higher than the ideal point of the dishonest sender of type — the supremum of the types of the dishonest sender who send . 21
Lemma 3 Suppose −1 6= . Then ( ) + Proof. Suppose = 1. Then obviously ( ) + . Suppose 1. Then and there must exist an such that is indifferent between sending and . I show by contradiction that ( ) + . Suppose ( ) ≥ + . Since ( ) is a weighted average of and ( ), it follows that ( ) ≤ max{ ( )}. Since ( ) , it follows that ( ) ≤ max{ }. So ( ) ≥ + implies that ( ) ≥ + . The indifference condition of type implies that ( − ) ( + − 2 − 2) + (1 − ) ( ( ) − ( )) ( ( ) + ( ) − 2 − 2) = 0. Since ≥ + and ( ) ≥ + , it must be true that ( ) ( ) and ( ) + ( ) 2 + 2. But since ( ), it follows that ( ), contradicting condition (∗). Hence, ( ) + . The next few lemmas are derived under the condition that the message space is sufficiently fine. Recall that ¯ = max{ ( )}. Note that if ¯ is small, then the distance between adjacent messages is also small. In particular, let = min{ () : ∈ [0 1])} R 0. Then ( ) = ∈ () ≥ (sup ( ) − inf ( )). Since +1 − sup (+1 ) − inf ( ) = (sup (+1 ) − inf (+1 )) + (sup ( ) − inf ( )), we have +1 − (+1 ) + ¯ ( ) ≤ 2 . So if ¯ ≤ , then (+1 − ) ≤ for all = 1 − 1. Also, if ¯ ≤ ,
2
then 1 and 1 − . It is possible that the sophisticated receiver’s best response to is higher than the ideal point of the type −1 (the infimum of the types of the dishonest sender who send ). Lemmas 4 and 5 show that it can happen only at the high end of the message space. Lemma 4 Suppsose −1 6= and ( ) ≥ −1 + . Then there exists a 1 0 such that if ¯ 1 , then ( ).
Proof. First, note that ( ) ≥ −1 + implies that ( ) ≥ −1 + . So − −1 . If ≤ ( ), then obviously ( ). Suppose ( ). ) +(1−) (|()= ) ( ) − ( ) = Note that ( ) − ( ) = (∈ (∈ )+(1−) (|()= ) ¢ ¡ (∈ ) − ( ) . (∈ )+(1−) (|()= ) R () ( − −1 ) and − ( ) 1 − , Since (| () = ) = −1 (∈ ) (1 − ). it follows that ( ) − ( ) (∈ )+(1−) R R () ( − ) ( )− ≥ −1 ( − ) = 12 ( − −1 )2 Note that − ( ) = −1 (−1 ) 1 2 0. 2 22
³ ´ ¡ ¢ ¡ ¢ (∈ ) Hence, − ( ) = − ( ) − ( ) − ( ) 12 2 − (∈ (1 − ) . ³ ´ ³ )+(1−) ´ 1 (1 − ) = 0. (If 12 2 − +(1−) (1 − ) Let 1 satisfy the condition 12 2 − 1 +(1−) 1 2 0, ∀, then let 1 = ∞. Note that as → 0, 1 → ∞) Since 2 0 and ³ ´ 1 2 − (1 − ) is decreasing in , it follows that 1 0. 2 +(1−) ³ ´ (∈ ) (1 − ) 0. So if ¯ 1 ,29 then If ( ∈ ) 1 , then 12 2 − (∈ )+(1−) − ( ) 0. Lemma 5 Suppose −1 6= and ( ) ≥ −1 + . Let = min{ : ( () = ) 0}. Then there exists a 2 0 such that if ¯ 2 , then ( ) −1 + and ( ). Proof. Note that = −1 . Suppose ¯ 1 . Then Lemma 4 implies that ( ). The indifference condition of the dishonest sender of type implies ( − − )2 + (1 − ) ( ( ) − − )2 = ( − − )2 + (1 − ) ( ( ) − − )2 . Consider the following cases. (i) Suppose ( − − )2 ≥ ( − − )2 . Then the indifference of type requires that ( ( ) − − )2 ≥ ( ( ) − − )2 2 . (ii) Suppose ( − − )2 ( − − )2 . Then there are two possible cases: either (a) ≥ and by Lemma 2 = +1 , or (b) , = and − 2 ( − ) 2 ( − −1 ). The indifference condition is ¢ ¡ ( − − )2 − ( − − )2 1− = ( + − ( ))2 + (( − ) (2 + 2 − − )) 1− ³ ´ ( ) As shown in the proof of Lemma 4, − ( ) 12 2 − ( (1 − ) . )+(1−)
( ( ) − − )2 = ( ( ) − − )2 +
Under case (iia), − = +1 − 2 ( − −1 )
2( ( )+ (+1 )) .
