Towards a Theory of Deception David Ettingeryand Philippe Jehielz 19th May 2006

Abstract Important aspects of deception are related to belief manipulation. This paper proposes an equilibrium approach to belief manipulation and deception. Speci…cally, a framework with boundedly rational players is considered in which agents are assumed to know only the average reaction function of other agents over groups of situations, and agents make their inferences based on a coarse information about others’ behaviors. Equilibrium requires that the coarse information available to agents is correct, and that inferences and optimizations are made based on the simplest theories compatible with the available information. The phenomenon of deception is illustrated through a series of stylized applications including a monitoring game, a bargaining game, and a repeated communication game. It is also suggested how reputation concerns may arise even in zero-sum games in which there is no value to commitment. The approach can be viewed as formalizing into a game theoretic setting a well documented bias in social psychology, the Fundamental Attribution Error.

1

Introduction

Deception and belief manipulation are key aspects of many strategic interactions including bargaining, poker games, military operations, politics or investment banking.1 Anecdotal We would like to thank K. Binmore, V. Crawford, D. Fudenberg, D. Laibson, A. Newman, A. Rubinstein, J. Sobel, the participants at ESSET 2004, Games 2004, ECCE 1, THEMA, Berkeley, Caltech, Institute for Advanced Study Jerusalem, the Harvard Behavioral/experimental seminar, Bonn University, the Game Theory Festival at Stony Brook 2005, the conference in honor of Ken Binmore UCL 2005, LSE internal workshop seminar, GATE 2005 workshop at Northwestern, Tel Aviv University, Stockholm School of Economics, for helpful comments. We are grateful to E. Kamenica for …rst noting the link of the approach to the literature on the FAE. y THEMA, Université de Cergy-Pontoise, 33 boulevard du Port, 95011 Cergy-Pontoise cedex, France ; [email protected] z PSE, 48 boulevard Jourdan, 75014 Paris, France and University College London ; [email protected] 1 Earlier game theoretic studies have developed other aspects of "deception" not related to belief manipulation. These include the ideas of playing mixed strategy (to avoid being detected) in zero-sum interactions

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evidences of belief manipulation and deception are very numerous, and Lewis’s (1990) bestseller "Liar’s Poker" reports colorful illustrations of such strategic behaviors in the world of investment banking in the late 80’s. Some aspects of the professional lives of investment bankers look more like a poker game than the popular view of …nancial engineering would suggest! From the viewpoint of game theory, belief manipulation and the deception that results from it are delicate to capture because traditional equilibrium approaches assume that players fully understand the strategy of their opponents. We depart from this tradition by assuming that players may have a partial rather than total understanding of the strategy of their opponents. This in turn allows us to propose an equilibrium approach to deception, where deception is de…ned to be the process by which actions are chosen to induce erroneous inferences so as to take advantage of them.2 To give a somewhat literary, yet popular, illustration of deception, consider Grimms’ fairy tale ”The wolf and the seven young kids”.3 Before going to the woods, an old goat told her seven kids: ”Be on your guard for the wolf... The villain often disguises himself, but you will recognize him at once by his rough voice and his black feet.” Soon after the goat left, someone knocked at the door and called out ”Open the door... I am your mother...” But, the little kids understood from the rough voice it was the wolf, and they cried out: ”We will not open. You are not our mother. She has a soft voice. You are the wolf.” After …nding a way to make his voice soft,4 the wolf went to the door for a second time and tried again. This time he was denied access because he failed to show white feet, as requested by the kids. The third time is the one of interest to us, the one where the wolf made his voice soft and his feet white.5 As in the …rst and second times, he went to the door and said (with a soft voice): ”Open the door... I am your mother...”The little kids cried out: ”First show us your paw so we may know you are our mother.” So he put his paw inside the window, and when they saw it was white, they believed that everything he had said was true, and they opened the door. The next step is, of course, dramatic for the kids, but for our purpose, the crucial feature in Grimms’ tale is the erroneous inference the kids make after seeing the white paw (and (von Neuman-Morgenstern), not playing a separating equilibrium (thereby not revealing one own’s type) in signaling games (Spence (1973)) or communication games (Sobel (1985) or Crawford (2003)) or repeated games (Kreps et al. (1982), Fudenberg and Levine (1989)). 2 Our view of deception is related but not identical to that of Vrij (2001) who de…nes deception as a successful or unsuccessful deliberate attempt, without forewarning, to create in another a belief that the communicator considers to be untrue in order to increase the communicator’s payo¤ at the expense of the other side (see also Gneezy (2004)). Vrij’s de…nition puts an emphasis on the lie aspect of deception whereas we put emphasis on the cognitive process through which the communicator manages to manipulate the belief of the receiver. 3 It seems that Grimms’ tales are better known in Europe than in the US. The following lines summarize the key features of the tale that are needed for our purpose. A reader interested in the tale should consult the full text (see, for example, http://www.‡n.vcu.edu/grimm/wolf_e.html). 4 According to Grimms’tale, eating a piece of chalk does that! 5 He has had some ‡our sprinkled on his feet.

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hearing the soft voice) of the wolf. At that point, the kids believe it is her mother and they open the door: they have been deceived by the wolf. In Grimms’ tale, the kids are de…nitely right that when the one knocking at the door has a soft voice and a white paw it is, in general, unlikely to be the wolf.6 (This is what the goat taught the kids.) But, in the special situation in which the wolf is being informed of the cues being used (as is the case for the third attempt of the wolf), the cues are not informative any longer (because even the wolf can pass the test). Yet, when the wolf knocks at the kids’door for the third time, the kids do not adapt their inference process using the extra information that the wolf has been told the cues.7 This erroneous inference process is what allows the wolf to deceive the kids. More generally, this paper will illustrate how deception may arise whenever agents use insu¢ ciently …ne cues to make their inferences. The cognitive bias this paper is related to has a long tradition in social psychology: it is referred to as the Fundamental Attribution Error (FAE) (see Jones and Davis (1965), Ross (1977), Ross, Amabile and Steinmetz (1977)).8 Roughly speaking, the FAE is "the tendency (in forming one own’s judgement about others) to underestimate the importance of the (speci…c) situation in which the observed behavior is occurring" (O’ Sullivan (2003)).9 In Grimms’tale, the FAE takes the form that the kids simply ignore in forming their judgement (about who is knocking at the door) that they have informed the wolf of the required tests. This paper formalizes the FAE into a game theoretic equilibrium approach, and it shows how deception can arise as a result of the FAE. In a nutshell, deception will be viewed as the exploitation (by rational agents) of the FAE. There are myriads of real life examples in which people do not take full advantage of the (…ne) details of the situation, and make their inferences based on a coarse understanding of others’ behaviors. For example, consider the popular belief that someone looking into another person’s eyes is unlikely be a liar.10 We will not dispute that it may be a correct view in general. But, if only this cue is used to detect lies, it opens the door to the possibility 6

