Sender s Small Concern for Credibility and Receiver s Dilemma

Transcription

1 April 2012 Sender s Small Concern for Credibility and Receiver s Dilemma Hanjoon Michael Jung The Institute of Economics, Academia Sinica Abstract We model a dilemma that receivers face when a sender has small concern for credibility. Consider a sender and multiple receivers. Suppose that the sender has concern for his credibility in reporting truthfully and his credibility concern is relatively small compared to his concern for the outcomes. Then, the sender could report untruthfully to achieve his favorite outcome. However, the sender s credibility concern still boosts his incentive to report truthfully which is referred to as Boosted Incentive Truthfulness (BIT). Next, due to this BIT, the receivers cannot completely ignore the sender s signals. As a result, the receivers get to play only the sender s favorite, even when they have their subjective priors. This is the Receiver s Dilemma. We find general situations where receivers would fall into this dilemma and formulate them as the ε-uncertain game. Then, we present the condition for the Receiver s Dilemma in this ε-uncertain game. Finally, we provide three examples of the Receiver s Dilemma. JEL Classification Number: C72, D82, D83. Keywords: Boosted Incentive Truthfulness, ε-uncertain Game, Information Manipulation, Receiver s Dilemma, Small Credibility, Subjective Prior. The Institute of Economics, Academia Sinica, 128, Sec. 2, Academia Road, Nangang, Taipei 115, Taiwan address: hanjoon@econ.sinica.edu.tw

2 1 Introduction We model a dilemma that receivers face when a sender has small concern for his credibility. Consider an information transmission situation between one sender and possibly many receivers. Suppose that the sender has concern for his credibility in reporting truthful information. Here, his credibility concern is relatively small compared to his concern for the outcomes. Accordingly, the sender could report untruthful information to influence the receivers to play his favorite outcome. However, the sender s credibility concern still boosts his incentive to report truthfully. Consequently, the sender would intend to reveal truthful information more, and hence the receivers cannot completely ignore the sender s reports. As a result, the receivers get to play only the sender s favorite outcome, as partially shown by Jung (2009a) in his specific models. This is the Receiver s Dilemma. In this study, we formulate general situations and conditions under which receivers fall into this dilemma. To understand the intuition behind the result of the Receiver s Dilemma, consider the following simple game. There are three players who are action taker 1, action taker 2, and a sender. Each action taker has two actions, L and R. Furthermore, there are two states which are a main state and an abnormal state. When the action takers choose their actions, they do not know what state has occurred. The sender, however, can observe the actual state and signals to the action takers either N, which notifies them that the main state has occurred, or A, which notifies them the abnormal state has occurred. The payoffs to action takers 1 and 2 are as follows. In these matrixes, action taker 1 chooses a row and action taker 2 a column, 1

3 When it is the main state When it is the abnormal state L R L R L 1, 1 0, 0 R 0, 0 ω, ω L 1, 1 1, 0 R 0, 0 0, 1 Table 1: Payoff matrixes of action takers 1 and 2 such that ω 10 9 where the first entry in each cell is action taker 1 s payoff for the corresponding actions and the second entry action taker 2 s. In the main state, the action takers play a Pareto-coordination game. Hence, they get positive payoffs if they succeed in coordinating their actions. In the abnormal state, action taker 1 has a dominant action L. In addition, each action taker has his subjective prior on the states. So, we assume that action taker 1 believes the abnormal state happens with probability p (0, 10 9 ] and action taker 2 believes it happens with probability q (0, 10 9 ]. Then, in the absence of the sender, there are two pure-strategy equilibrium outcomes which are LL and RR, where the outcome LL denotes that both players choose the action L and the outcome RR denotes that both players choose the action R. Here, both action takers have a strong preference for the outcome RR to the outcome LL because they would get at least in the outcome RR and would get 1 in the outcome LL. Regarding the sender s payoff related to the outcomes, he gets 1 for each taken action L. That is, he gets 2 if both action takers choose L, he gets 1 if only one action taker chooses L, and he gets zero if none chooses L. In addition, the sender has relatively small 2

4 concern for his credibility in reporting truthful information. Hence, if he reports truthfully, he would get α 0 extra payoff and, if he reports untruthfully, he would lose β 0 extra payoff. Here, we assume 2 > α + β > 0. That is, the sender cares for his credibility, which is formulated as α + β > 0. However, his concern for his credibility, which is denoted by α + β, is relatively small compared to his concern for the outcomes, which is as great as 2 since this is his highest payoff related to the outcomes. Accordingly, the sender could report untruthfully to achieve his favorite outcome LL. Figure 1 below depicts the extensive form of this simple game with the three players. In the figure, the first entry in each parenthesis denotes action taker 1 s payoff for the corresponding actions, the second entry action taker 2 s, and the third entry the sender s. Figure 1: Extensive form with the sender 3

