Reputation with a Bounded Memory Receiver

Transcription

1 Reputation with a Bounded Memory Receiver Daniel Monte Yale University- Department of Economics 28 Hillhouse Avenue New Haven, CT UA April 22, 2005 Abstract This paper studies the implications of bounded memory in a strategic context. In particular, we look at a model with a bounded memory receiver in a repeated cheap talk game with incomplete information on the sender s type. We find that there are multiple equilibria in this game, including many equilibria in which the receiver wastes memory sets. We describe necessary conditions for equilibrium and find that the receiver s memory will have some intuitive properties. Namely that the receiver never updates to a lower belief after a true signal, he stops updating after he is sufficiently convinced about the sender s type and he will never ignore lies, unless he is sure about the sender s type. There are, however, some important differences in this reputation game with a bounded memory receiver from the standard version with a Bayesian receiver. In particular, we find that the bounded memory receiver might test the sender before updating, in contrast to a Bayesian receiver that will always update as long as he is not certain of the sender s type. Finally, the bounded memory receiver might choose (given his memory constraint) to stop updating atapointwhereheisstillnotsureaboutthesender stype. Inthis case, he will never entirely believe the sender, even after an arbitrarily long sequence of good signals. I am very grateful to Ben Polak and tephen Morris for their guidance and advices. I am also thankful to Attila Ambrus, Itzhak Gilboa and Pei-Yu Lo for very helpful discussions. daniel.monte@yale.edu 1

2 1 Introduction This paper studies the implications of bounded memory in a strategic context. In particular, we look at a model with a bounded memory receiver in a repeated cheap talk game with incomplete information on the sender s type. We use the model first studied by obel (1985) and add this constraint on the receiver s information storage capacity. At every stage game of obel s model the sender observes the true state of the world in that period (the states are ex-ante equally likely) and sends a message to the receiver. The receiver then takes an action and the payoffs are realized. At this point the state of the world is verified and the receiver knows whether the sender has lied to him or not. The receiver then updates his belief (Bayesian updating) on the sender s type. The bounded memory receiver, on the other hand, has to choose an action rule and an updating rule given his memory constraint. In other words, he will choose how to update the beliefs after receiving signals, given his current memory state, and a map from the memory states and messages received to actions. The updating rule for this bounded memory receiver is interpreted as a rule of thumb on updating beliefs. Thus, this model captures the heuristics on how agents keep track of information in this context. We then find necessary conditions for this updating rule to be an equilibrium. The interim receiver knows the rule of thumb and decides on which action to take and whether or not to follow this rule. We describe rules in which the interim receiver will find in his best interest to follow the rule. The way we model the bounded memory receiver is based on Wilson (2004). Wilson has an extensive game where a decision maker receives a sequence of signals and has to make a decision about the true state of the world after a very long sequence of signals. However, the decision maker cannot recall the sequence of signals and has, instead, to choose the best way to store information given his finite set of memory states 1. We extend Wilson s analysis to a repeated two player game setting, where one of the players (the receiver) exhibits the same type of memory constraint as Wilson s decision maker, while the other player (the sender) is unconstrained. There are several possible applications to the model studied in this paper. 1 A similar problem was studied by Hellman and Cover (1970). Their model describes the best way to keep track of the signals with a finite state machine for the two-hypothesis testing problem. 2

3 One interpretation is in the context of a policy maker that listens to an informed advisor in every different area. Each advisor can be one of two types. He could be an honest type, a very professional employee that will always inform the policy maker about the true state of the world (a behavioral type), or he could be an advisor with a secret interest in becoming the next election s running up candidate (a strategic type). This strategic type of advisor gains whenever the policy maker is worse off. However, the policy maker can be cheated at most once when he is a Bayesian player. What we study in this model is the case where the policy maker is dealing with several different advisors, i.e. with advisors in all differentareas. Itbecomeshard for the policy maker to keep track of all the signals that the advisors have send. We describe a rule of thumb on how this policy maker will keep track of the information using only a finite number (possibly small) of memory states. We have two main goals in this project. First, to study games with imperfect recall in a strategic context. Games with imperfect recall have only been studied on single player games 2. We want to discuss how some issues that arise in these games, such as the way we compute beliefs and also the equilibrium concepts, carry over to a multi- player game. The second goal of this paper is to characterize how reputation transmission takes place when the sender is facing a bounded memory receiver. We compare the reputation transmission in this context to the one with a Bayesian receiver (obel (1985)). The reason for studying the implication of a finite memory player in this particular setting is that this game has been studied before 3 so the results due to finite memory can be analyzed in comparison to the Bayesian game. We characterize the equilibria for a general setting with infinite horizon and where the receiver has N memory states. The Markov equilibrium in this game is a memory rule (action rule, transition rule and initial distribution over memory states) together with a strategy for the sender such that the memory rule is incentive compatible, meaning that the interim receivers find it to be in their best interest to follow the memory rule given that his future selves are also following the equilibrium rule (as in Piccione and Rubinstein (1997) and Gilboa (1997)) and also such that the strategy of the sender is a best response for him. 2 ee Gilboa (1997), Piccione and Rubinstein (1997) and Wilson (2004). 3 ee also Benabou and Laroque (1992) and Morris (2001). 3