In most situations, the kids would not have informed the wolf of the cues being used. An alternative possible (however, to us, less appealing) explanation would be that the kids do not remember having told the cues to the wolf. But, there is no (explicit) reference to such imperfect recall in the tale. 8 For further discussion and references, see http://en.wikipedia.org/wiki/Fundamental_attribution_error 9 Ross et al. (1977) report a striking example in support of the FAE. "Questioners" were requested to ask di¢ cult questions to answerers. Every questioner was matched to a single answerer. After the quizz (answerers and questioners then knew how many good answers were made in their match), it was observed that answerers consistently thought they were less good than questioners, thereby ignoring that the pool of questions on which they had a relatively poor performance was not generated at random but drawn from the esoteric knowledge of the questioner. (The same study suggests that there was no observed bias on the questioner’s side.) 10 There is a long tradition starting with Darwin that tries to elicit the link between emotions and facial expressions (with a special focus on whether subjects can control their own facial expressions). Ekman (2003) who extends the study to deception and lie detection suggests that a number of facial expressions can be controlled. He also suggests that subjects do not, in general, consider the cues that are the best predictors of lies. 7

3

of deception (at least in some situations). For the sake of illustration, suppose that there is an equal share of truth-tellers and manipulative agents in the population, and suppose that there are two types of situations A or B. Truth-tellers are assumed to look into another person’s eye both in situations A and B whereas manipulative agents are assumed to look into another person’s eye in situations A but not in situations B. Suppose further that judgements (about agents’characteristics) are based on whether the agent looks into another person’s eye, but not on the type of situation A or B.11 By looking into another person’s eye in situations A, a manipulative agent conveys the (false) belief that he is more likely to be a truth-teller because looking into another person’s eye is more typical of truth-tellers than of manipulative agents’behaviors over all situations A and B. This is a false belief because in situations A all agents (whether truth-telling or manipulative) look into another person’s eye. Thus, looking into another person’s eye is not informative in situations A, and the correct belief should be that the two types of agents are equally likely (this is the prior belief). Clearly, this inference error may turn to the advantage of the manipulative agent, which is the essence of deception as considered in this paper. This paper provides the formal tools to analyze such inference errors (and their strategic exploitations) in a game-theoretic setting. The model combines the classic approach of Sequential Equilibria (Kreps and Wilson (1982)) and the recent approach of Analogy-based Expectation Equilibria (Jehiel (2005)). Speci…cally, we consider two-player multi-stage games with incomplete information and observable actions. Players may be of several types, and past actions are assumed to be observable by everyone. Types may a¤ect the preference relations, but the main novelty lies in the introduction of cognitive types. In addition to their preference types, players are characterized by their ability to understand (or learn) the strategy of their opponent. Cognitive types are modelled by assuming that players partition the decision nodes of their opponents into various sets referred to as analogy classes, and that players understand only the average behavior of their opponent over the various decision nodes forming their analogy classes. Cognitive types are further di¤erentiated according to whether or not the player distinguishes between the types of his opponent. Thus, cognitive types may vary in two dimensions: A player may be more or less …ne on the partition of the decision nodes of his opponent (the analogy part), and a player may or may not distinguish the behaviors of the various types of his opponent (we refer to the latter as the sophistication part). The equilibrium concept we use is the Analogy-based Sequential Equilibrium. It is formally described in Section 3. In equilibrium, players have correct expectations about the average behavior of their opponents in the various analogy classes - these are referred to as 11

The person assessing the agent’s type may however observe whether he is in situation A or B (her action space or signal structure may be di¤erent in A and B). So the framework is very di¤erent from the one arising with imperfect recall (à la Piccione and Rubinstein (1997)).

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analogy-based expectations. Besides, whenever they move, players play best-responses to their analogy-based expectations and to their belief about the type of their opponent. As the game proceeds, players update their beliefs about the type of their opponent according to Bayes’rule as derived from their analogy-based expectations. Compared to Sequential Equilibria, the main novelty lies in the introduction of cognitive types who may only know partial aspects of the strategy of their opponent. Compared to Analogy-based Expectation Equilibria (Jehiel (2005)), the main novelty lies in the introduction of players’uncertainty about the type of their opponent. It is the combination of the two features that allows us to speak of inference errors of the type referred to as FAE by social psychologists.12 We think of the Analogy-based Expectation Sequential Equilibrium as the limiting outcome of a learning process involving populations of players who are a priori unaware of the structure of the payo¤s of their opponents (so that introspective reasoning about the play of opponents is limited). Each individual player plays the game just a few times, and those individuals who must play in a given round have a limited access to the database that records the past behaviors of individuals engaged in similar interactions. The cognitive type of a player as de…ned by his analogy coarseness and his sophistication dimension (whether or not he distinguishes between the behavior of each type of his opponent) can be interpreted as standing for how limited his access is to the behaviors of players (previously) engaged in similar interactions. For example, a player with a coarse analogy partition has access only to the average past behaviors of the opponents within each of his classes. A sophisticated player has access to the average behaviors for each type of the opponents while a non-sophisticated player has only access to the average behaviors across all types in each of his classes. The Analogy-based Sequential Equilibrium concept assumes that the underlying learning process with data processing as just described has converged. Deception as a result of belief manipulation may arise in such a setup whenever there are several possible types of player i (including a type who is fully rational) and at least one of player j’s cognitive types is a sophisticated coarse type, by which we mean that this type of player j di¤erentiates between the behaviors of each possible type of player i (the sophisticated part), but she does not distinguish the behaviors at each possible decision node (i.e. she uses a coarse analogy partition and she knows only the average behaviors of player 12

Recently, Eyster and Rabin (2005) have proposed a concept for static games of incomplete information, called cursed equilibrium, in which players do not fully take into account how other people’s actions depend on these other people’s information. Eyster and Rabin’s fully cursed equilibrium has some connection with the analogy-based expectation equilibrium (see Jehiel (2005), Eyster and Rabin (2005) and Jehiel and Koessler (2006) for further discussion). The cursed equilibrium of Eyster-Rabin gives rise to some erroneous equilibrium belief about the relation between the strategy and the signal of the opponent, which is relevant for problems with interdependent preferences (common value uncertainty). Yet, by the very static nature of the games considered by Eyster-Rabin, no belief manipulation can be captured by their approach (players’beliefs cannot change in a static framework, hence they cannot be purposedly manipulated). See also Fudenberg (2006) for further discussion on Eyster-Rabin’s concept.