5 We focus on pure-strategy perfect Bayesian equilibria for simplicity s sake. Then, this game has equilibria and, in all these equilibria, the sender s favorite outcome LL is the unique equilibrium outcome. Therefore, in equilibrium, the action takers lose their favorite outcome RR, which indeed they could achieve in the absence of the sender. To see why they cannot achieve the outcome RR, suppose first that both action takers try to ignore the sender s signals and to play only the action R. That is, they plan to choose the action R regardless of the sender s signals. Then, the outcome under these strategies is only RR. In this case, if the sender reports untruthfully, he would gain nothing but lose his credibility. Concretely, responding to these action takers strategies, the sender would get α for reporting truthfully and would get β for reporting untruthfully. Since α + β > 0, which embodies that the sender cares for his credibility, he would report truthfully. That is, the sender signals N when it is the main state, and he signals A when it is the abnormal state. Next, by observing the sender s signal, the action takers can figure out the true state. Hence, when action taker 1 observes the sender s signal A, he can be sure that it is the abnormal state. Thus, action taker 1 has an incentive to deviate from the action R to the action L since he has a dominant action L in the abnormal state. As a consequence, the action takers strategies suggested here cannot be parts of an equilibrium. Second, suppose that both action takers try to play the game according to the sender s signals. That is, they plan to choose the action R when the sender signals N and plan to choose L when the sender signals A. Then, the outcome would be RR if the sender signals N and it would be LL if the sender signals A. In this case, if the sender signals A, he might 4

6 lose his credibility, but he can achieve his favorite outcome LL. If he signals N, he could maintain his credibility, but he would be faced with his worst outcome RR. Concretely, the sender would get at least 2 β by signalling A and would get at most α by signalling N. Hence, the sender would signal only A in every state because his credibility concern is relatively small compared to his concern for the outcomes, which is formulated by 2 > α + β. Next, responding to this sender s strategy, the action takers might have no incentive to deviate from their strategies. This is because the sender signals only A, and thus he does not reveal any information through his signals. In this case, each action taker would be better off choosing L since the other action taker would play L. Furthermore, the sender never signals N, so the action takers cannot employ Bayes rule to update their beliefs conditional on the signal N. In this case, according to the solution concept of the perfect Bayesian equilibrium, the action takers may adopt any arbitrary beliefs. As a result, the action takers can be better off choosing the action R responding to the sender s signal N. Therefore, the strategies suggested here can be an equilibrium and, in this equilibrium, the unique equilibrium outcome is LL in which the action takers choose the action L only. Likewise, we can show that, if a strategy profile is a pure-strategy perfect Bayesian equilibrium, it has the unique equilibrium outcome LL. Therefore, in pure-strategy 1 equilibrium, the action takers lose their favorite outcome RR while the sender always achieves his favorite LL. We refer to this situation that the action takers are faced with as the Receiver s Dilemma. 1 Note that, in mixed-strategy equilibrium, the action takers can achieve their favorite outcome RR with positive probability that is less than However, the action takers strictly prefer the LL outcome to the mixed-strategy equilibrium. This is because the action takers expected payoffs in the mixed-strategy equilibrium are less than one while their payoffs in the outcome LL are one for sure. 5

7 In this simple game, only the action takers can choose actions. Then, their actions determine outcomes and the action takers payoffs depend only on the outcomes. Hence, seemingly, they can choose their own payoffs. Moreover, the action takers are supposed to clearly comprehend the setting of the game, and thus they know both that the sender can report untruthfully and that, in fact, they do not benefit from the sender s signals at all. Nevertheless, they still cannot ignore the sender s signals and thus cannot choose their actions according to their preferences. As a result, they cannot help playing the sender s favorite outcome 2 in equilibrium. This is the dilemma that the action takers, or receivers in general, are confronted with. This Receiver s Dilemma results from the sender s incentive to report truthfully, which has been boosted by his concern for the credibility. In the simple game, the sender has concern for his credibility in reporting truthful information. This sender s credibility concern boosts his incentive to report truthfully, which is embodied by the condition α + β > 0. We refer to this boosted incentive to report truthfully as Boosted Incentive Truthfulness or simply as BIT. Then, this BIT makes the sender report truthfully when responding to some ignoring strategies of the action takers, such as playing R regardless of the sender s signals under which the action takers ignore the sender s signals and try to play the action R only. Next, these sender s truthful signals reveal true information, and as a result these signals give rise to the incentives for the action takers to follow the sender s signals, which means that the action takers cannot completely ignore the sender s signals. On the other 2 Note that, even when we introduce a cheap-talk process between the action takers, this result does not change. That is, still the sender s favorite is the unique equilibrium outcome. 6

8 hand, the sender s Boosted Incentive Truthfulness does not dominate his incentive to achieve his favorite outcome, which is formulated by the condition 2 > α + β. Hence, he would report untruthfully if he can influence the action takers to play his favorite LL. Then, since the sender does not reveal any information through his signals, the action takers could not find any incentive to change their strategies, and thus they can play the outcome LL in equilibrium. Consequently, the sender s BIT restricts the equilibrium outcomes, and therefore it results in the Receiver s Dilemma. Originally, this study on the Receiver s Dilemma stems from Siddiqi (2007) and Jung (2009a and 2009b). Jung (2009a) studied information manipulation through the media and showed that, if the media outlet cares about its credibility, then it can successfully manipulate information without being detected. Siddiqi (2007) applied this Jung s idea to stock markets to explain stock price manipulation by intermediaries who can be regarded as senders. In addition, Jung (2009b) presented a simple model that includes only two players to show the paradoxical role of the sender s credibility concern. These studies, however, considered only particular models whose application would be limited to specific cases. Our study, therefore, intends to find general situations and conditions under which action takers would fall into the Receiver s Dilemma. We formulate these general situations as the ε-uncertain game and find the condition for the Receiver s Dilemma in this ε-uncertain game. The rest of the paper is organized as follows. Section 2 discusses some related literature. Section 3 formally defines the ε-uncertain game. Section 4 reveals the properties of the ε-uncertain game and presents the condition for the Receiver s Dilemma. Here, we consider 7