4 We find that there are multiple equilibria in this game, in particular there is always a trivial equilibrium in which all the memory states have the same beliefs. In this trivial case any transition rule is consistent with equilibrium and this game becomes in fact an infinite sequence of one-shot games. There are many equilibria in which the receiver wastes memory sets like in this trivial case. We describe necessary conditions for equilibrium and find that the receiver s memory will have some intuitive properties when the equilibrium is not the trivial one described above. Namely that the receiver never updates to a lower belief after a true signal, he stops updating after he is sufficiently convinced about the sender s type and he will never ignore lies, unless he is sure about the sender s type. However, there are some important differences in this reputation game with a bounded memory receiver from the standard version with a Bayesian receiver. In particular, we find that the bounded memory receiver might test the sender before updating, in contrast to a Bayesian receiver that always updates as long as he is not certain of the sender s type. Also, the bounded memory receiver might choose (given his memory constraint) to stop updatingatapointwhereheisstillnotsureaboutthesender stype. Inthis case, he will never entirely believe the sender, even after an arbitrarily long sequence of good signals. The structure of this paper will be as follows. On section 2 we describe the model and define memory and strategies for this game. On section 3 we solve for the two period and two memory states case. Then, on section 4 we compute the equilibrium for three memory states and three periods. ection 5 shows the results for the general setting with N states and infinite periods. ection 6 concludes. All the proofs are in the appendix. 2 Model This is a repeated cheap talk game with incomplete information on the sender s type. The model is based on obel (1985). The senders can be one of two possible types. With probability ρ we assume that the sender is a behavioral type, which we alternate and also call honest type (H), that always tells the truth. Nothing will be said about the preferences or strategy of this behavioral type of sender, we just assume that at every period the message that this type sends is equal to the true state of the world, i.e. 4

5 m i = ω i. With probability (1 ρ) the sender is actually a strategic type () 4, with utility opposite to the receiver s. The timing of every stage game is described below. The sender observes the true state of the world Ω = {0, 1} and sends a message m = {0, 1} to the receiver. This message has no direct influence on the player s payoffs, thus a cheap talk game. The receiver then observes this message and takes an action. After he takes the action, the payoffs are realized and the states are verified. At this point, the receiver can tell whether the sender has lied to him or if the sender told him the truth about the state of the world. Based on this information, the receiver updates his belief on the sender s type. ThusthegameishΓ, (H, ), (ρ, 1 ρ), (U,U R )i. The utilities of the players for every stage game are given by U R = (a t ω t ) 2 and U = (a t ω t ) 2. The receiver is minimizing a quadratic loss function. With this particular functional form of utility we get a unique best response for every message send. The action rule will be uniquely determined given the message. Another important consequence of these utility functions is that the payoffs are symmetric to the states of the world, meaning that it only matters whether a truth or a lie was told. Formally: U R (m i =1 ω i =1)= U R (m i =0 ω i =0). Finally, we assume no discounting. 2.1 Memory and trategies We define the memory of the receiver to be a finite set where each element corresponds to a different memory state. Let I be the memory of the receiver. Thus, I = {1, 2,..., N} means that the receiver is constrained to N memory states. The transition rule is a map from the memory of the receiver and the signal (true or lie) received, to a subset of the his memory. In other words, this transition rule σ : I {T,L} (I) is the receiver s updating rule. It is also part of the receiver s strategy to decide at time t =0, before the first period starts, at which memory set he will start the game. Let g 0 be this initial distribution over the memory states: g 0 (I) Finally, the receiver s strategy also includes an action rule a : I {0, 1} R which is a map from memory and messages to actions. We call the tuple (σ,a,g 0 ) the memory rule of the receiver. To define the sender s strategy we need two additional assumptions. First, 4 obel (1985) calls the honest type the Friend and the strategic type, the Enemy. 5

6 we assume that at every period of the game the sender knows in which memory state the receiver is. This assumption will make the analysis easier without changing qualitatively the results. If we wanted to drop this assumption, we would have to condition the sender s strategy on his beliefs over the memory states rather than on the exact memory state that the receiver is in. Note, however, that because of the action taken by the receiver, the sender would be able to infer the memory state that he was in the previous period. o his beliefs would be restricted to the states that have positive probability to be reached by the equilibrium transition rule. We also assume that since payoffs are symmetric to the states of the world, the sender s strategy will also be symmetric. Define q it to be the probability that the sender tells the truth at memory state i and period t. Then, q it Pr (m it =1 ω it =1)=Pr(m it =0 ω it =0). In principle we should condition the strategy of the sender on the histories of the game, however we concentrate on Markov strategies, i.e. on strategies that depend only on the state of the economy which are the memory states in this game. Thus, the strategy of the sender (type ) isgivenby: q : I Z + [0, 1]. 3 Two periods In this section we study the case where this cheap talk game is played for two periods. We want to compare this game when you have a Bayesian receiver to the restricted version of this game where the receiver is constrained to two memory states only. The proposition below shows the equilibrium when the receiver is using Bayesian updating to update his beliefs on the sender s type. For the proof of this proposition see obel (1985). [include here for completeness]. Proposition 1 For each prior ρ there is a unique equilibrium in the two period game with a Bayesian receiver. Moreover, for any ρ > 1 the equilibrium is such that in the initial period the receiver believes the sender with 4 probability greater than 1. 2 The equilibrium is the following. If the probability of a behavioral type is very high, ρ 7, the receiver will have a very high belief that the truth 8 will be told in the first period and therefore it will be very costly for the 6