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i’s types in each analogy class). Deception is the exploitation (by rational player is) of the erroneous inference process of this type of player j that arises due to her coarse analogy grouping on the one hand and her di¤erentiation of the behaviors of the various types of player i on the other.13 Observe that deceiving type j of player j requires that this type j be neither fully rational (otherwise her belief could not be manipulated as it would have to be correct) nor fully irrational (in the sense of not distinguishing the behaviors of the various types of player i) as this would forbid the possibility of inference process as the interaction proceeds. Our theory thus provides some content to the common sense idea that the best candidates for belief manipulation and deception are individuals who are neither too smart nor too dumb.14 Beyond providing a theoretical framework to cope with the phenomenon of deception as a result of belief manipulation, the paper also suggests how various standard economic insights may be a¤ected by the possibility of deception. We show how the analysis of incentives is substantially altered in a simple monitoring game, in which due to the coarseness of the supervisor’s cues, incentives are designed as if the supervisor were facing an adverse selection problem whereas she is, in fact, facing a moral hazard problem. We next consider a zero-sum application for which our approach suggests a new perspective on the theory of reputation. While reputation is generally associated with the idea of commitment, we show that reputational concerns may arise even in multi-stage zero-sum games in which there is no value to commitment. We also suggest how, in a bargaining application, deception may make some poor alternatives look credible in situations in which they would traditionally be considered as irrelevant. Finally, we suggest how deception and belief manipulation can be magni…ed as one increases the duration of the interaction in a simple trust game in which a sender can either be a friend or an enemy and who repeatedly communicates his knowledge about the state of the world. The rest of the paper is organized as follows. Section 2 presents an illustration of our approach in a simple game in which one agent communicates his intended play before playing a normal form game (this allows us to di¤erentiate our approach from that of Crawford (2003)). Section 3 presents the formal model, which is further applied in Sections 4 to 7. Section 8 concludes. 13 Exploitations of boundedly rational agents arise in very di¤erent contexts (not involving mistaken updating) in Piccione and Rubinstein (2003), Gabaix and Laibson (2005) or Spiegler (2005). 14 It should be noted that the inference process of our sophisticated coarse player j is erroneous here only to the extent that the cues used by player j (her analogy classes) are not …ne enough. Player j is in all other respects perfectly standard in that she has correct expectations conditional on the cues she uses, and she relies on Bayes’law (as derived from her analogy-based expectations) to update her belief. Deception would a fortiori arise in our setup if we were to explicitly introduce non-Bayesian elements in the updating process (see Kahneman et al. (1982) or Thaler (1991) for an exposition of such biases). But such non-Bayesian elements are not the key element of our theory. The key element is the coarseness of the cues used by the players (which, as explained above, is related to a well founded psychological bias, the FAE).

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2

Belief manipulation in a simple pre-play communication game

Consider the following framework. There are two possible underlying games A or B, where game A is chosen (by nature) with probability , and game B is chosen with probability 1 . Game A is the introductory example considered by Crawford (2003) to model Operation Fortitude in World War II.15 It will be described shortly. Game B is here to introduce the possibility of confusion in the interpretation of messages. In both A and B, two players a Sender and a Receiver (who observe the game they are in) choose simultaneously between two pure actions U for U p or D for Down for the Sender and L for Lef t or R for Right for the Receiver. Before playing the game (A or B), the Sender sends the Receiver a costless, nonbinding message u or d about his intended action, with u (d) representing action U (D) in a commonly understood language. The message is observed without noise by the Receiver. There are two types of Senders: d-truthtellers who always (in both A and B) send message d and then play D,16 and rational Senders whose payo¤s will be described shortly, and who, as rational agents, have a perfect understanding of the strategy of the Receiver. Their behavior will be endogeneized later. The type of the Sender is not directly observed by the Receiver who knows only that the prior probability that a Sender is a d-truthteller is 0. There is one type of Receiver. The Receiver is boundedly rational in the sense that she does not fully understand the strategy of the Sender. Yet, she is assumed to di¤erentiate the behaviors of the two types of Senders, which allows her to update her belief (about the Sender’s type) after observing the message sent by the Sender. Speci…cally, we assume that the Receiver knows only the average communication strategy of the two types of Senders (over A and B), but not the communication strategy game by game (i.e., in A and B separately). She also understands the distribution of actions chosen by each type of Senders in each game A and B separately. The payo¤s of the Receiver and of the rational Sender in A and B are as follows. From their perspective, both A and B are zero-sum games. In game A their payo¤s are given in Figure 1 with a > 1, and they correspond to a perturbed Matching Pennies game such as the one considered by Crawford (2003). 15

Crawford’s model is related to that of Hendricks and McAfee (2005). Yet, it adds the essential element of "crazy" types. 16 Crawford (2003) considers more general crazy types, but for the sake of our illustration the presence of d-truthtellers is su¢ cient.

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L

R

U

(a,-a) (0, 0)

D

(0, 0) (1,-1)

Figure 1. Game A Payo¤s in B are given in Figure 2. L

R

U

(a,-a) (1,-1)

D

(1,-1) (0,-0)

Figure 2. Game B We assume that the proportion

of games A is not too large and satis…es:17 a

0

1

(1) 0

We will now check that the following strategy pro…le is an Analogy-based Expectation Sequential Equilibrium (see Section 3 for a formal de…nition). d-truthellers: In games A and B, Send d and Play D. Rational Sender: In game A, Send d and Play U ; In game B, Send u and Play U . Receiver: In game A, Play L if d is received (and R if u is received); in game B, Play R always. The behaviors of d-truthtellers is dictated by the de…nition of their types. Besides, the behaviors in game B are optimal because in this game U is a dominant strategy for (rational) Senders, and R is a dominant strategy for Receivers (and the message sent has no e¤ect). So let us concentrate on game A. In game A, the Rational Sender gets an overall payo¤ of a, which is clearly the maximum payo¤ he can hope to get. So his behavior is clearly optimal. The Receiver gets a payo¤ equal to a when she faces a rational Sender and 0 when she faces a d-truthteller, so an expected payo¤ equal to (1 0 )( a). a , which is obtained when the Sender Observe that the value of the Receiver in A is 1+a plays U with probability 1=(1+a) and the Receiver plays L with probability 1=(1+a). Thus, whenever (1 + a)(1 0 ) < 1 (which is compatible with (1)) the Receiver gets less than her value in A! This is so because the Receiver is being deceived by the (rational) Sender in game A. After seeing message d, the Receiver puts an excessively high probability on the Sender being a d-truthteller (because d is more typical of d-truthellers’ communication strategies when no distinction is made between A and B). This erroneous belief leads her in turn to 17

This constraint is binding when

0

is not too large.