9 only pure-strategy perfect Bayesian equilibria for simplicity. Finally, Section 5 provides three examples to show the practical application possibility of the results in this study. 2 Related literature The ε-uncertain game can be categorized as a signalling game, such as Spence (1973), Bhattacharya (1979), and Banks and Sobel (1987). Its contribution to this literature is to formally model general signalling situations under which, due to the credibility concern, the sender can influence the receivers decision and he could achieve his favorite outcome. Consequently, it is intended to provide a systematic study on the role of the sender s credibility concern in a signalling game. The ε-uncertain game, however, differs from the classical signalling game developed by Spence (1973). This is because, in his model, a sender uses information only about his own type while, in the ε-uncertain game, the sender can use any information 3 that receivers need so as to find their optimal strategies. In addition, the ε-uncertain game differs from cheap-talk games, such as Crawford and Sobel (1982), Farrell and Rabin (1996), Aumann and Hart (2003), and Goltsman et al. (2007). In cheap-talk games, talk is irrelevant to a sender s payoff. So, the sender does not have an incentive to report truthfully when responding to receiver s ignoring strategies, which in turn implies that the receiver can ignore the sender s talk. As a result, only if the sender and the receiver have a common interest, can the cheap-talk between them have influence on the outcomes. In the ε-uncertain game, on the other hand, reports are relevant to the 3 In the ε-uncertain game, since the sender does not necessarily report the information about himself, we could not employ the solution concept of the forward induction by Kohlberg and Mertens (1986) and Van Damme (1989). 8

10 sender s payoff. Thus, the sender is endowed with his Boosted Incentive Truthfulness. Then, because of this BIT, receivers cannot completely ignore the sender s signals. Therefore, even when the sender and the receivers have contradictory preferences, the sender s signals still have influence on the outcomes. Kartik et al. (2007), Kartik (2009), and Hodler et al. (2010) considered a sender s credibility concern in information transmission models like in the ε-uncertain game. Concretely, they assumed that the sender would be punished for untruthful reports. In their models, however, the sender s influence on receivers was limited so that the sender might not achieve his favorite outcome. First, in the models by Kartik et al. (2007) and Kartik (2009), the sender has different favorite outcomes in different states, thus he tries to influence receivers to play different actions in different states. Accordingly, the sender s influence on rational receivers is limited, and as a result, he cannot make the rational receivers play his favorite. Second, in the model by Hodler et al. (2010), the sender has the same favorite in all states, but his favorite is not an equilibrium outcome. Since the sender is not an action taker, his influence is limited within equilibrium outcomes. That is, he could influence a receiver to play one of the equilibrium outcomes, but he cannot make the receiver play non-equilibrium outcomes unless he reveals truthful information. As a consequence, the sender partially achieves his favorite by revealing part of the truthful information. In the ε-uncertain game, we organize the sender s preference over the states so that he has the same favorite in most of the states. In addition, the sender s favorite is an equilibrium outcome. Therefore, the sender can influence the receivers to play his favorite. 9

11 Furthermore, Sobel (1985), Celetani et al. (1996), and Ely and Välimäki (2003) studied a sender s credibility. In Sobel s model, a sender has multiple types and has an incentive to pretend to be a truth-telling type in order to improve his future payoffs. Then, because of his concern for the future payoffs, the sender adopts truth-telling signals. This is how a sender appears to care about his credibility without any direct concern for his credibility. Consequently, he studied the situation in which a sender s concern for his credibility is endogenously generated without any intrinsic concern for it. Celetani et al. (1996) and Ely and Välimäki (2003) also modeled this endogenously generated credibility concern. The current study, on the other hand, assumes that the sender directly cares about his credibility and focuses on the effect of the sender s credibility concern on the receivers behavior. Finally, in the ε-uncertain game, the uniqueness of the equilibrium outcome depends on the setting about the abnormal states that occur with small probability indeed. Rubinstein (1989), Carlsson and van Damme (1993), and Morris and Shin (1998) also showed that the small possibility of uncertainties could lead to a unique equilibrium in their models. In addition, van Damme (1989) proved that introducing even useless actions could eliminate all the equilibria except one. The underlying principle of the uniqueness in their models, however, differs from that in the ε-uncertain game. In their models, the uncertainties or the useless actions make some of the actions dominated. Hence, they could have a unique equilibrium by iteratively eliminating those dominated actions. On the other hand, in the ε-uncertain game, the abnormal states would not make any action dominated. Nevertheless, the abnormal states give rise to an incentive for the receivers to play the sender s favorite. 10

12 Therefore, by strategically revealing the information about the abnormal states, the sender could influence the receivers to play the unique outcome, which is the sender s favorite. 3 ε-uncertain game The ε-uncertain game consists of multiple states. As Osborne and Rubinstein (1994) pointed out, the notion of a state is often given various meanings in the economic literature. At one extreme, it represents a description of the contingencies which can affect players optimal decisions. This, for example, is the case when a state determines payoff parameters or types of the players. At the other extreme, a state represents a full description of the uncertainties which can affect players optimal decisions. In this case, the uncertainties can be placed not only on payoff parameters or types, just as in the former, but also on the players taken and hidden actions, as shown in Epstein and Wang (1996), Asheim and Dufwenberg (2003), and Baliga and Sjöström (2009). In this study, we adopt the latter. Therefore, a state in the ε-uncertain game is viewed as a full description of the uncertainties. Then, a non-empty metric space Θ is defined as the set of the states. Players in the game have their own subjective priors (or subjective probability measures) on the class of the Borel subsets ß(Θ) of Θ. The set of the states Θ includes the main state, denoted by θ m, and abnormal states, denoted by θ ab. The main state θ m represents the state that happens with major probability so that it has the main influence on the equilibrium outcomes. Accordingly, we assume that every subjective prior assigns the main state θ m the probability which is in the interval [1 ε, 1) for some small ε > 0. That is, every player believes that the main state 11