7 strategic sender to build a reputation. This implies that the best response for the sender is to lie on the first period, q 1 =0. If the prior on the sender s type is instead very low, ρ 1, then the 4 receiver believes with high probability that he is dealing with a strategic type. Therefore the only possible equilibrium is babbling in both periods. The equilibrium with reputational concerns occurs for intermediate priors, in particular for ρ 1, In this case, the proposition above shows that there will also be a unique equilibrium. On the first period the sender will be mixing with some probability q>0. This implies that the probability that the receiver thinks that the message is true is π 1 Pr (T/1) = ρ +(1 ρ) q. On the second period after having observed a true signal, the sender will lie with probability 1, q 2T =0, since it will be the last period and there will be no reputational concern in the last period. The belief in the second period following a true signal in the first period is the proportion of behavior type in the last period, i.e. π 2T =. Finally, on the second period after ρ ρ+(1 ρ)q observing a lie, babbling is the only possible outcome (q 2L = 1, and π 2 2L = 1). 2 Note that if we had a two period game and a receiver with bounded memory but restricted to three memory sets, we would be able to exactly reproduce the equilibrium described above. Just denote the initial state by state 2, and attach the belief ρ+(1 ρ) q to that state. To the state labeled 3, ρ ρ+(1 ρ)q attach the belief and finally to state number 1 we associate babbling. Formally, if the memory of the receiver is I = {1, 2, 3} then consider the following transition function: σ 1T 1 =1, σ 2T 3 =1, and σ 3T 3 = 1 together with σ il1 =1, i I. This memory, with this transition rule together with g 0 (2) = 1 (i.e. the probability of starting the game at memory state 2 is equal to 1) will reproduce the equilibrium above. Thus in this section we restrict attention to the two memory states and two periods case, since this is the simplest setting where you depart from the Bayesian updating. Denote this game with only two periods and allowing the receiver to use only two memory states by Γ 2. We ask two questions in this setting: which are the possible equilibria in this game and also, among these equilibria, which one gives the receiver a higher ex-ante expected payoff. The first issue in this game before we can answer the questions above is how to compute the beliefs on the memory states. Given the equilibrium strategies, we can compute the beliefs in each state and the posterior on the sender s type. Let I = {A, B} and denote π i as the probability of truth given 7

8 Figure 1: Updating Rule > rule of thumb that the receiver is in memory state i. Inthiscasewehaveπ A Pr (T A) and π B Pr (T B). For the posterior on the sender s type, denote p H i as the probability that the receiver believes that the sender is a behavioral type after verifying if the sender lied or told the truth in state i p H A Pr (H T,A).Notethatsince there is no noise on the sender s information structure, whenever the receiver observes a lie, he can be sure that the sender is a strategic type. To compute the beliefs in each memory set, we use the same approach as in Piccione and Rubinstein (1997). [write the formal definition here] The intuition is to think of the probabilities of time periods as long run frequencies. Consider as an example the case where the transition rule is: σ AT A =1 and σ AF B =1. This transition rule is depicted in figure 2. If this is the transition rule, the best response for the sender is: q A =0. Figure 2: Memory rule separating liars 8

9 ince the transition rule is completely separating the liars in the second period, whenever the receiver reaches memory set B he can be sure that he is dealing with a strategic type of sender, thus the only possible belief in that state is the one associated to babbling: π B = 1 2. To compute the belief in memory state A we have to compute the probabilities of the time periods. π A =Pr(t =1 A)Pr(T t =1,A)+Pr(t =2 A)Pr(T t =2,A) (1) If we think of these probabilities as long run frequencies, then Pr (t =2 A) = ρ Pr (t =1 A) but we also have that the probabilities of the time periods should sum to 1. olving for these two equations we get that: Pr (t =1 A) = 1 ρ and Pr (t =1 B) =. Moreover, Pr (T t =1,A)=ρ which is the proportion of people telling the truth in time 1, and Pr (T t =2,A)=1, which 1+ρ 1+ρ means that if the receiver reaches the second period and is still in state 1, then it must mean that he is dealing with the behavioral type of sender, therefore he may expect to receive a true signal with probability one. Thus, equation (1) gives us π A = 2ρ 1+ρ. There are three classes of equilibria in this two state game. To find the equilibrium we use the notion of incentive compatibility as described by Piccione and Rubinstein (1997) and Wilson (2004). A strategy is incentive compatible if an agent cannot gain by one shot deviations from his equilibrium strategy, given the payoffs and beliefs induced by his strategy. Assuming that all other selves are playing the equilibrium strategy. Below we show the definition of incentive compatibility for the transition rule σ in this game Γ 2. Define p H i Pr(H/T, i, 1) and similarly p i Pr(/T, i, 1), then we have the following definition for an incentive compatible strategy. Definition 1 A transition rule σ in Γ 2 is said to be incentive compatible if: σ it j > 0 p H i (1 π j ) 2 p i π 2 j p H i (1 π j 0) 2 p i π 2 j 0 For i, j, j 0 I = {A, B}. σ ilj > 0 π 2 j π 2 j 0 If, in equilibrium π A = π B then any transition rule can support this equilibrium. Thus, there will always be a trivial equilibrium in which the states have equal beliefs. In this trivial equilibrium, the receiver is in fact 9