8

choose a strategy that in her mind performs better (and yet, in reality, performs worse) than the minmax strategy that guarantees the value.18 More precisely, the Receiver’s inference process goes as follows. The Receiver knows the prior probability 0 that a Sender is a d-truthteller. She also knows that on average over all games (A and B), a d-truthtelling Sender sends message d with probability 1 while a rational Sender sends message d with probability (rational Senders send message d only in A and the frequency of games A is ). Thus, after seeing message d, the Receiver believes that she faces a d-truthtelling Sender with probability +(1 0 ) (this is the posterior belief 0 0 assuming that rational Senders send message d with probability uniformly over games A and B while d-truthellers always send d). Based on her perceived posterior +(1 0 ) , the 0 0 Receiver chooses optimally to play L.19 Crawford (2003) considers a related model, but in line with the previous literature, he assumes that those players whose beliefs are made endogenous are perfectly rational. Thus, in Crawford’s model, if the rational Sender were to play according to the strategy shown above, the (rational) Receiver would believe that the probability that the Sender is a dtruthteller is the prior 0 (because in game A message d is not informative). Accordingly, the Receiver would …nd it optimal to play L whenever a 1 0 . In this case, she would get 0 a an expected payo¤ of (1 0 )( a), which is, of course, greater than the value 1+a . Crawford’s model makes good progress in modelling the idea of lying for strategic advantage (in his model as in ours rational Senders send messages that do not correspond to their actual play and they take advantage of it).20 Yet, the belief of rational Receivers in Crawford’s model cannot be manipulated as equilibrium requires that rational Receivers are not mistaken about the distribution of types nor about their strategies. By contrast, in our model, the Receiver ends up with a mistaken belief, which in turn leads her to choose a suboptimal strategy that she thinks (in her mind) to be very smart. The exploitation of such mistakes is the essence of deception as a result of belief manipulation, and as far as we are aware, it does not have its counterpart in earlier game-theoretic approaches. 18

>From the learning perspective suggested in Introduction, it is important that a speci…c Receiver does not play herself too often the game as the observation of past performance might trigger the belief that there is something wrong with the Receiver’s theory. The view that deception is more likely to occur when agents do not play a game too often seems in accordance with the public opinion. 19 0 This is because a ensures that (1+(1 0 ) ) ( a) > +(1 0 ) ( 1), and the Receiver correctly 1 0 0 0 0 0 expects a d-truthteller to play D and a rational Sender to play U in game A. 20 It is interesting to observe that in Crawford’s model rational Senders take advantage of the presence of truthtellers only if the latter are su¢ ciently numerous (a > 1 0 whenever 0 is small enough). In our 0 model, lying for strategic advantage arises whatever 0 , as long as is small enough (condition 1 need to be satis…ed).

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3

A general framework

In this Section we provide a general framework to analyze the interactions of agents with limited cognitive abilities of the type described above. The construction follows those of the Sequential Equilibrium (Kreps and Wilson (1982)), and the Analogy-based Expectation Equilibrium (Jehiel (2005)). A reader interested in applications may jump into Section 4.

3.1

The class of games

We consider multi-stage two-player games with observed actions and incomplete information. Extension to more than two players raises no conceptual di¢ culties. Each player i = 1; 2 can be one of …nitely many types i 2 i . Player i knows his own type i , but not that of player j, j 6= i. We assume that the distribution of types is independent across players, and we let p i > 0 denote the prior probability that player i is of type i . These prior probabilities pi = (p i ) i are assumed to be known to the players. Players observe past actions and earlier moves by nature (except for the choice of their opponent’s type). Moreover, there is a …nite number of stages, and, at every stage and for every player (including nature), the set of pure actions is …nite. Player i whatever his type is assumed to face the same choices throughout the game. That is, player i whatever his type plays in the same set Hi of histories.21 Moreover, the action space of player i at history h 2 Hi is common to all types i , and it is denoted by Ai (h). The set of all histories is denoted by H and the set of terminal histories is denoted by Z. The set of players who must move at history h is denoted by I(h), and ha is the history starting with h and followed by a where a 2 Ai (h) is the action pro…le played (by the i2I(h)

players who must move) at node h. Each player i is endowed with a VNM utility function de…ned on lotteries over terminal histories h 2 Z. Player i’s VNM utility is denoted by ui and it may depend on the types of players i and j together with the terminal history. That is, ui (h; i ; j ) is player i’s payo¤ if the terminal history h 2 Z is reached, and players i and j are of type i and j , respectively. So far the description of the strategic environment (referred to as ) is perfectly standard. The novelty lies in the de…nition of the types i . Types i are made of two components i = (ti ; ci ) where ti is the preference type of player i that acts on players’preferences - this is the standard component in the type - and ci is the cognitive type of player i that stands for how …nely player i understands the strategy of player j - this is the novel non-standard component in the type. 21

A history in this class of games refers to the earlier moves made by the players and possibly the earlier moves made by nature except for the choice of players’types (which is not included in the history). Given our observability assumptions, histories are commonly known to the players.

10

As common sense suggests, the cognitive type of players do not a¤ect players’assessments of the various terminal nodes. That is, for every terminal history h 2 Z, we have that 0 ui (h; i ; j ) = ui (h; 0i ; 0j ) whenever i and i have the same preference type ti , and j and 0 j have the same preference tj . With a slight abuse of notation, this common utility will be denoted by ui (h; ti ; tj ) whenever convenient. Cognitive types ci are de…ned as follows. Each player i forms an expectation about the behavior of player j by pooling together several histories h 2 Hj at which player j must move. Each such pool of histories is referred to as a class of analogy. Players are also di¤erentiated according to whether or not they distinguish between the behaviors of the various types of their opponent. Formally, a cognitive type ci of player i is characterized by (Ani ; i ) where Ani stands for player i’s analogy partition and i is a dummy variable that speci…es whether or not type i distinguishes between the behaviors of the various types j of player j. We let i = 1 when type i distinguishes between types j ’s behaviors and i = 0 otherwise. Following Jehiel (2005), type i ’s analogy grouping Ani is de…ned as a partition of the set Hj of histories at which player j must move into subsets i called analogy classes.22 When h and h0 are in the same analogy class i , it is required that Aj (h) = Aj (h0 ). That is, at two histories h and h0 that player i treats by analogy, the action space of player j should be the same, and A( i ) denotes the common action space in i .