(+1 )+ ( )
and under case (iib), −
Also, since ( − − )2 ( − − )2 , it follows
29
The bound 1 is not tight. The other bounds found in the following lemmas are not tight either, but since the goal is to establish properties of the equilibrium strategies if the message space is suffciently fine, the bounds suffice.
23
that 0 2 + 2 − − 2 − 2. So under case (iia), 2
( ( ) − − )
¶¶2 µ µ 1 2 ( ) (1 − ) + − 2 ( ) + (1 − ) ¶ µ (+1 ) + ( ) (2 − 2) . + 1−
Under case (iib), 2
( ( ) − − )
¶¶2 µ µ 1 2 ( ) (1 − ) + − 2 ( ) + (1 − ) ¶ µ 2 ( ( ) + (−1 )) (2 − 2) . + 1−
³ ´ ³ ³ ´´2 01 2×201 1 2 (2 − 2) = 2 . (Note satisfy + 2 − 0 +(1−) (1 − ) + 1− Let 1 ³ ³ ´´2 ¡ ¢ 0 1 1 2 2 2 2 and + 2 − +(1−) (1 − ) + that as → 0, 1 → ∞.) Since + 2 ³ ´ 2 (2 − 2) is decreasing in , it follows that 01 0. If ¯ 01 , then ( ( ) − − )2 1− 2 . So in all three cases (i), (iia), (iib), ( ( ) − − )2 2 . To satisfy ( ( ) − − )2 2 either ( ) or ( ) + 2. Suppose ( ) . Since ( ) is a weighted average of and ( ) where ( ) , it follows that ( ) . Since + and ( ) + , the indifference of type implies that ( ) ( ). So we have ( ) ( ) + and ( ) , ( ) , contradicting condition (∗). Hence ( ) + 2 = −1 + 2. Below, I show that ( ) −1 + for ¯ sufficiently small. Consider the following two cases: (1) Suppose ≤ ( ). Then ( ) ≥ ( ) −1 + 2. (2) Suppose ( ). Then ( ) ( ) By Lemma 3, + ( ). Since ( ) −1 + 2, we have + and (| () = ) ≥ . 01
(∈ ) +(1−) ( ) (∈ ) +(1−) (|()= ) ( ) ≤ and (∈ )+(1−) (|()= ) (∈ )+(1−) (∈ )+(1−) ( ) 1 we have −1 + 2 (∈ . )+(1−) 00 +(1−) ( +) −1 Let 001 (−1 ) satisfy 1 00 +(1−) = −1 + 2. (Note that as → 0, 001 (−1 ) → 1 ∞.) So 001 (−1 ) 0 and if ( ∈ ) 001 (−1 ), then ( ) −1 +. As a function of −1 , 001 (−1 ) is increasing. Note that −1 −1 + ≥ . So 001 (−1 ) ≥ 001 (). Let 2 = min{ 1 01 001 ()} 0. If ¯ 2 , then in both cases (1) and (2), ( )
Since ( ) =
−1 + and by Lemma 4, ( )
24
Let ¯ = max{ ∈ : −1 6= , ( ) −1 + }. Then, if ¯ and ¯ = = . Suppose −1 6= , then ( ) ≥ −1 + . Of course, if ≥ 1, then 1. Note that the number of messages that can possibly have ( ) ≥ −1 + is finite. Let () be the number of messages that can have ( ) ≥ −1 + . If ¯ is sufficiently small, then there will be more than () messages above . In particular, ¯
recall that (+1 − )
2¯ ∀.