10 wasting memory sets. The more interesting cases come when we look at memory rules where the receiver is not wasting memory states. Proposition 2 The only equilibrium rules in Γ 2 when π A 6= π B are the ones that involve no mixing. Among the equilibrium rules, the one that gives the receiver the highest ex-ante payoff is the one that we described in the example. Proposition 3 For ρ > 1, the equilibrium that gives the receiver highest 3 payoff is: (σ AT A =1, σ AF B =1,q 1 =0,q 2 = 1). 2 For ρ 1,babbling in both states is the only possible equilibrium. Where 3 babbling is characterized by a belief of 1 in both states and with the strategic 2 sender telling the truth with probability 1 in both periods. 2 The properties of this good equilibrium for the receiver (which corresponds to the transition rule depicted in figure 2) are that there are no randomizations on memory rule, and that the receiver keeps track of the liars. Also worth noting that the strategic sender will gain not because the receiver will forget in case he lies, but because the receiver doesn t know the period that he is in when he starts the game. In other words, the receiver is confused about the time period when he is in state A, sohedoesn tknowif he has already separated all the liars or not. This inflates the belief in state A and gives the sender a high payoff in the initial period. 4 Example We compute an example with three periods and three memory states to show that there will be equilibria where the transition rule of the receiver involves mixing. In particular, we find in this example that the best equilibrium for the receiver is the one depicted in figure 3. 5 Infinite Horizon To analyze the general setting with N memory sates and infinite horizon we need an additional modification in the game. We will assume that at every period there is an exogenous probability η that the game will end. We need 10

11 Figure 3: Three periods and three states thisassumptionbecauseaswedepartfromthefinite horizon case, we need well defined priors over the time periods. This death rate will give us this distribution. The only condition on η is that it is bounded away from zero. We keep the assumption of no discounting (other than the death rate). We also need extra notation to proceed in the analysis. The expected continuation payoff for the receiver at memory state i given that the sender is a behavioral type is denoted by vi H, where vi H is equal to: v H i = (1 π i ) 2 +(1 η) Σ j σ it j v H j The expected continuation payoff for the receiver given a strategic sender is denoted by vi. incewedon tknowthebehaviorofthestrategicsender,the only thing that we can tell is that vi = U (i). Where U (i) is the expected continuation payoff for the strategic sender and U (i) =max{u (T i),u (L i)}. 5.1 Incentive Compatibility In this section we define incentive compatibility for a general memory with N memory states. It is an extension of the definition presented on section 3. Definition 2 The memory rule is incentive compatible if for i, j we have that: σ it j > 0 p H i vj H + p i vj p H i vj H + 0 p i vj 0, j0 σ ilj > 0 v j vj 0, j0 11

12 The action rule of the receiver is: max P (ω =1 m)(a 1) 2 P (ω =0 m) a 2 a a i (m) =P (ω =1 m) 5.2 Equilibrium We define the equilibrium in this game to be a memory rule for the receiver together with a strategy for the sender such that the strategy for the sender is a best response for him and the strategy for the receiver (his memory rule) is incentive compatible. Definition 3 A Markov equilibrium is a tuple (σ,a,g 0 ; q) that satisfies the following conditions: q is such that: q it < 1 U (L i) U (T i) q it > 0 U (T i) U (L i) (E1) The memory rule (σ,a,g 0 ) is incentive compatible, given q. (E2) Proposition 4 below is the main result of this paper. We find necessary conditions for the equilibrium. As we have pointed out, there are multiple equilibria in this game, in particular, there are many rules in which the receiver has redundant states. We wrote the proposition below to allow for these bad equilibria, but the most intuitive way to understand the proposition is to have in mind a rule without the redundant states, i.e. with N different memory states (holding different beliefs in equilibrium). Before looking at the proposition below, first note that in obel s game with the Bayesian receiver, for every prior there will be a critical time period in which all the strategic sender s will have lied. From that period onwards, the receiver is sure of the sender s type. Thus, in the game with a bounded memory receiver, for every (η, ρ) there will be a critical N such that if the receiver has N or more memory states, we can reproduce obel s equilibrium, by using deterministic rules. Therefore, the interesting case is when we have a receiver with less than N memory states. For the proposition below, label the states in non-decreasing order. I.e. if, in equilibrium, π j > π i then label j>i. 12

13 Proposition 4 If (σ,a,g 0,q) is an equilibrium, then: 1) After Lie: σ jll 0 = 0 where l 0 / {l π l =min i π i } (there is always a dumping state) 2) If U (L i) >U (T i) σ it h 0 =0where h 0 / {h π h =max i π i } 3) After True: π j > π i σ jti =0(don t go back after a True signal) 4) g 0 (i) =0, π i > π (2) 5) ingle crossing (for states where p H i 6= p H j ) σ it k > 0,σ it m > 0 and σ jtk > 0 σ jtm =0. k, m such that π k 6= π m. 6) No Jumps (for states where p H i 6= p H j ) σ it k 1 > 0,σ it k+1 > 0 σ jtk =0 i, j such that p H i 6= p H j The proposition above shows that any memory rule, in equilibrium, has to be such that the receiver separates the liars after a lie is observed. We have two possible rules then. One in which there is an absorbing state associated to the babbling, and another in which the state where the liars are placed is the initial one. Thus, in this case where the memory rule does not have an absorbing state the strategic senders might constantly cheat the receiver. Another result is that while the receiver might ignore true signals, by not updating after receiving them, he will never update to a worse belief after a true. One interpretation of this result is that the receiver might not pay attention (update) to some signals, but he will never forget the information that he already holds. At this point, (parts 1-4 in the proposition) we have ruled out some memory rules that could never be played in equilibrium. In particular, rules with loops and rules that don t separate the liars. However, it would be good to understand how this bounded memory receiver updates after true signals. Parts 5 and 6 in the proposition tells us part of the story. One rule that is possible and satisfies proposition 4 is depicted in figure 4. Again there are some inefficiencies for the receiver that could not be ruled out by equilibrium conditions only. We could still have rules that split into two different paths, for example, and also we can still have rules with redundant states. 13