3.2

Solution Concept

Analogy-based expectations: An analogy-based expectation for player i of type i is denoted by i . It speci…es for every analogy class i of type i of player i a probability measure over the action space A( i ) of player j. Types j of player j are distinguished or not according to whether i = 1 or 0. If i = 1, i is a function of j and i , and i ( j ; i ) is type i -player i’s expectation about the average behavior of player j with type j in class i . If i = 0, player i merges the behaviors of all types j of player j, and i is a sole function of i : i ( i ) is player i’s expectation about the average behavior of player j in class i (where the average is taken over all possible types). We let i = ( i ) i 2 i denote the analogy-based expectation of player i for the various possible types i 2 i . Strategy: A behavioral strategy of player i is denoted by si . It is a mapping that assigns to every history h 2 Hi at which player i must move a distribution over player i’s action space 22

A partition of a set X is a collection of subsets xk

X such that

S k

11

xk = X and xk \ xk0 = ; for k 6= k 0 .

Ai (h).23 We let i denote the behavioral strategy of type i , and for every n 2 Ni we let (h) 2 Ai (h) denote the distribution over Ai (h) according to which player i of type i i selects actions in Ai (h) when at h. We let i (h)[ai ] be the corresponding probability that type i plays ai 2 Ai (h) when at h, and we let i = ( i ) i denote the strategy of player i for the various possible types i ; will denote the strategy pro…le of the two players. Belief system: When player i distinguishes the types of player j, i.e. i = 1, he holds a belief about the type of his opponent and this belief may typically change from one history to another. Formally, we let i denote the belief system of player i of type i = (Ani ; i ), where i (h)[ j ] is the probability that player i of type i assigns to the event ”player j is of type j ” conditional on the history h being realized. When player i does not distinguish the types of player j, no belief system is required. To save on notation, we assume that in this case player i ’s belief coincides with the prior pj throughout the game. We call i the belief system of player i for the various possible types be the pro…le of belief systems for the two players i = 1; 2. i , and we let Sequential rationality: From his analogy-based expectation i , player i of type i derives the following representation of player j’s strategy: At all histories h of the analogy class i player j is perceived to behave according to the average behavior in class i as given by i .24 The induced strategy depends on the type j of player j whenever i = 1 (but not when i = 0). At every history h 2 Hi where he must play, player i is assumed to play a best-response to this perceived strategy of player j. Formally, we de…ne the i -perceived strategy of player j, j i , as If

i

= 1

If

i

= 0

i j i j

(h) =

i

(h) =

i

( j;

i)

for every h 2

( i ) for every h 2

i

i

and

and j

2

j

2

j

j

Given the strategy si of player i and given history h, we let si jh denote the continuation strategy of player i induced by si from history h onwards. We also let uhi (si jh ; sj jh ; i ; j ) denote the expected payo¤ obtained by player i when history h has been realized, players i and j’s types are given by i and j , respectively, and players i and j behave according to si and sj , respectively.25 De…nition 1 (Criterion) Player i’s strategy

i

23

is a sequential best-response to ( i ;

i)

if and

Mixed strategies and behavioral strategies are equivalent since we consider games of perfect recall. This is the simplest representation compatible with type i ’s knowledge. 25 Observe that this expected utility depends only on the preference types ti and tj of players i and j. 24

12

only if for all X j2

i

2

i,

for all strategies si and all histories h 2 Hi ,

(h)[ j ]uhi ( i

i

j

jh ;

i j

jh ; i ;

X

j)

j2

i

j

(h)[ j ]uhi (si jh ;

i j

jh ; i ;

j ):

Consistency: In equilibrium, two notions of consistency are required: the …rst consistency requirement relates the analogy-based expectations to the strategy pro…le; the second one relates the belief systems to the analogy-based expectations. We start with the consistency of the analogy-based expectations. Analogy-based expectations are required in equilibrium to coincide with the real average behaviors in every considered class and for every possible type (if types are di¤erentiated) where the weight given to the various elements of an analogy class must itself be consistent with the real probabilities of visits of these various elements. A learning interpretation of this consistency requirement will be suggested. Here is the formal de…nition of consistency where P ( i ; j ; h) denotes the probability that history h is reached when players i and j are of types i and j respectively, and players play according to .

De…nition 2 Player i’s analogy-based expectation if and only if: For any ( i ;

j)

2

such that ( j; i

whenever there exist For any

i

2

0 i

i)

whenever there exist

i

( i) = 0 i,

=

= 1, and for all

P

( 0i ;h)2

and h 2

such that

i

i

0 j

i

P

i

i

( 0i ;h)2

i

i

0 j ;h)2

P

( 0i ;

and h 2

i 0 j ;h)2

i

2 Ani ,

p 0i P

such that P ( 0i ;

i

( 0i ;

i

p 0i P ( 0i ;

= 0, and for all P

is consistent with the strategy pro…le

i

(h) j ; h) j ( 0i ; j ; h)

j ; h)

2 Ani ,

p 0i p 0j P ( 0i ; i

> 0:

p 0i p 0j P

such that P ( 0i ;

0 0 (h) j ; h) j 0 0 ( i ; j ; h) 0 j ; h)

> 0.

The consistency of the analogy-based expectations should be thought of as the limiting outcome of a learning process. Speci…cally, assume that there are populations of players i and j who are repeatedly and randomly matched to play the game. In the population of players i, there is a proportion p i of players of type i . After the end of a session, the behaviors of all individual players and their types are revealed. All pieces of information are 13