(1−) ), 2(()+1) (1−) min{ 2 2(()+1) 2 } 0.
Since 1 −
¯
¯
and + , if
¯
1− − + 2¯
() (i.e., if ¯
then there are at least () messages above . Let
=
We have
Lemma 6 If ¯ , then ¯ ≥ and for all , ¯ −1 6= , ( ) ≥ −1 + ¯ , then . ( ), ( ) ( ); moreover, if ¯ ≥ . Proof. Lemma 2 says that if ≥ , then −1 6= . So if ¯ , then ¯ ¯ Lemmas 4 and 5 immediately imply the following: if , then for all , ( ) ≥ −1 + and ( ). To show that ( ), note that if ¯ then sup[ ] inf[ ] + . Also, ( ) ≥ −1 + implies that ≥ −1 + . Since sup[ ] = , it follows that if ¯ , then inf[ ] −1 and hence ( ). Also, if ¯ 2 , then −1 −. So −1 ( ) − implies that −1 −1 . Induction shows that ( ) ≥ −1 + implies ( ). Since ( ) is a weighted average of and ( ), it follows ¯ , then that ( ) ( ). Induction also shows that if The proof of Lemma 5 shows that a similar result can be established when − ( ) 0. This result will be useful in proving Proposition 2, so I state it here. Corollary 1 Suppose −1 6= and − ( ) 0. Let = min{ : ( () = ) 0}. There exists a ˜ 2 () 0 such that if ¯ ˜ 2 () , then ( ) −1 + and ( ). 0 Proof. Define ˜ 1 in the ³ same ´ way as 1 was defined in the proof of Lemma 5: ˜ 1 satisfies ( + )2 + 2×2˜1 (2 − 2) = 2 . (Note that ˜ 1 is a function of and 1− ˜ 1 () 0, ∀ 0. Fix c, as → 0, ˜ 1 → ∞) Let ˜2 () = min{ 1 ˜ 1 () 00 ()} 0. 1
The proof used in Lemma 5 goes through. Lemmas 5 and 6 establish properties for messages above . ¯ For messages below , ¯ the following result holds. ¯ with , −1 6= , −1 6= and = −1 . Lemma 7 Suppose , ≤ Then ( ) ( ) . 25
Proof. Since ≤ , ¯ ( ) −1 + . If ( ) ≥ −1 + , then ( ) ( ). Since ( ) is a weighted average of ( ) and , it follows that ( ) ( ) . Now suppose ( ) −1 + . Lemma 3 implies that ( ) + . Since the dishonest sender of type is indifferent between sending and , condition (∗) implies that ( ) + ( ) ≤ + . Suppose ( ) ≥ . Since ( ) + , we have + . Type ’s indifference condition implies that ( ) ( ). But this implies that ( ) + ( ) + , a contradiction. Hence ( ) . Since ( ) is a weighted average of and ( ), it follows that ( ) ( ) . Next, I establish a number of results on (0) and ( (0)). Claim 1 (0) ≤ . Proof. If = , then clearly (0) ≤ . ¡ ¢ Suppose and suppose (0) . Then = ≥ and the dishonest ¡ ¢ sender with type 0 = − gets his highest possible payoff by sending = . So the dihonest sender of type 0 has a strict incentive