13 θ m happens with at least the probability 1 ε and all the other uncertain states happen with at most the probability ε, which shows why this game is referred to as the ε-uncertain game. Note that, since the main state happens with probability at least 1 ε, its occurrence cannot depend on the players taken and hidden actions. In addition, an abnormal state θ ab represents the state in which a sender s favorite outcome is the unique equilibrium outcome if this state actually occurs and the players correctly expect their occurrence. The existence of the abnormal states is an essential condition for the receiver s dilemma in the ε-uncertain game because it enables the sender to influence receivers decision-making process. Namely, because of their existence, the receivers cannot completely ignore the sender s signals if he reports truthfully, and as a consequence, the sender could influence the receivers to play his favorite outcome by using this information. A non-empty Borel set Θ ab is defined as the set of abnormal states 4 that the sender can detect. Then, to make this existence condition effective, we assume that every player believes that Θ ab happens with positive probability. Since a state is a full description of the uncertainties, this assumption means both that players actions which condition the abnormal states in Θ ab are feasible and that every subjective prior assigns the abnormal state set Θ ab positive probability conditional on the players taking those actions that condition the states in Θ ab. These settings on the states make the ε-uncertain game a perturbed game of the main state. In the ε-uncertain game, the main state happens with large probability, which is at 4 Note that, if the sender cannot detect a state, then the state does not belong to Θ ab even when it is an abnormal state. 12

14 least 1 ε, and abnormal states occur with small probability, which is at most ε. Hence, we can view the ε-uncertain game as resulting from disturbing the main state so that it may also have the abnormal states with small, but positive, probability. The form of disturbance in the ε-uncertain game, however, is different from the forms in existing literature, such as Harsanyi (1973), Selten (1975), and Carlsson and van Damme (1993). This is because the ε-uncertain game only requires the abnormal states to occur with positive probability while those other perturbed games designate specific forms of disturbing either payoffs or strategies. The setting of the players is formulated in Subsections 3.1 and 3.2. The players consist of action takers and a sender. In this study, we are interested in general circumstances of the action takers in which the sender can influence the action takers to play his favorite outcome. Therefore, the ε-uncertain game is designed to capture the action takers setting as general, particularly in the main state θ m, as possible while maintaining the sender s influence on them. To focus on this generality of the action takers setting, we consider a relatively simple setting of the sender. In addition, we take the standard approach to the players information about the setting in the economic literature, except that about the subjective priors. Hence, we assume that all players fully understand the setting of the ε-uncertain game and they know their own priors. However, regarding the other players priors, they know only the assumption that every prior assigns the main state θ m at least probability 1 ε and the abnormal states in Θ ab can happen with positive probability. The timeline of the ε-uncertain game is presented in Subsection

15 3.1 Setting of the action takers There are a finite number of action takers, denoted by i = 1, 2,..., I. Out of the action takers, action taker i {1,..., R} such that 1 R I does not have the information on the actual states when he takes his action and thus is referred to as a receiver, which means an action taker who needs to receive the sender s information. Action taker i {R + 1,..., D 1} such that R D 1 I, on the other hand, can figure out some of the actual states including abnormal states θ ab Θ ab when he takes his action and so is referred to as a non-receiver. More concretely, if a subset Θ i ( Θ) is the set of states whose occurrence non-receiver i can observe, then non-receiver i can observe the occurrence of individual states in Θ i and we always have 5 Θ ab Θ i. Furthermore, action taker i {D,..., I} has actions that condition the states in Θ and hence is referred to as a decider. Note that either R = D 1, which denotes no non-receiver in the game, or D 1 = I, which denotes no decider in the game, or both are possible. In addition, the sender could be one of the non-receivers so that he tries to influence the other action takers to play his favorite outcome by using his private information as exemplified by Jung (2009b). Given a state θ Θ, each action taker i has a non-empty set of feasible actions A θ i. Since receiver i ( R) and decider i ( D) do not know the actual states, they have A θ i = A θm i and A θ i =, respectively, for every θ Θ where A Aθm θm i i and A θm i are the sets of their feasible actions in the main state θ m. Non-receiver i ( {R + 1,..., D 1}), on the other hand, can figure out some of the actual states, including the abnormal states, hence 5 Hence, it is possible that a non-receiver could figure out no state other than the abnormal states in Θ ab. 14