14 6 Conclusion Figure 4: An Equilibrium Rule In this paper we looked for the equilibrium in a repeated cheap talk game with incomplete information and with a bounded memory receiver. We saw that there are multiple equilibria in this game, in particular that there are rules in which the receiver is wasting memory states. obel (1985) studied this game, but with a Bayesian receiver and showed that for every prior on the sender s type, the equilibrium is unique. This contrasts with our result because when the receiver is updating his beliefs with Bayes rule, we don t allow him to choose a memory rule or how to update, but we fix the updating rule (Bayes ). And with Bayes rule there is not only a sense that the player updates after a new signal, but also the requirement that he must update after every signal. In other words, there is a compulsion in updating, meaning that the player using Bayes rule is not allowed to forget and is not allowed to ignore information, even if it were in hisbestinteresttodoso.thisiswhyinagamelikeobel,thereisnoroom for redundant states (before the receiver is sure about the sender s type). In the reputation game with a bounded memory receiver, however, this feature of Bayes rule is not imposed, that is why we can t get rid of redundant states in equilibrium, and there are always bad equilibria for the receiver. Nevertheless, we derived necessary conditions for equilibrium and showed that the receiver s memory rule will have intuitive properties in equilibrium, for example that there will never be updating to a lower belief state and there 14

15 will no be transition rules involving two non consecutive memory states, for states with different posteriors. Finally, although the memory rule will have intuitive properties, there will be important differences in updating between the Bayesian receiver and the bounded memory receiver. One of the biases in information processing that we observe is that updating is possibly much more infrequent than for a Bayesian receiver. In particular, the bounded memory receiver might test the sender and will ignore information not only when he is sufficiently convinced of the sender s type (as in Wilson(2004)). Another important departure from Bayesian updating is that the bounded memory receiver will stop updating not necessarily when he is certain of the sender s type. Finally, while the receiver ignores true signals, it will be too costly to ignore false signals, which he only does when he is certain of the sender being a strategic type. 7 Appendix 7.1 Two Memory tates Two periods In this section we prove propositions 2 and 3 in the paper. Proposition 2 shows that the only possible equilibria in the two state two period game where the receiver does not waste memory states are the ones that involve no mixing in the transition rule. Proposition 2 If π A 6= π B The only rules that are incentive compatibles are the ones that involve no mixing. Proof. To prove this lemma, first note that since there are only two periods, and both states are such that the belief is greater or equal than 0.5, the strategic sender will lie when he reaches period 2 (or will be indifferent between lying and telling the truth). If in equilibrium π i > π j σ if j = 1. This comes directly from the definition of incentive compatibility. We consider the two cases separately, first when π A > π B and then when π A < π B. First note that if in equilibrium π A > π B q 1 =0. ince: U (LL) = π 2 A + σ ALA π 2 A + σ ALB π 2 B and U (TL)=(1 π A ) 2 + σ AT A π 2 A + σ AT B π 2 B But, π 2 A σ AT Aπ 2 A + σ AT Bπ 2 B and σ ALAπ 2 A + σ ALBπ 2 B (1 π A) 2. Thus, U (LL) U (TL) and therefore, q 1 =0. Therefore, p H A = 1 which, in turn, implies by incentive compatibility that σ AT A =1. 15

16 Finally, we want to show that if in equilibrium π B > π A σ AT B =1. p H ρ A = ρ+(1 ρ)q 1 and π B = p H A, since only the behavioral type will tell the truth in the second period when the state is informative (has a belief higher than 1 ). Thus: 2 uppose the memory rule is incentive compatible and σ AT A > 0 p H A (1 π A ) 2 p Aπ 2 A p H A (1 π B ) 2 p Aπ 2 B, Using the fact that π B = p H A, we have that: π B (1 π A ) 2 (1 π B ) π 2 A π B (1 π B ) 2 (1 π B ) π 2 B = π B (1 π B ) π B +2π B π A π B π 2 A π2 A + π Bπ 2 A π B + π 2 B 2π B π A π 2 A π 2 B π 2 B + π 2 A 2π B π A 0 (π B π A ) 2 0( ) Therefore, if the memory rule is incentive compatible, then it must be that σ AT A =0. Proposition 3 For ρ > 1, the equilibrium that gives the receiver highest 3 payoff is: (σ AT A =1, σ AF B =1,q 1 =0,q 2 = 1). 2 [stillworkingonabetterproof,pleasedisregardthispart] Proof. We only look at priors in the range ρ 1, 1, since if ρ < 1, 3 3 babbling is the only possible equilibrium. Define M1 as the memory rule corresponding to a transition rule: σ AT A = 1, σ AF B =1 µ U R (M1) = 2ρ (1 π A ) 2 (1 ρ) π 2 A + 1 (2) 4 However, since π A = 2ρ 1+ρ we have that: U R (M1) = 1 4 ( 1+ρ) 9ρ +1 1+ρ (3) Define M3 as the memory rule corresponding to a transition rule: σ AT B = 1, σ AF A =1 U R (M3) = ρ (1 π A ) 2 +(1 π B ) 2 (1 ρ) q 1 (1 πa ) 2 + πb 2 +(1 q1 )2π 2 A (4) The beliefs in states A and B are given by:π A = ρ+(1 ρ)q 1 (1 ρ)(1 q 1 )+1 and π B = ρ ρ+(1 ρ)q 1 Where q 1 comes from the following indifference condition: U (LL) = U (TL)whichis:2π 2 A =(1 π A ) 2 + π 2 B. 16