to deviate, a contradiction. Claim 2 If ¯ , then ( (0)) ( (0)) (0). Proof. Suppose (0) = ≥ . Lemma 6 implies that ( (0)) 0 + . So it follows that ( (0)) ( (0)) (0). Suppose (0) . Let 0 = sup (| () = (0)) and (0 ) = (0). So 0 = −1 . Lemma 2 implies that ≤ . Lemma 7 implies that ( ) ( ) . Note that the dishonest sender of type −1 is indifferent between sending (0) and . Consider the following three cases. (i) Suppose ( − −1 − )2 ≤ ( (0) − −1 − )2 and ( ) ≤ −1 + . Since ( (0)) −1 + by Lemma 3, type −1 ’s indifference implies that ( (0)) ≥ ( ). Note that ( (0)) ( ) and ( ) ( ) by Lemma 7. Since ( (0)) is a weighted average of ( (0)) and (0), it follows that (0) ( (0)) ( (0)). (ii) Suppose ( − −1 − )2 ≤ ( (0) − −1 − )2 and ( ) −1 + . Since ( (0)) −1 + , type −1 ’s indifference implies that ( ) − −1 − ≥ −1 + − ( (0)). Since ( ) ≤ , it follows that − −1 − −1 + − ( (0)), which implies that ( (0)) 2−1 + 2 − . Since − −1 when ¯ 2 it follows that 2 − 0. Hence ( (0)) 2−1 ( (0)) and therefore (0) ( (0)) ( (0)). 26
(iii) Suppose ( − −1 − )2 ( (0) − −1 − )2 . Since (0), (0) and hence (0) − −1 − 0, it follows that −1 + . Since ≤ = min { ∈ : ≥ }, it follows that = and −1 − . Since −1 + − (0) − −1 − , it follows that 2−1 + 2 − (0). Since 2 − 0 when ¯ 2 we have −1 (0). Since ( (0)) −1 , we have ( (0)) (0). So (0) ( (0)) ( (0)). Claim 3 For all ≥ (0), −1 6= . Proof. Lemma 2 says that if ≥ , then −1 6= . Suppose (0) ≤ . Since a message sent by some type of the dishonest sender in equilibrium is sent by a positive measure of the dishonest sender, if = (0), ¢ ¡ then 6= −1 . Now suppose there exists an ∈ (0) such that = −1 . Then ( ) = ∈ ( (0) ). Claim 2 says that ( (0)) (0). So, the dishonest sender of type 0 has a strict incentive to deviate from sending (0) to sending , a contradiction. Lemma 8 Suppose ¯ . If (0) ≤ ≤ , ¯ then −1 6= , ( ) ( ) ; moreover, if , then . Proof. Lemma 7 and Claims 1, 2 and 3 imply that that for all (0) ≤ ≤ , ¯ it holds that −1 6= , ( ) ( ) . So I only need to show that if . Suppose not. Then ≥ . Since , we have +1 ≤ . Since +1 + when ¯ 2 , it follows that + +1 . Since +1 + , it follows ¯ So (+1 ) + and (+1 ) + . Since from Lemma 6 that +1 ≤ . +1 + and ( ) + by Lemma 3, type ’s indifference between sending and +1 implies that (+1 ) ( ). Since (+1 ) ≥ ( ), it follows that (+1 ) (+1 ), contradicting Lemma 7. Hence if .