16 he could have A θ i A θ i for some distinct states θ and θ. In addition, action taker i has a set of his strategies Π i. Then, in the absence of the sender, receiver i ( R) and decider i ( D) have Π i = A θm i and Π i = A θm, respectively. Next, action taker i s prior is a measure i µ i (, ) : I k=d Π k ß(Θ) [0, 1] such that, given each strategy profile of the deciders a d ( ) I k=d Π k, the measure µ i (a d ( ), ) is a probability measure on ß(Θ). Then, according to the setting of the ε-uncertain game, we have µ i (a d ( ), {θ m }) 1 ε for every a d ( ) I k=d Π k and µ i (a d ( ), Θ ab) > 0 for some a d ( ) I k=d Π k. Action taker i s payoff function is defined as, given a state θ Θ, U i ( ; θ) : I k=1 Aθ k R. Finally, we can define action takers conditional expected payoff functionals as follows. Let a non-empty set Π s be the set of the sender s strategies. Here, for notational convenience, we use payoff functions that are defined on the set of the players strategy profiles. That is, for each action taker i, a function Ūi(,, ) : Θ ( I k=1 Π k) Π s R is defined as follows. Given (θ, a( ), s( )) Θ ( I k=1 Π k) Π s, suppose that the action takers strategy profile a( ) and the sender s strategy s( ) induce an outcome a I k=1 Aθ k in the state θ. Then, we have Ūi(θ, a( ), s( )) = U i (a; θ). We assume that, given any strategy profile (a( ), s( )) ( I k=1 Π k) Π s, each Ūi(, a( ), s( )) is bounded above or bounded below and ß(Θ) measurable, which guarantees that Ūi(, a( ), s( )) is integrable. Then, given a Borel set Θ ( ß(Θ)), we define action taker i s expected payoff functional conditional on reaching the set of states Θ as a function E i (, ; Θ ) : ( I k=1 Π k) Π s R (= R {, }) such that, for each (a( ), s( )) ( I k=1 Π k) Π s, µ i (a d ( ), Θ ) E i (a( ), s( ); Θ ) = Ūi(θ, a( ), s( ))µ Θ i (a d ( ), dθ) where a d ( ) is the deciders strategies in a( ). For notational simplicity, in the absence of 15

17 the sender, we denote that, given Θ ß(Θ), action taker i s conditional expected payoff functional is E i (a( ); Θ ) for any a( ) I k=1 Π k. Regarding the setting related to the main state θ m, we assume that, in the absence of the sender, there exist pure-strategy equilibrium outcomes in θ m and these are indeed main parts of the pure-strategy equilibria in the whole set of the states Θ. In the ε-uncertain game, we introduce the sender, who has small concern for his credibility in reporting truthful information, into the setting of the action takers and study his influence on the action takers behavior. Here, we focus on the change in the action takers behavior in purestrategy equilibrium outcomes of the main state θ m. However, before we introduce the sender, that is, in the absence of the sender, the ε-uncertain game might not have a purestrategy equilibrium. Moreover, the ε-uncertain game in the absence of the sender can be an infinite game, which contains infinite states or strategies. An infinite game 6 might not have any equilibrium, which in turn means that we cannot study the sender s influence on the action takers decisions by introducing him into the action takers setting. Therefore, to ensure the existence of a pure-strategy equilibrium so that we can properly examine the sender s influence, we assume that, in the absence of the sender, there exist equilibrium outcomes in θ m and the ε-uncertain game has these outcomes as main parts of the purestrategy equilibria in Θ. Formally, we assume as follows. Let a j I i=1a θm i for each j N be an action profile in θ m. Then, for some J N, a non-empty set {a j } J j=1 is assumed to be the set of all possible pure-strategy equilibrium 6 For the information about the equilibrium existence condition in infinite games, please refer to Milgrom and Weber (1985) and Balder (1988). 16

18 outcomes in the main state θ m of the ε-uncertain game in the absence of the sender. More specifically, this assumption is made through the following two assumptions. That is, the set {a j } J j=1 contains all the action profiles in θ m that satisfy the two assumptions. First, given any action profile a j {a k } J k=1, we assume that, in the absence of the sender, there exists the action takers strategy profile a ( ) that contains the actions a j in the main state θ m such that 1) we have E i (a ( ); Θ ) = max a i ( ) Π i E i (a i ( ), a i( ); Θ ) for every action taker i and for every set of states Θ containing θ m, that is, θ m Θ Θ, where (a i ( ), a i( )) = a ( ) and 2) if non-receiver i (> R and < D) can observe the occurrence of a state θ Θ, we have U i (a ; θ) = max U i (a i, a i; θ) a i Aθ i where a is the action takers actions in the state θ according to a ( ) and (a i, a i) = a. Second, given any action takers strategy profile a( ) I k=1 Π k that contains their actions a I i=1a θm i in the main state θ m such that a / {a j } J j=1, we assume that, in the absence of the sender, there exists action taker i such that sup E i (a i( ), a i ( ); Θ ) > E i (a( ); Θ ) (1) a i ( ) Π i for any set of states Θ containing θ m where (a i ( ), a i ( )) = a( ). An abnormal state θ ab Θ ab requires a special setting as well. An abnormal state θ ab is defined as the state that contains only one equilibrium outcome, which is the sender s favorite outcome, if θ ab actually occurs and the occurrence of the abnormal states in Θ ab is 17

19 expected. In the ε-uncertain game, we assume 7 that a 1 is the sender s favorite outcome. Then, we model the action takers setting in the abnormal states through the following three conditions. First, in each abnormal state θ ab Θ ab, the non-receivers can figure out the occurrence of θ ab. Thus, the non-receivers are regarded as having an incentive to play actions in a 1. That is, in each abnormal state θ ab, every non-receiver i (> R and < D) can play a 1 i, which means a 1 i A θ ab i, and this action a 1 i is indeed his dominant action. Second, responding to the non-receivers actions a 1 nr and the deciders actions a 1 d, the receivers have the best responses a 1 r in each abnormal state θ ab Θ ab where (a 1 r, a 1 nr, a 1 d ) = a1. Formally, we assume that, in the absence of the sender, there exists the action takers strategy profile a 1 ( ) that contains the actions a 1 in each θ ab Θ ab such that E i (a 1 ( ); Θ ab ) = max a i ( ) Π i E i (a i ( ), a 1 i( ); Θ ab ) for every receiver i ( R) where (a 1 i ( ), a 1 i( )) = a 1 ( ). Third, given any action takers strategy profile a( ) I k=1 Π k that contains the non-receivers actions a 1 nr and the deciders actions a 1 d in each abnormal state θ ab Θ ab and that contains the receivers actions a r R k=1 Aθm k such that a r a 1 r, there exists receiver i in the absence of the sender such that sup E i (a i( ), a i ( ); Θ ab ) > E i (a( ); Θ ab ). a i ( ) Π i 7 The reason for this assumption will be discussed in the next subsection. In addition, since the abnormal states can occur only after the deciders choose their actions a 1 d, these states do not require an additional condition on the deciders setting, except the feasibility of a 1 d. 18