gathered in a general data set, but players have di¤erent access to this data set depending on their types. A player i with cognitive type ci = (Ani ; i ) such that i = 0 has access to the average empirical distribution of behavior in every analogy class i 2 Ani where the average is taken over all histories h 2 i and over the entire population of players j. A player with cognitive type ci = (Ani ; i ) such that i = 1 has access to the average empirical distribution of behavior in every i 2 Ani for each subpopulation of types j of players j. At each round of the learning process, players choose their strategy as a function of the feedback they received, which in turn generates new data for the next round. Each individual player need not be present in every round. Actually, a more compelling scenario (for the present theory) is one in which each individual player plays just a few rounds, and he is then replaced by another player with the same type. 26 Now suppose that the true pattern of behavior adopted by the players is that described by the strategy pro…le . A player i with cognitive type ci = (Ani ; i ) such that i = 1 will collect data about the average behavior of types j in every class i 2 Ani as soon as a player j with type j reaches some history h 2 i with positive probability (according to ). In the long run, every such statistic should converge (in Cesaro’ s sense) and the limit point should be an average of what player j with type j actually does at each of the histories h where h 2 i , that is, j (h). The weighting of j (h) should also coincide with the frequency with which h is visited (according to ) relative to other elements in i , hence the above expression for i ( j ; i ). A similar argument applies when i = 0 for i ( i ). The second consistency requirement relates players’ belief systems to their analogybased expectations. The analogy-based expectation i of player i with type i = (ti; ci ), ci = (Ani ; i ), i = 1 allows him to distinguish between the behaviors of players j with di¤erent types. As the game proceeds, player i updates his belief about the type of player j using Bayes’rule (whenever applicable) and assuming that type j behaves according to i above). i ’s perception (see j De…nition 3 Player i’s belief system i is consistent with the analogy-based expectation if and only if for any ( i ; j ) 2 such that i = 1 i

i

( j )(;) = p j :

26

This ensures that players have little opportunity to re…ne their understanding of the strategy of their opponents based on their own experience, and the replacement assumption guarantees the stationarity of the distribution of types in the population. The replacement scenario is reminiscent of the recurring game framework studied by Jackson and Kalai (1997) who assume that each individual player plays only once. This is to be contrasted with a recent paper by Esponda (2005) who, in static games of incomplete information, elaborates on Eyster-Rabin’s fully cursed equilibrium by assuming that player is have access both to the empirical distribution of actions of player js (but not how these actions are related to j’s private information) and i’s own distribution of payo¤s.

14

And for all histories h, ha i

(ha)[ j ] =

i

( j )(ha) = P i

(h)[ j ] whenever h 2 = Hj i 0 j2

(h)[ j ]

j

i

i j

(h)[aj ]

(h)[ 0j ]

whenever h 2 Hj , there exists

0 j

0 j

i

(h)[aj ]

s.t.

0 j

i

(h)[aj ] > 0 and player j plays aj at h:

Comment: The consistency of the belief system i with the analogy-based expectation i should be thought of as resulting from an introspective calculus of player i. Based on his representation of the strategy of the various types of his opponent he makes inferences (using Bayes’ law) as to the likelihood of the various possible types he is facing. This should be contrasted with our learning interpretation of the consistency requirement for the analogybased expectations (see above De…nition 2).27 Equilibrium: At every history, players play best-responses to their analogy-based expectations (sequential rationality) and both analogy-based expectations and belief systems are consistent. In line with the Sequential Equilibrium (Kreps and Wilson (1982)), we require the analogybased expectations and belief systems to be consistent with respect to slight totally mixed perturbations of the strategy pro…le where a totally mixed strategy for player i is a strategy that assigns strictly positive probability to every action ai 2 Ai (h) and for every history h 2 Hi . This in turn puts additional structure on the expectations and beliefs at histories that belong to analogy classes that are never reached in equilibrium.28 De…nition 4 A strategy pro…le is an Analogy-based Sequential Equilibrium if and only if there exist analogy-based expectations i , belief systems i for i = 1; 2; and sequences ( k )k , ( ki )k , ( ki )k converging to , , , respectively such that each k is a totally mixed strategy pro…le and for every i and k : 1. i is a sequential best-response to ( i ; i ) 2. ki is consistent with k and 3. ki is consistent with ki . Compared to Sequential Equilibria, the main novelty lies in the introduction of cognitive types who may only know partial aspects of the strategy of their opponent. Compared to 27

A frequentist interpretation of this consistency requirement is possible in some cases, but it is less appealing to us (and it would be problematic in cases in which player i has to update his belief about the type of player j following several moves of player j). 28 It should be noted that these perturbations (sometimes referred to as trembles) have less bite with coarser analogy partitions because for an analogy class to be reached with positive probability it is enough that one of the histories in the analogy class is reached with positive probability - a requirement that is weaker when the analogy class is larger.

15

Analogy-based Expectation Equilibria (Jehiel (2005)), the main novelty lies in the introduction of players’uncertainty about the type of their opponent. Is is the combination of the two features that allows us to speak of deception in the sense discussed in Introduction. More precisely, deception requires the presence of players who are both uncertain about their opponent’s type (so that there is room for inference processes) and are partially knowledgeable of the strategy of their opponent so that the inferences may be erroneous. We believe that the kind of inference errors made by our boundedly rational players is closely related to what social psychologists refer to as the Fundamental Attribution Error. Indeed, our players make their inferences by incorrectly assuming that the behavior of their opponent is not sensitive to the situation (as long as it belongs to the same analogy class). So in line with the literature on the FAE, one can argue that they pay insu¢ cient attention to the situation that drives their opponent’s behavior.29 In the next two sections, we apply the approach to several economic problems. Before moving to these applications, we make three preliminary observations. Proposition 1 In …nite environments, there always exists at least one Analogy-based Sequential Equilibrium. Proof: The proof follows standard methods, …rst noting the existence of equilibria in which each player i is constrained to play any action ai 2 Ai (h) at any history h 2 Hi with a probability no less than ", and then by showing that the limit as " tends to 0 of such strategy pro…les is an Analogy-based Sequential Equilibrium. Q. E. D. Our second observation concerns the link of the Analogy-based Sequential Equilibrium to standard equilibrium concepts when players’cognitive types are as …ne as possible. Speci…cally, given a strategic environment as described above, we de…ne the associated (standard) game st in which for each player i = 1; 2, each type i = (ti ; ci ) of player i is replaced by the (preference) type ti only.30 It is readily veri…ed that: Proposition 2 Consider an environment such that for each player i = 1; 2 and all types i = (ti ; ci ) of player i, the cognitive type ci = (Ani ; i ) is such that Ani is the …nest analogy S partition fhg, and player i distinguishes between player j’s types, i = 1. If is an h2Hj

Analogy-based Sequential Equilibrium of , then there exists a belief system in st such that ( ; ) is a Sequential Equilibrium of st . Conversely, if ( ; ) is a Sequential Equilibrium of st , then is an Analogy-based Sequential Equilibrium of . 29

Our approach o¤ers a new perspective on this well documented psychological bias by suggesting that the FAE may be the result of agents’imperfections in information gathering at the learning stage. 30 In st players are perfectly rational as in the Sequential Equilibrium approach, and there are no cognitive types.