Lemma 9 Suppose ¯ and (0) ≤ +1 ≤ . ¯ If (+1 − − )2 ( − − )2 , then (+1 ) ( ); if (+1 − − )2 ( − − )2 , then (+1 ) ( ). Proof. Claim 3 implies that the dishonest sender of type is indifferent between sending and +1 . Then ( (+1 ) − − )2 Suppose (+1 − − )2 ( − − )2 . ( ( ) − − )2 . So | ( (+1 ) − − ) | | ( ) − − |. Since ( ) + by Lemma 3, it follows that (+1 ) ( ). 27
Suppose (+1 − − )2 ( − − )2 . Then ( (+1 ) − − )2 ( ( ) − − )2 . So | ( (+1 ) − − ) | | ( ) − − |. Since ( ) + , it follows that either (i) (+1 ) ( ) + or (ii) (+1 ) + ( ) with (+1 ) − − + − ( ). Suppose (+1 ) + ( ). Lemma 8 implies that +1 (+1 ) and ( ) Hence +1 − − 0 and |+1 − − | | + − |, but this contradicts (+1 − − )2 ( − − )2 . Hence it must be the case that (+1 ) ( ) + . To summarize, Claim 1 and Claim 3 imply part (1) of Proposition 1. For part (2) of Proposition 1, Lemma 3 and Claim 3 imply that ( ) + for all (0) ≤ ≤ ; Lemma 6 and Lemma 8 imply that if (0) ≤ ≤ , then ( ) ( ) , ( ) + and if (0) ≤ , then . Part (3) of Proposition 1 comes from Lemma 6. Part (4) of Proposition 1 comes from Lemma 9. So if ¯ , then the message-monotone equilibrium strategies () and () satisfy the properties in Proposition 1. Proof of Proposition 2. First, I show that ˆ is a most informative equilibrium outcome in Γ by using the selection criterion introduced in Chen, Kartik and Sobel (2008), the “no incentive to separate” (NITS) condition. This condition is satisfied if the type-0 sender’s equilibrium payoff is at least as high as the payoff he gets if the receiver knew his type and responded optimally. In my model, this condition implies that the type-0 sender’s equilibrium payoff is at least as high as − (0 − 0 − )2 = −2 . Proposition 3 in Chen, Kartik and Sobel shows that under the regularity condition (M), only the most informative equilibrium outcome satisfies NITS. Below, I show that ˆ satisfies NITS. Suppose the sequence of message-monotone equilibrium outcomes {ˆ } converges to ˆ. Let ( () ()) be the equilibrium strategy profile that generates ˆ . Let 0 be such that if ¯ , then the properties in Proposition 1 holds for the game Γ . Let 0 = inf { }. The proof of Proposition 1 shows that 0 is bounded away from 0 as and approach 0. Hence 0 0. So if ¯ 0 , Proposition 1 implies that for any either (i) 1 (0) and (1 ) = 1 ∈ (0 ) or (ii) 1 = (0) and (1 ) = ( (0)) ≤ 1 ∈ (0 ). So (1 ) ∈ (0 ). Since converges uniformly to ˆ, it follows that ˆ (1 ) ∈ [0 ]. So the type-0 dishonest sender’s payoff in the limit equilibrium outcome ˆ is at least as high as the payoff he gets by sending 1 and inducing ˆ (1 ) ∈ [0 ]. So in the limit distribution ˆ, the type-0 dishonest sender’s payoff is at least as high as −2 , satisfying NITS. Hence ˆ is a most informative equilibrium outcome in Γ . Let ( 0 1 ,.., ) be the partition of Ω in the most informative equilibrium in the CS game Γ . Suppose ( = 1 ) is the receiver’s best response to step , i.e., 28