20 3.2 Setting of the sender The basic setting of the sender is simpler than that of the action takers. That is, in the ε-uncertain game, there is only one sender, and the sender has only two signals. This basic setting, however, can be extended to cover general circumstances while preserving the same results as in the ε-uncertain game, as partially shown by Jung 8 (2009a). The sender is denoted by s. The sender s two signals are normal and abnormal. Here, the abnormal signal signifies that an abnormal state θ ab in Θ ab occurs, and the normal signal signifies that a state θ other than the abnormal states in Θ ab, that is, θ Θ \ Θ ab, occurs. For notational simplicity, we refer to a state θ Θ\Θ ab as a normal state. Then, the main state θ m belongs to normal states and the normal signal implies that the main state happens with at least probability 1 ε. To lay the foundation of the two-signal setting, we assume that the sender can observe the occurrence of individual abnormal states only in Θ ab or can read only the symptoms that indicate the occurrence of the states Θ ab. Given a state θ Θ, the sender s payoff function related to the outcomes is defined as U s ( ; θ) : I i=1a θ i R. Like in the case of the action takers, we assume that, when we define this sender s payoff function on the set of the players strategy profiles, it is ß(Θ) measurable given each strategy profile of the players. Here, to focus on the equilibrium outcomes in the main state θ m, we consider the case in which equilibrium outcomes in a state θ other than θ m have bounded influence on the total outcomes. As a consequence, we assume that there 8 Jung (2009a) considered two senders and three signals in his models and found the conditions under which the senders could still successfully influence action takers to play the senders favored outcomes, which represents the same result as the Receiver s Dilemma in the ε-uncertain game. 19

21 exists a positive real B s such that sup a I i=1 A U s(a; θ) B θ i s for each θ Θ \ {θ m }. This setting will be used to make the model robust against the sender s subjective priors such that the priors can assign {θ m } and Θ ab at least probability 1 ε and positive probability, respectively. In addition to the payoffs related to the outcomes, the sender can get or lose extra payoffs related to his credibility in reporting truthful information. That is, in a normal state θ Θ \ Θ ab, the sender would get an extra payoff α n 0 for truthful signals and would lose an extra payoff β n 0 for untruthful signals such that α n +β n > 0, which formulates that the sender cares for his credibility. Likewise, in an abnormal state θ ab Θ ab, the sender would get α ab 0 for truthful signals and would lose extra β ab 0 for untruthful signals such that α ab +β ab > 0. Regarding the relationship between these two kinds of the payoffs, we restrict attention to the cases in which the sender has relatively small concern for his credibility compared to his concern for the outcomes. Accordingly, in a normal state θ Θ \ Θ ab, we assume 9 that (1 ε){ sup U s (a; θ m ) inf U s (a ; θ m )} 2εB s > α n + β n, (2) a I i=1 Aθm i a I i=1 Aθm i which formulates that the payoffs related to the credibility, α n +β n, is less than the supremum of the payoffdifferences in the main state θ m regardless of the outcomes in the other normal states Θ\(Θ ab {θ m }). Here, the payoffs sup a I i=1 A θm i U s (a; θ m ) and inf a I i=1 Aθm i U s (a ; θ m ) denote the sender s best payoff and his worst payoff, respectively, in the main state. Hence, the inequality (2) implies that, if the action takers plan to choose either the sender s best 9 In this inequality, we use the positive real B s for simplicity s sake. However, it can be replaced with 1 2 sup θ Θ\({θ m} Θ ab ) and a,a {U s(a; θ) U I i=1 Aθ i s (a ; θ)}, which could be less than B s. 20

22 outcome or his worst outcome, then the sender would surely influence them to choose his best outcome regardless of his concern for credibility in reporting truthful information. Note that the inequality (2) holds regardless of the outcomes in a normal state other than the main state. This setting enables us to adopt arbitrary subjective priors that can assign {θ m } and Θ ab at least probability 1 ε and positive probability, respectively. Likewise, in an abnormal state θ ab Θ ab, we assume that sup a r R i=1 Aθ ab i U s (a r, a 1 r; θ ab ) inf U s (a r, a 1 r; θ ab ) > α ab + β ab (3) a r R i=1 Aθ ab i where (a 1 r, a 1 r) = a 1. Note that, in the inequality (3), the actions of the non-receivers and the deciders are fixed as a 1 r (= (a 1 nr, a 1 d )). This is because, in an abnormal state θ ab, the deciders took their actions a 1 d already and the non-receivers have dominant actions a1 nr. Thus, in equilibrium, they always choose the actions a 1 r in θ ab. Now, we make three assumptions to properly measure the sender s influence on the action takers decisions and to ensure the existence of a pure-strategy equilibrium. First, we organize the sender s preferences on the equilibrium outcomes {a j } J j=1 to measure the sender s influence. In the ε-uncertain game, the sender can directly influence only the receivers, but not the non-receivers or the deciders. Hence, we assume that, for any a j r, a j+1 r {a k r} J k=1 where {(a k r, a k r)} J k=1 = {ak } J k=1, the sender prefers the equilibrium outcomes related to aj r over the equilibrium outcomes related to a j+1 r Formally, in a normal state θ Θ \ Θ ab, we have regardless of the outcomes in the other states. (1 ε)u s (a j ; θ m ) εb s > (1 ε)u s (a j+1 ; θ m ) + εb s 21