17 Thus we have to solve the following system to obtain the result: π A = π B = ρ +(1 ρ) q 1 (1 ρ)(1 q 1 )+1 ρ ρ +(1 ρ) q 1 (5) 2π 2 A = (1 π A ) 2 + π 2 B olving the system above, we find q 1 andthenplugbackin(4)tofind the receiver s utility. olving (5) we get the following: 1 4 (1 ρ) ³ 2 ³ 2 ρ+(1 ρ)q1 (1 ρ)(1 q 1 )+1 = 1 ρ+(1 ρ)q 1 (1 ρ)(1 q 1 )+1 µ ³ 9ρ+1 ( 1+ρ) < ρ 1 ρ+(1 ρ)q 1 1+ρ µ ³ q 1 1 ρ+(1 ρ)q 2 ³ 1 (1 ρ)(1 q 1 )+1 + (1 ρ)(1 q 1 ) ³ 2 + ³ 1 2 ρ ρ+(1 ρ)q 1 ρ ρ+(1 ρ)q 1 2 +(1 q 1 )2 ρ ρ+(1 ρ)q 1 2 ³ ρ+(1 ρ)q1 (1 ρ)(1 q 1 )+1 2 The solution is : n q 1 = o n λ, ρ < λ, 1 < ρ, q 1+ρ 3 1 = o n λ, ρ < λ, λ < ρ, q 1+ρ 1 = o λ, λ < ρ 1+ρ where λ is a root of: 132Z Z Z Z Z 4 +32Z Z Z +17 All roots above are positive, meaning that U R (M1) <U R (M3) only if q 1 < Infinite Horizon In this section we prove proposition 4. Define l as the state with highest expected continuation payoff if the receiver is facing a strategic sender. Formally: D {l I v l v i, i I}, similarly define: U {u I v H u v H i, i I}. The first result comes from incentive compatibility. ince Pr (H/i, F) = 0, i, we must have that after a lie, the receiver moves to a state with highest expected continuation payoff given that the sender is strategic. As defined 17

18 above, the receiver moves to a state where the expected continuation payoff for the receiver conditional on the bad type of sender is equal to vl and for the sender is U (l). Before we state the first lemma, denote ½ j I after a true p j H I (j) j vj H + p j vj ¾ ph j vj0 H + p j vj0; after a lie: vj v j0, j 0 I Thus, the payoff of the sender after lying is:u (F i) =π 2 i +(1 η) P i σ ili U (l). imilarly, the payoff of the sender after telling the truth is: U (T i) = (1 π i ) 2 +(1 η) P j σ it j U (j ) Lemma 1 j/ D σ if j =0, i I. Proof. By incentive compatibility, σ if j > 0 vj vj 0, j0 Thereforewecanwritethepayoff of the sender after lying as: U (F i) =π 2 i +(1 η) U (l) We now show a lemma that will be very helpful in subsequent results. The lemma is that whenever the sender reaches a state where π i =1, i.e., the highest possible belief,then the sender will strictly prefer to lie. This is because by lying the sender gets the highest possible current payoff and is then placed on the lowest state l. However, lying or telling the truth in l is strictly better to the sender than telling the truth in a state with belief higher than 1 2. Lemma 2 In the highest state the strategic sender lies with probability one (except for the trivial equilibrium where all the states are the same). U (L N) >U (T N) Proof. U (L i) =π 2 N +(1 η) U (l) U (T i) =(1 π N ) 2 +(1 η) X σ it j U (j ) j We can write the expected continuation payoff of the sender as: U (j) =(1 π j ) 2 +(1 η) X j σ jtj (1 π j ) (1 η) t π 2 k +(1 η) t+1 U (l) 18

19 Note also that telling the truth in any state gives the strategic sender a lower current payoff than the babbling payoff and lying at state N gives the strategic sender the highest current payoff among all other states. (1 π j ) 2 π 2 l, j π 2 j π 2 N, j U (L N) =π 2 N +(1 η)t π 2 l +(1 η) π2 l (1 η)t+1 U (l) (1 π j ) 2 +(1 η) t π 2 k < 1 4 +(1 η)t π 2 N < π2 N +(1 η)t 1 4 π2 N + (1 η) t π 2 l Thus,wehavethat: U (j) U (L N), j In particular this holds for j = N. Corollary 1 If the state has belief 1 then the sender strictly prefers to lie. π i =1 U (L i) >U (T i) Lemma 3 ender weakly prefers to lie in all the states. U (L i) U (T i), i Proof. uppose U (T i) >U (L i) q i =1 π i =1. By the lemma above, we have a contradiction. We show that the best state to place a strategic sender are the states with lowest beliefs. In other words, that π l = π 1. The proof is by showing that by placing a strategic sender on state 1 gives the receiver a higher payoff than if the sender is placed on state l (l >1). Remember that after a lie, the receiver knows with probability one that the sender is strategic. From now on, we write q i instead of q it. We do this w.l.o.g. because the argument has to hold for any time period. ending the bad sender to vl gives the receiver the following payoff: v l = q l (1 πl ) 2 +(1 η) Σ j σ lt j v j ª +(1 ql ) π 2 l +(1 η) v l ª (6) However, in this state i the strategic sender weakly prefers lying than telling the truth. For if is this not the case, q i =1 π i =1, which implies that lying is actually better for the sender. o we have to consider only the case where (1 π i ) 2 +(1 η) Σ j σ it j U (j ) π 2 i +(1 η) U (i) 19