R = arg max −1 − ( − )2 (). Note that ( 1 − 1 ) is bounded away from 0. That is, there exists a 0 such that 1 − 1 0. Suppose ˆ () and ˆ () are the sequential equilibrium strategies that generate the outcome ˆ and the vector ˆ represents ˆ (). Suppose is the lowest message that ˆ ( ) = 1 and is sent with positive probability in ˆ. Then ˆ−1 = 0 and ˆ = 1 , ˆ − ˆ ( ) . Recall that ( () ()) is the equilibrium strategy profile that generates ˆ and {ˆ } converges to ˆ. Let represent (). Since converges to ˆ and converges uniformly to ˆ, it follows that for any ∈ (0 ), there exists a such that if and , then − ( ) − 0. Let ˜ = min{ 0 ˜ 2 ( − )} 0. (˜ 2 is defined in the proof of Corollary 1.) By Corollary 1 and Lemma 6, if ¯ ˜, then for all and , implies that ˆ ˆ ≥ ˆ−1 + for all , i.e., −1 + . Hence in the limit distribution every message above is sent with positive probability. Since only messages are sent with positive probability in the limit distribution, it follows that = −+1 and only the messages at the top of are sent with positive probability. Proof of Proposition 4. Since only the most informative equilibrium outcome in the CS game satisfies the NITS condition, it suffices to show that NITS is satisfied in any message-monotone equilibrium in Γ0 for 0. Suppose not. Then there exists a message-monotone equilibrium in Γ0 that violates NITS. Let () and () be the strategies in this equilibrium. Since NITS is violated, () 2 for any ∈ . This implies that for any , −1 6= . The argument in Lemma 2 still applies when = 0, which implies that for any message ∈ , −1 6= . Let ˆ = 2 . Then, if ¯ ≤ ˆ, then (+1 − ) ≤ for all = 1 − 1 and 1 , 1 − . Since (1 ) 2 and 1 , it follows that 1 (1 ) (1 ) 2 1 + . Note that if −1 (−1 ), then to satisfy the indifference condition of type −1 , either ( ) = (−1 ) or ( ) −1 + 2. Let 1 = min{ ∈ : () (1 )}. Then (1 −1 ) = (1 ) and (1 ) 1 −1 + 2 (1 −1 ) + 2. Since (1 −1 ) is a weighted average of 1 −1 and (1 −1 ) and (1 −1 ) (1 −1 ), we have 1 −1 (1 −1 ). Hence 1 (1 −1 ) + (1 ) − . Since (1 ) is a weighted average of 1 and (1 ), it follows that 1 (1 ) (1 ) 1 −1 + 2. ¢ ¡ Let denote the th jump in (). We can show, by using arguments similar to those for 1 , that whenever () jumps, it jumps by more than 2. Moreover, ¡ ¡ ¢ ¡ ¢ ¢ −1 + 2 and − . 29
Let ¯ be the last jump point. Since ¯ ¯ − and 1 − , it follows that + 2, it follows that (¯ ) (¯ ) ¯ 6= . So ¯ . Since ¯ −1 ¯ if ¯ ≤ 2 . But since ( ) is a weighted average of ( ) and , it follows that ( ) (¯ ), a contradiction. Hence, if ¯ ≤ ˆ, any message-monotone equilibrium in Γ0 must satisfy NITS. Proof of Proposition 5. Step 1. The sophisticated receiver’s strategy is (·) : → . Say that a vector ∈ represents the strategy (·) if and only if = ( ) for = 1 . For a given (·), define () = + (1 − ) (). Let be the P collection of strategies (·) such that ( ) ≥ ( ) for ≥ . Define = { ∈ P [0 1] : ∃ (·) ∈ such that represents (·)}. Note that is convex. P The dishonest sender’s strategy is (·) : Ω → . Let = { ∈ [0 1]+1 0 = 0 0 ≤ 1 ≤ ≤ −1 ≤ = 1} be the