23 for each j {1,..., J 1}. In addition, in an abnormal state θ ab Θ ab, we have U s (a j r, a 1 r; θ ab ) > U s (a j+1 r, a 1 r; θ ab ) for each j {1,..., J 1}. As a result, this assumption enables us to measure how much the sender can influence the action takers to choose his favored outcomes in the main state θ m. That is, the sender can be considered to be more influential if, in θ m, he can achieve lower ranked equilibrium outcomes out of {a j } J j=1. Second, we differentiate the equilibrium outcomes {a j } J j=1 with respect to the sender s influence in order to properly measure the sender s influence on the equilibrium outcomes {a j } J j=1. Since the sender s direct influence is limited to the receivers actions, if two distinct elements a j, a j in {a k } J k=1 contain the same actions of the receivers, then the sender cannot influence the action takers when they plan to choose between these two elements, which then implies that we cannot examine the sender s influence between these two outcomes. Therefore, to properly measure the sender s influence on each of the equilibrium outcomes {a j } J j=1, we assume that different elements in {a j } J j=1 contain different actions of the receivers, that is, a j r a j r for any distinct a j, a j {a k } J k=1 where (aj r, a j r) = a j and (a j r, a j r) = a j. Finally, if the equilibrium outcome a 1 ( {a j } J j=1) is not the sender s favorite in states Θ ab {θ m }, then there might not exist a pure-strategy equilibrium 10 in the ε-uncertain 10 For example, consider the following sender-receiver game. Nature chooses a good state with probability 1 ε and a bad state with probability ε for suffi ciently small ε > 0. The sender observes the choice of Nature, then he sends the receiver a signal, either good or bad. After observing the sender s signal, the receiver chooses his action either G or B. In the good state, the receiver and the sender get 10 and 0, respectively, when the receiver takes the action G, and they get 0 and 1, respectively, when the receiver takes B. In the bad state, the receiver and the sender get 0 and 0, respectively, when the receiver takes the action G and get 1 and 1, respectively, when he takes B. In addition to these payoffs, the sender gets an 22

24 game. Therefore, to ensure the existence of a pure-strategy equilibrium, we assume that a 1 is the sender s favorite outcome in those states, that is, U s (a 1 ; θ) = max a I i=1 A U s(a; θ) for θ i each θ Θ ab {θ m }. In addition, if a 1 would be the only element in {a j } J j=1, that is, J = 1, then this a 1 would be the unique equilibrium outcome in the main state. Hence, in this case, the sender can achieve his favorite outcome a 1 even without influencing the action takers. In the ε-uncertain game, since we are interested in the role of the sender s influence in the decision-making process of the action takers, we consider only J 2 cases. 3.3 Timeline The ε-uncertain game proceeds as follows. At stage zero, Nature and the deciders, whose actions condition the states, choose a state θ in Θ. Here, when the deciders choose their actions, they are supposed to independently choose them without the information about the other deciders actions. Next, the sender and all the non-receivers acquire their own information about the actual state θ. Note that, in this ε-uncertain game, different players could get different amounts of information. At stage one, the sender sends the action takers a signal, then all the action takers observe the sender s signal. At stage two, the receivers and the non-receivers independently choose their own actions without the information about extra payoff α 0 for truthful signals and would lose an extra payoff β 0 for untruthful signals such that 1 > α + β > 0. Here, in the absence of the sender, there is the unique equilibrium in which the receiver chooses the action G only, which is not the sender s favorite. In this game with the sender, however, there is no pure-strategy equilibrium. This is because, when the receiver ignores the sender s signals, the sender has an incentive to report truthfully, then, responding to this sender s strategy, the receiver would have an incentive to respond to the sender s signals. But, when the receiver responds to the sender s signals, the sender has an incentive to manipulate the information, then, responding to this strategy, the receiver would have an incentive to ignore the sender s signals. As a consequence, the game represents a situation similar to that in the discoordination game, which does not have a pure-strategy equilibrium as shown by Coate (1995) and Rasmusen (2007, Ch 3.3). 23

25 the actions taken by the other action takers at this stage. Finally, the actual state is revealed to all the action takers and their payoffs are realized accordingly. 4 Results Proposition 1 ensures the existence of a pure-strategy perfect Bayesian equilibrium 11 in the ε-uncertain game. Concretely, it shows that there always exists a pure-strategy equilibrium that contains the sender s favorite outcome a 1 in the main state θ m. Note that, in this study, we consider only pure-strategy equilibrium outcomes. Therefore, Proposition 1 establishes that the ε-uncertain game is well-defined so that we can properly study the sender s influence on the action takers in this ε-uncertain game. Proposition 1 There exists a pure-strategy perfect Bayesian equilibrium in the ε-uncertain game. Proof. Suppose that the deciders plan to choose the actions a 1 d. Next, the receivers plan to choose the actions a 1 r regardless of the sender s signals. For the non-receivers, suppose that they plan to choose the actions a 1 nr regardless of the sender s signals in the main state θ m and in an abnormal state θ ab Θ ab. In addition, in a state θ / {θ m } Θ ab, responding to the other action takers actions a 1 nr = (a 1 r, a 1 d ), suppose that the non-receivers plan to choose their best responses, which are designated in the setting of the ε-uncertain game, regardless of the sender s signals. Note that all these strategies are pure strategies. Then, responding 11 Fudenberg and Tirole (1991) presented a version of the perfect Bayesian equilibrium in the game with observed actions. In this study, we adopt their definition and adapt it to the ε-uncertain game because it might contain the setting of unobserved actions. A formal definition of this perfect Bayesian equilibrium in the ε-uncertain game is provided in Appendix. 24