16

Our third observation is that in environments with a mix of rational and boundedly rational players, rational players do no worse than their boundedly rational fellow players with the same preference type.31 Proposition 3 Consider an Analogy-based Sequential Equilibrium of an environment including two types i = (ti ; ci ), 0i = (ti ; c0i ) of player i with the same preference type ti but not the same cognitive type ci = c0i . Assume further that type i is rational in the sense that S ci = (ti ; ci ), ci = (Ani ; i ) and Ani = fhg, i = 1 . Then type i of player i gets an h2Hj

equilibrium expected payo¤ that is no smaller than that of type

0 i.

Proof: Player i with type i can always mimic the behavior of type 0i , thereby ensuring that he can get at least as much as the expected payo¤ obtained by 0i . Q. E. D.

4

The monitoring game

We consider the following stylized monitoring game in which agents di¤er only in their cognitive types (not their preference types). At date t = 0, an employee decides whether to Work (exert e¤ort) or Shirk. After observing the employee’s date t = 0 decision, the supervisor decides at t = 1 whether to Delegate the decision making to the employee (give him discretion) or Control him. At each of the next two dates t = 2; 3, the employee decides whether to Shirk or Work. We assume that payo¤s are speci…ed so that the employee does not like working unless he is controlled, and that the cost of control (for the supervisor) is less than the bene…t that results from the employee working. Speci…cally, if the employee works at t = 0, he gets 0 and the supervisor gets 1. If the employee shirks at t = 0, he gets 1 and the supervisor gets 0. Whenever the supervisor chooses to control (C), shirking at t = 2; 3 is costly to the employee but not to the supervisor who gets 2 whatever the employee’s choices at t = 2; 3. The employee’s pay-o¤ is strictly decreasing in the number of times he shirks at t = 2; 3: He gets 2 if he works twice; 1 if he works once and shirks once; and 0 if he shirks twice.32 31

This should be contrasted with results suggesting that irrational types may perform better in equilibrium. Here it is a comparison of the equilibrium payo¤s obtained by di¤erent types within the same equilibrium. It is not a comparison between the equilibrium payo¤s obtained by the rational types and the irrational ones when one switches from an environment with only rational types to an environment with only irrational types. 32 An interpretation of the control technology is that it is such that the employee always ful…lls his task. If he shirks, he is punished and eventually does what he should do.

17

1 W 2

HH HH S H HH HH 2 @

@ @C @C D @ @ @ @ @1 @1 1 1 A A A A W AS W AS W AS W AS A A A A A1 A1 A1 A1 1 1 1 1 A A A A A A A A W S W AS W S W AS W S W AS W S W AS A A A A A A A A (1,4)(2,3)(2,3)(4,1)(2,3)(1,3)(1,3)(0,3)(2,3)(3,2)(3,2)(5,0)(3,2)(2,2)(2,2)(1,2)

D

Fig 3. The monitoring game Whenever the supervisor chooses to delegate (D), her payo¤ is strictly increasing in the number of times the employee works at t = 2; 3. If the employee shirks twice, the supervisor gets 0, if he shirks once and works once, the supervisor gets 2 and if he works twice, the supervisor gets 3. The pay-o¤ of the employee is strictly decreasing in the number of times he works at t = 2; 3: He gets 1 if he never shirks; 2 if he shirks once; and 4 if he shirks twice. The corresponding game with perfectly rational players is represented in …gure 3. The standard analysis of this monitoring game is as follows. The date t = 0 employee’s decision to work (or shirk) is sunk. So he should optimally decide to Shirk. Then the supervisor should decide to Control so as to make the employee Work (the employee would not work otherwise). The employee gets 3 and the supervisor gets 2. Consider now the following cognitive environment. There are two types of employees: the Coarse employees and the Rational employees. Supervisors are assumed to be Sophisticated Coarse. That is, supervisors make their inferences about the type of their employees based solely on the overall frequency with which the various types of employees shirk. In view of the psychology literature mentioned in introduction, supervisors are subject to the Fundamental Attribution Error: they base their judgement (about the type of their employees) based on the overall working attitude without paying attention to the situation (date t = 0 or 2; 3) in which the work/shirk decision is made. Formally, Coarse employees put in the same analogy class all the histories at which the supervisor has to make a decision, and Rational employees use two analogy classes, one for each history at which the supervisor must play. The employee is Coarse with probability 2=3 and Rational with probability 1=3. The Sophisticated Coarse supervisor uses a unique analogy class that contains all the histories at which the employee must play, and she distinguishes between the behaviors of the two di¤erent types of employees, i.e. 2 = 1. 18

We …rst observe that the behaviors generated by the standard rationality framework are no longer part of an equilibrium in this cognitive environment. If it were, then the belief of the supervisor should be that the employee works with probability 2=3 whatever his type. But, with such a belief, the supervisor would choose to Delegate and not to Control.33 Equilibrium is characterized in the next Proposition: Proposition 4 The game has a unique Analogy-based Sequential Equilibrium in pure strategies. At t = 0, the employee shirks when Coarse and works when Rational. In the last two periods, the employee, whatever his type and his behavior at t = 0, shirks if the supervisor chooses to delegate (D) and works if the supervisor chooses to control (C) at t = 1. The supervisor chooses to delegate (D) if he observes that the employee works at t = 0 and to control (C) if she observes that the employee shirks in period 1. In equilibrium, whenever the employee is Rational, he works at date t = 0, the supervisor chooses to delegate D at date t = 0, and the employee shirks at dates t = 2; 3. Whenever the employee is Coarse, he shirks in the …rst period, the supervisor chooses C and the employee works in periods 2 and 3. A Rational employee gets 4, a Coarse employee gets 3 and the expected payo¤ of the supervisor is 5=3. Compared to what happens when all agents are rational, a Coarse employee gets the same payo¤ as in the rational paradigm (!), a Rational employee obtains a higher payo¤ and the supervisor gets a lower expected payo¤. How do we rationalize these behaviors? In particular, why does the supervisor choose to delegate to the Rational employee given the cost attached to not controlling? Remember that our premise is that the supervisor is not aware of the structure of the game including the preferences of the employee. In particular, she is not aware that the employee would not work at dates t = 2; 3 unless he is controlled. The supervisor knows only that a Coarse employee works two thirds of the time, and a Rational employee one third of the time (these are the frequencies that result from the proposed strategies). Accordingly, she perceives a Coarse employee to be a (relatively) working employee and a Rational employee to be a (relatively) shirking employee. When the supervisor chooses between D and C, she cares about the type of her employee insofar as it is indicative of whether the employee is perceived to be (relatively) working or shirking. A key aspect of her decision is governed by her updated belief after she observes the employee’s action at date t = 0. When she observes that the employee works at t = 0, she puts more weight on the probability that the employee is the working type. The supervisor chooses to delegate, i.e. D, because she is su¢ ciently con…dent that her employee will work next with a high probability. By a symmetric argument, when the employee shirks at t = 0, the supervisor puts more weight on the probability that the employee is a shirking type, and she chooses optimally to control. 33

He would choose D and not C because ( 23 )2

3 + ( 13 )2

19

0 + 2( 13 )( 23 )

2 > 2.