collection of that represents a nondeP creasing (·). Note that is convex. Fix (·) ∈ and . Let ( ) = ( () ). Suppose , ∈ , . Then 4 ( ) = ( ) − ( ) = − ( − ) = ( + − 2 − 2)−(1 − ) ( ( ) − ( )) ( ( ) + ( ) − 2 − 2). So (4) 2 ( − ) + 2 (1 − ) ( ( ) − ( )) = 2 ( ( ) − ( )). If (·) ∈ , then (4) ≥ 0 and (·) satisfies the single crossing condition in (; ). The type- dishonest sender’s optimization problem is max∈ ( ). Let (| (·)) be his best response correspondence. Since ( ) satisfies the single crossing property in (; ) for (·) ∈ , (| (·)) is nondecreasing in the strong set order (Lemma 1, Athey (2001)). This implies that there exists a selection (| (·)) ∈ (| (·)) for each ∈ Ω such that (·| (·)) is increasing and (·| (·)) can be represented by an P ∈ . Given (·), let ( (·) (·)) be the sophisticated receiver’s expected payoff if she P plays (·). That is ( (·) (·)) = =1 ( (·) ( )) where ( (·) ( )) = R R − ∈ ( ( ) − )2 ()−(1 − ) {:()= } − ( ( ) − )2 (). Let (x y) = P P ( (·) (·)) where ∈ represents (·) and ∈ represents (·). Define the sophisticated receiver’s “restricted” best response (·) to (·) as follows: (·| (·)) maximizes ( (·) (·)) subject to (·) ∈ . Note that ex ante optimality implies interim optimality because each ∈ is sent with strictly positive probability. Note also that the strict concavity of ( (·) ) in also implies that given (·), (·| (·)) is unique. (This can be shown by contradiction. Suppose there are two functions 1 (·) and 2 (·) ∈ and both are restricted best responses. Define 3 () = 1 () + (1 − ) 2 () ∀ ∈ ∈ (0 1). Then 3 (·) ∈ and the strict concavity of ( (·) ) in implies that 3 (·) is a better 30
response, a contradiction.) Now define the set of vectors that represent best response strategies. Γ (y) = { ∈ P : represents (·) and ∀ ∈ Ω, () ∈ (| (·)) where (·) is represented by P P ∈ }. Γ (x) = { ∈ : represents (·| (·)) where (·) is represented by P ∈ }. ³P P ´ P P . is a compact, convex subset of +1+ . Below, I apply Let = Kakutani’s fixed point theorem to show that the “restricted” best response correspondence (Γ (·) Γ (·)) has a fixed point. By Lemma 2 in Athey (2001), Γ is convex since (·| (·)) is nondecreasing in the strong set order when (·) ∈ . The argument in Lemma 3 in Athey (2001) also shows that Γ has a closed graph. Since ( | (·)) is unique for = 1 , Γ is convex. That Γ is continuous ¢ ¡ in can be shown by contradiction. Suppose a sequence x y converges to (x y) ¡ ¢ P Γ (x). Then there exists an 0 0 ∈ 0 6= and where ∈ Γ x and ∈ (x y0 ) (x y)+2. Since (·) is continuous in the elements of and , there exists ¯ ¯ ¡ ¯ ¯ ¡ ¢ ¢ a such that for all , ¯ x y0 − (x y0 )¯ and ¯ x y − (x y)¯ . ¡ ¢ ¢ ¡ Hence, for , x y0 (x y0 ) − (x y) + x y , a contradiction. So the best response correspondence (Γ (·) Γ (·)) is convex and has a closed graph and therefore has a fixed point by Kakutani’s fixed point theorem. Step 2: Let (x∗ y∗ ) be a fixed point found in step 1 where ∗ represents ∗ (·) and ∗ represents ∗ (·). Let ˜ (·) be the