26 to these actions, the sender s best response would be reporting truthfully, that is, reporting normal when it is normal and reporting abnormal when it is abnormal. Regarding the action takers incentives, consider an arbitrary action taker i. Then, responding to both the sender s signals and the other action takers actions, we can see that action taker i s actions proposed above can be one of the best responses. If action taker i ( R) is a receiver, then he would have no incentive to deviate from a 1 i when the sender signals normal because the other action takers would play a 1 i in the main state θ m, which happens with fairly large probability at least 1 ε, and the non-receivers would play the designated actions in a state θ other than the main state θ m, that is, θ Θ \ ({θ m } Θ ab ). When the sender signals abnormal, all the other action takers would play a 1 i and thus receiver i has no incentive to deviate from a 1 i as well. If action taker i ( {R+1,..., D 1}) is a non-receiver, then he would play a 1 i either when he cannot figure out the actual state or when it is a state in {θ m } Θ ab. This is because, when he does not know the actual state, the main state θ m happens with large probability and the other action takers would play a 1 i in θ m regardless of the sender s signals. In an abnormal state θ ab Θ ab, his dominant action is a 1 i. When it is a state θ / {θ m } Θ ab, he is playing his own best response to the other action takers actions regardless of the sender s signals. Finally, if action taker i ( D) is a decider, whose actions condition the states, then he would play a 1 i as well. This is because, responding to the actions a 1 i in the main state θ m and the non-receivers designated actions in a state θ / {θ m } Θ ab, decider i has no 25

27 incentive to deviate from a 1 i. Consequently, all players are making their best responses in each of their decision places, that is, in each of their information sets, with respect to their own subjective priors, and hence the strategy profile proposed above is a perfect Bayesian equilibrium. Note that we consider only pure strategies here. Therefore, this strategy profile is indeed a pure-strategy perfect Bayesian equilibrium. The next result reveals the feature of the equilibrium outcomes in the main state θ m. In the absence of the sender, the ε-uncertain game has the equilibrium outcomes {a j } J j=1 in θ m. Lemma 1 proves that these outcomes {a j } J j=1 survive as the equilibrium outcomes in θ m even when the sender is introduced into the ε-uncertain game. As a result, Lemma 1 enables us to focus only on the outcomes {a j } J j=1 to study the sender s influence on the action takers behavior in θ m. Lemma 1 The action takers play action profiles only out of {a j } J j=1 outcomes of the main state θ m. in the equilibrium Proof. By way of contradiction, suppose that the ε-uncertain game has an equilibrium outcome a 0 in the main state θ m such that a 0 ( I i=1a θm i ) \ {a j } J j=1. Then, there exists an equilibrium strategy profile of the action takers a 0 ( ) I i=1π i such that a 0 ( ) designates the action takers actions a 0 in θ m. In equilibrium, if the sender makes the same signal in every state θ Θ, then the outcomes in θ m are the same as those in the absence of the sender, which are {a j } J j=1. Hence, it suffi ces to show the contradiction of this hypothesis when the sender makes two distinct signals in equilibrium. Let a set of states Θ 0 Θ contain θ m and suppose that the sender makes the same signal in every state θ in Θ 0. Then, we show the contradiction by using the inequality (1), which 26

28 says as follows. In the absence of the sender, given any action takers strategy profile a( ) I k=1 Π k that does not contain actions out of {a j } J j=1 in θ m, there exists action taker i such that sup E i (a i( ), a i ( ); Θ ) > E i (a( ); Θ ) a i ( ) Π i for any set of states Θ containing θ m where (a i ( ), a i ( )) = a( ). When we replace a( ) and Θ with a 0 ( ) and Θ 0, respectively, in the inequality (1), if this action taker i is either a receiver or a non-receiver, then action taker i has an incentive to deviate from a 0 i ( ). This is because, in the ε-uncertain game with the sender, either a receiver or a non-receiver can simply change his strategy and can raise his payoff without considering outcomes in states Θ \ Θ 0, which accordingly shows the contradiction. When we replace them, if this action taker i is a decider, then we need to replace the set Θ 0 with the whole set Θ again. This is because the deciders take their actions before observing the occurrence of the actual states and the sender s signals, and thus they need to consider the whole states Θ no matter what the sender plans to signal. Then, we have sup a i ( ) Π i E i (a i( ), a 0 i( ); Θ) > E i (a 0 ( ); Θ) according to the inequality (1). Therefore, decider i has an incentive to deviate from a 0 i ( ), which consequently completes the proof. Now, we present the main results about the Receiver s Dilemma. In the ε-uncertain game, the sender has his concern for credibility in reporting truthful information that is relatively small compared to his concern for the outcomes. boosts the sender s incentive to report truthfully. This credibility concern, nevertheless, We refer to this sender s incentive as Boosted Incentive Truthfulness (or simply BIT). Then, the action takers cannot completely 27