20 Thus equation (6) can be written as: v l = π 2 l +(1 η) v l (7) Now consider a deviation where the receiver receives a lie and decides to place the sender in the lowest belief state instead of moving to the state where the expected continuation payoff is vl. This deviation gives the receiver a payoff of: v1 = q 1 (1 π1 ) 2 ª +(1 η) Σ j σ 1Tj vj +(1 q1 ) ª π 2 1 +(1 η) v i Again, we have only to consider the case where: (1 π i ) 2 +(1 η) Σ j σ it j U (j ) π 2 i +(1 η) U (i) for if this is not true q 1 = 1 and state 1 would not be the lowest belief state. Thus, again we can write: v 1 = π 2 1 +(1 η) v l (8) However we can compare the expected payoff on equations (7) and (8) to see that: v1 vl, since : π2 1 +(1 η) vl π 2 l +(1 η) vl. This means that after a lie, the receiver always prefers to place the bad sender on state 1. σ if 1 =1, i. Lemma 4 Memory state 1 has highest expected payoff given a strategic sender. 1 D. Proof. vl = q l (1 πl ) 2 ª +(1 η) Σ j σ lt j vj +(1 ql ) π 2 l +(1 η) ª v l However,(1 π l ) 2 +(1 η) Σ j σ lt j U (j ) π 2 l +(1 η) U (l), for if the sender strictly prefers to tell the truth in state l, then we would have that π l = 1 and lying would be strictly preferred as we saw in lemma (?), which would be a contradiction. Thus we can write vl as : v l = π 2 l +(1 η) v l Now consider the expected continuation payoff of placing a strategic sender on state 1. Again, we need only to consider the case where (1 π 1 ) 2 + (1 η) Σ j σ 1Tj U (j ) π 2 1 +(1 η) U (1). Thus, we can write v 1 as: v 1 = π 2 1 +(1 η) v l 20

21 π 1 π l π 2 1 π 2 l, and finally: π2 1 +(1 η) vl π 2 l +(1 η) v l. Thus, v1 vl. ince by definition of v l,v 1 vl, we proved this lemma. The corollary below shows an immediate consequence of this lemma is that unless there is a state π 2 such that π 2 = π 1 and v2 = v1, we must have that σ if 1 =1. Corollary 2 i D π i = π 1. Proof. ince we ordered the states by π i, by definition π 1 π l. uppose π l > π 1. As shown in the lemma above: v l = π 2 l +(1 η) v l If π l > π 1 v l <v 1 ( ). v 1 = π 2 1 +(1 η) v l Corollary 3 For any state j such that π j > π 1 by incentive compatibility σ ilj =0. Proof. ince Pr (H i,l) = 0, i Then by incentive compatibility: p i v1 >p i vj σ ilj =0. In the following lemma we show that, in equilibrium, the order of the states is exactly the opposite of the order by vi. This means that a state with higher belief has lower expected continuation payoff given that the sender is strategic. The proof relies on the fact that after lying the sender is placed to a state where his expected payoff is v1. Again, this lemma relies on the first result of this section, that says that lying is always weakly preferred by the sender. Lemma 5 π i and v i have the exact opposite ordering. Proof. vi = π 2 i +(1 η) v1, i If π j > π i (<) then π 2 i +(1 η) v1 > π 2 j +(1 η) v1 vi >vj. This lemma leads us to the following corollary: the order of states will be the same as the order by vi H. This means that states with higher beliefs have higher expected continuation payoff for the receiver given that the sender is a behavioral type. The proof of this corollary relies on incentive compatibility. If a state is reached with positive probability, than there must not exist 21

22 another state that has higher expected continuation payoff for the receiver for both types of sender (i.e. higher vi and vi H ). ince a state with lower belief has higher vi it must be that this state with lower belief has lower vi H, otherwise for whatever posterior the receiver holds, it is always strictly better to move to this lower belief state than to the original state. Corollary 4 For the states reached with positive probability, π i and vi H the exact same ordering. have Proof. uppose π k > π j, and vj H vk H. If j is reached with positive probability, then i such that: p H i vh j + p i v j p H i vh j + 0 p i v j0, j0 ince π k > π j, we already know that vj >vk. Thus, p H i 0 vh j + p i 0v j p H i 0 vh k + p i 0v k, i0 N, in particular for i 0 = i k is never reached with positive probability. Lemma 6 If the receiver knows with probability one that the sender is behavioral type, she will update to the state with highest expected continuation payoff given a behavioral type of sender. If U (L i) >U (T i)( q it =0, t) by incentive compatibility σ it h = 1. Proof. q i =0 Pr (H i, T )=1. ince we know that vh H vh i, 0 i0 and also that vi H and π i have the same ordering, we must have that: N =argmax i 0 p H i vi H + 0 p i vi =argmax 0 i 0 vh i 0 Thus, σ it N =1 Lemma 7 N U and π h = π N, h U. Proof. First we show that vn H = vh H,h U. q N =0. uppose vh H >vh N σ NTh =1(sinceq N =0). v H h = (1 π h ) 2 +(1 η) X h σ ht h v H h (1 π N ) 2 +(1 η) X h σ ht h v H h (1 π N ) 2 +(1 η) v H h = v H N 22