sophisticated receiver’s unrestricted best reply to ∗ (·). That is, ˜ (·) ˜ (·) is unique maximizes (∗ (·) (·)) without the constraint that (·) ∈ . Note that ˜ (1 ), since the receiver’s payoff function is strictly concave. Observe that if ∗ (1 ) 6= ∗ ∗ ∗ ˜ (1 ); if ( ) 6= ˜ ( ), then ( ) ˜ ( ). For any ∈ , then (1 ) ˜ ( ), then ∗ ( ) = if ∗ ( ) ˜ ( ), then ∗ ( ) = ∗ (+1 ) ; if ∗ ( ) ∗ (−1 ). ˜ (), ∀ ∈ . Below, I show by contradiction that ∗ () = Suppose not. Then there exist adjacent messages , +1 ∈ such that ∗ ( ) = ∗ (+1 ). Since +1 , it follows that ∗ (+1 ) ∗ ( ). Let 4∗ (+1 ) = ∗ (+1 ) − ∗ ( ). Since ∗ ( ) = ∗ (+1 ), we have (+1 − ) + (1 − ) (∗ (+1 ) − ∗ ( )) = 0. So 4 ∗ (+1 ) = − (+1 − ) (+1 + − ∗ (+1 ) − ∗ ( )). Note that 4∗ () 0 if and only if − ∗ ( ) () ∗ (+1 ) − +1 . If 4 ∗ 0, then ∀ ∈ Ω, the dishonest sender strictly prefers ( ∗ ( )) to (+1 ∗ (+1 )). Therefore (|∗ () = +1 ) = 0 and ˜ (+1 ) = +1 . Likewise if 4 ∗ 0, then ˜ ( ) = . (|∗ () = ) = 0 and 31
Consider the following cases. (1) Suppose ≥ ∗ ( ). Since +1 and ∗ ( ) ∗ (+1 ), we have ˜ (+1 ) = +1 ≥ ∗ ( ) − ∗ ( ) ∗ (+1 ) − +1 and 4 ∗ 0. So ∗ (+1 ), which implies that ∗ (+1 ) = ∗ (+2 ). By repeating the argument, we ˜ ( ) = ∗ ( ), a contradiction. have ˜ (+2 ) = +2 ∗ (+2 ) and so on. So (2) Suppose ∗ ( ) and (|∗ () = ) = 0. Then ˜ ( ) = . ˜ ( ) = . Then ∗ ( ) − +1 − ∗ ( ) +1 − (2a) Suppose ∗ ( ) ≤ ˜ (+1 ) = +1 ≥ ∗ ( ) ∗ (+1 ), which implies ∗ (+1 ), i.e., 4∗ 0. So that ∗ (+1 ) = ∗ (+2 ). Since +1 ∗ (+1 ), we are back to case (1). ˜ ( ) = . Then ∗ ( ) = ∗ (−1 ) and −1 (2b) Suppose ∗ ( ) implies that ∗ (−1 ) ∗ ( ). Then ∗ (−1 ) − −1 − ∗ ( ). So ˜ (−1 ) = ˜ (−2 ) = −2 ∗ (−2 ) and −1 ∗ (−1 ). By repeating the argument, we have so on. So ˜ (0 ) = 0 ∗ (0 ), a contradiction. ˜ (+1 ) = +1 (3) Suppose ∗ ( ) and (|∗ () = +1 ) = 0. Then ∗ ∗ ˜ (+1 ) = +1 . Then ( ) − +1 − ∗ (+1 ). (3a) Suppose (+1 ) ≥ ˜ ( ) = +1 ≤ ∗ (+1 ) ∗ ( ) and we are back to case (2b). So 4 ∗ 0 (3b) Suppose ∗ (+1 ) ˜ (+1 ) = +1 . Then ∗ (+1 ) = ∗ (+2 ) and we are back to case (1). (4) Suppose ∗ ( ) and (|∗ () = ) 0 (|∗ () = +1 ) 0. Then the dishonest sender of type ∗ is indifferent between ( ∗ ( )) and (+1 ∗ (+1 )). That is, 4∗ (+1 ∗ ) = 0. So + +1 = ∗ ( ) + ∗ (+1 ). Since ∗ (+1 ) ∗ ( ) and +1 , this implies that ∗ (+1 ) +1 and ∗ ( ) . (4a) Suppose ˜ (+1 ) ∗ (+1 ). Then either +1 = and we have a contradiction or ∗ (+1 ) = ∗ (+2 ). Since ∗ (+1 ) +1 , we are back to case (1). (4b) Suppose ˜ (+1 ) ≤ ∗ (+1 ). Note that ˜ (+1 ) min{+1 ∗ }. Since ˜ (+1 ) ∗ . Since ∗ ( ) ∗ (+1 ), +1 ∗ (+1 ), we have +1 ∗ (+1 ) ≥ ˜ ( ) max{ ∗ }. Since ∗ ( ) ∗ and we also have ∗ ( ) ∗ . Note that ˜ ( ). Hence, either = 1 and we have a ∗ ( ) , this implies that ∗ ( ) contradiction or ∗ ( ) = ∗ (−1 ) and we are back to one of the above cases. To summarize, ∗ () = ˜ (), ∀ ∈ . Since ∗ (·) and ∗ (·) are best responses to each other and ∗ (·) is increasing, a message-monotone equilibrium exists in Γ .
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