23 Thus, vh H >vh N cannot happen. The proof that π h = π N is analogous to corollary (2). The next lemma will be important in order to show that the receiver will not move to a lower state after a true signal. Lemma 8 If the sender strictly prefers to lie on state i and is indifferent in state j, thenπ i > π j. U (L i) >U (T i) and U (L j) =U (T j) π i > π j Proof. uppose U (L i) >U (T i), U (L j) =U (T j) and π i π j. U (L i) =π 2 i +(1 η) U (1) U (T i) =(1 π i ) 2 +(1 η) U (h) π 2 i +(1 η) U (1) > (1 π i ) 2 +(1 η) U (h) (9) But, we also have that: π 2 j +(1 η) U (1) = (1 π j ) 2 +(1 η) Σ j σ it j U (j ) (10) ince, π i π j, we have that: π 2 j +(1 η) U (1) π 2 i +(1 η) U (1) > (1 π i ) 2 +(1 η) U (h) However: U (h) U (i), i and (1 π i ) 2 > (1 π j ) 2.Thus, (1 π i ) 2 +(1 η) U (h) > (1 π j ) 2 +(1 η) Σ j σ it j U (j ) Finally, from (9) and (10) we have that: π 2 j +(1 η) U (1) > (1 π j ) 2 +(1 η) Σ j σ it j U (j ). Which is a contradiction with equation (10). The lemma below shows that the receiver will not walk backwards after receiving a true signal. This is true because after receiving this true signal, the receiver does better staying in the same place rather than degrading the sender. The current payoff is higher and also the future payoff. Lemma 9 The Receiver will only go up chain after a true signal. π j > π i σ jti =0. Proof. uppose π j > π i and σ jti > 0. First note that By incentive compatibility: p H j vi H + p j vi p H j vj H + p j vj However, we have that: p H j vi H + p j vi = p H j (1 πi ) 2 +(1 η) P i σ it i vi H + p j vi 23

24 But vi = U (i) =U (L i) U (T i), with strict inequality [only] if q i > 0. If U (L i) >U (T i) σ it N =1, implying that π i > π j [see lemma that says that if q i =0and q j > 0 π i > π j ]. Thus, we conclude that U (L i) =U (T i). Therefore, vi = U (T i) = (1 π i ) 2 + P i σ it i U (i ) vi = (1 π i ) 2 + P i σ it i vi p H j vi H + p j vi = (1 π i ) 2 + P i σ it i p H j vi H + p j vi If, instead of going to state i after a truth the receiver decides to stay on state j for one more period, he gains from that: p H j vj H + p j vj = (1 π j ) 2 + P j σ jtj p H j vj H + p j vj By incentive compatibility: p H j vj H + p j vj ph j vi H + p j vi. Thus: p H j vj H + p j vj p H j vi H + p j vi. Lemma 10 ingle Crossing (for states where p H i 6= p H j ) σ it k > 0,σ it l > 0 and σ jtk > 0 σ jtl =0. Proof. ProofofingleCrossing:σ it k > 0 and σ it m > 0 p H i v H k vm H + p i v k vm =0 (I) uppose σ jtk > 0andσ jtm > 0 p H j v H k vm H + p j v k vm =0 (II) If p H i 6= p H j then (I) and (II) cannot hold at the same time. The lemma below shows a no jump result for states where p H i and p H j are different. Lemma 11 Nojumps(forstateswherep H i 6= p H j ) σ it k 1 > 0,σ it k+1 > 0 σ jtk =0 i, j such that p H i 6= p H j Proof. σ it k+1 > 0 and σ it k 1 > 0 p H i v H k+1 vk H + p i v k+1 vk 0 (11) p H i v H k vk 1 H + p i v k vk 1 0 (12) If σ jtk > 0 p H j p H j v H k+1 vk H + p j v k+1 vk 0 (13) v H k vk 1 H + p j v k vk 1 0 (14) The equations above cannot hold for π k+1 > π k > π k 1 and p H i 6= p H j. 24

25 References [1] Benabou, R. and Laroque, G. Using Privileged Information to Manipulate Markets: Insiders, Gurus, and Credibility, Quarterly Journal of Economics, August [2] Conlisk, J. Why Bounded Rationality?, Journal of Economic Literature, XXXIVJune,pp ,1996. [3] Gilboa, I. A Comment on the Absent Minded Driver s Paradox, Games and Economic Behavior. 20, [4] Greenberg, J. Avoiding Tax Avoidance: A (Repeated) Game Theoretic Approach, Journal of Economic Theory, 32 no. 1, pp1-13, [5] Harrington, W. Enforcement Leverage when Penalties are Restricted, Journal of Public Economics, 37, pp29-53, [6] Hellman, M. and Cover, T. M. Learning with Finite Memory, The Annals of Mathematical tatistics, vol. 41, no. 3, [7] Morris,. Political Correctness, Journal of Political Economy, vol.109, no.2, [8] Piccione, M. and Rubinstein, A. On the Interpretation of Decision Problems with Imperfect Recall, Games and Economic Behavior 20, [9] obel, A Theory of Credibility, Review of Economic tudies, LII, [10] Wilson, A. Bounded Memory and Biases in Information Processing, Job market paper, Princeton,