Rationalizing the behavior of the employee at date t = 0 is easily derived. A Coarse employee puts the two histories at which the supervisor must move in the same analogy class. Accordingly, he decides to shirk in period 1 because he fails to recognize that the supervisor’s decision to Control depends on the date t = 0 decision to Work.34 A Rational employee perceives that by working in the …rst period, he will deceive the supervisor who will believe that he is more likely to be of the working type. Even though working induces an immediate loss of 1 at date t = 0, it is worth doing because it leads the supervisor to delegate next, thereby yielding an extra pro…t of 2 at t = 2; 3. A striking feature of our monitoring game is that the Coarse employee (rightly considered to be a relatively working type) does not even work at t = 0, he shirks. The Rational employee in order to be confused with a Coarse employee follows at t = 0 the most frequent behavior of a Coarse employee, even though in this speci…c situation a Coarse employee would not even behave that way. We recognize here standard swindlers’ stratagems. The swindler tries initially to build a con…dence relationship with his prey. To do so, in the …rst interactions, he follows an excessively honest behavior (even a standard honest agent would not behave that way). The coarse prey infers from this behavior that the agent he is facing is honest. He drops his guard and the swindler takes advantage of it in the following periods. The swindler’s strategy relies on the coarseness of his prey who wrongly interprets his initial extreme honesty.35 A rational prey would rightly interpret this excessively honest behavior of the swindler in the initial periods, “too good to be true”or “too nice to be honest”, and would not believe in the honesty of the swindler. But, a boundedly rational prey can be deceived as the supervisor is in our monitoring game. Another perspective on our monitoring game is that because the supervisor is boundedly rational she does not perceive the interaction with the employee as one with moral hazard. Instead she thinks she is facing an adverse selection problem (with two (preference) types of workers determining their propensity to work), and the supervisor’s concern is about how to adapt the monitoring scheme (control or delegate) to the type of the employee. It should be stressed that the incorrect model used by the supervisor is not being assumed exogenously. It is being derived endogenously from the limited access of the supervisor to the statistics about the working attitude of employees. From this perspective, it is key in our analysis that the supervisor does not have access to the statistics relating the employee’s working attitude to the monitoring decision (which …ts better with less experienced supervisors). If such conditional statistics were available, the supervisor would know that monitored employees work more than unmonitored employees, and she would optimally choose to control, leaving no room for deception. 34

Ironically, this perception is the right one in the standard rationality paradigm. This phenomenon is well illustrated in many movies such as ”The House of Games” (1987) by David Mamet, “The Sting” (1973) by George Roy Hill, The Hustler (1961) by Robert Rossen or “The Color of Money” (1986) by Martin Scorsese (in these two movies, honest is replace by bad pool player ). 35

20

5

Deception, Reputation, and Commitment Value

The traditional game-theoretic approach to reputation is associated with the idea of commitment. This formalizes an intuition appearing as early as in Schelling (1960). From this perspective, reputation is successful whenever an agent manages to convey the belief that he will behave in a certain way. Making such commitments credible may be valuable in a number of situations including the chain store game (Selten (1978)), the …nitely repeated prisoner’s dilemma (Kreps et al. (1982)) and other multi-stage interactions (see, in particular, Fudenberg and Levine (1989)). Yet, in zero-sum games, there is no value to commitment because the opponent can always guarantee her value if she is rational irrespective of the agent’s own strategy (this follows from the celebrated minmax theorem, see von NeumannMorgenstern (1944)). Thus, the standard approach would conclude that there is no room for reputation building in zero-sum interactions whether repeated or not.36 In our framework, reputation concerns may arise even in repeated zero-sum games, thus illustrating how the possibility of deception may deeply a¤ect the phenomenon of reputation.37 For the sake of illustration, we consider the simplest setup of a zero-sum game G that is repeated twice. In stage game G the Row player chooses an action U , D, or B, the Column player chooses an action L or R, and stage game payo¤s are as represented in Figure 4. Players do not discount payo¤s between the two periods, and their overall payo¤ is simply the sum of the payo¤s obtained in the two periods. U D B

L R 5, -5 3, -3 0, 0 7, -7 11/2, -11/2 0, 0

Figure 4. The stage game G

When players are rational, they play the unique Nash equilibrium of the stage game in every period. The Row player plays U with probability 7=9 and D with probability 2=9 and the Column player plays L with probability 4=9 and R with probability 5=9. The overall value of the two-period game is 70=9 for the Row player and 70=9 for the Column player. In equilibrium players play in mixed strategies in order to avoid being predictable. But, note that no player is ever deceived by his opponent: Whatever players do in the …rst period they are expected to play according to the same mixed strategy in the second period, and players do behave according to that expected mixed strategy in period two. As a matter of 36

The view that deception and reputation concerns cannot arise in zero-sum games seems in con‡ict with the public opinion on poker games for which it is generally believed that the best gains come when you do have a strong hand, but you get others to believe that you are blu¢ ng (an elaborate form of deception and reputation building). 37 The example of Section 2 was already suggestive of this. Yet, the interaction there was not truly a zero-sum game because of the presence of d-truthtellers (see Crawford (2003) for a similar observation).

21

fact, in a zero-sum game like the one considered here a player can secure his value no matter what the other player does. Thus, with the standard approach there is no point in deceiving the opponent (assumed to be fully rational) as this could only lower one’s own payo¤. Consider now the following setup with boundedly rational players. There are two types of Row players, the Rational type and the Coarse type assumed to be equally likely. The Rational Row player has a perfect understanding of the strategy of the Column player, as in the standard case. The Coarse Row player knows only the average behavioral strategy of the Column player all over the two time periods. That is, the Coarse Row player has only an expectation about the average behavior of the Column player all over the game (i.e., he bundles the two time periods into one analogy class). There is one type for the Column player. The Column player is assumed to be Sophisticated in the sense that he distinguishes between the behaviors of the Rational Row player and the Coarse Row player. But, he is assumed to be Coarse in the sense that for each type of the Row player he knows only the average behavior of this type over the two time periods, i.e. all histories are bundled into one analogy class. In short, we say that the Column player is a Sophisticated Coarse player. Proposition 5 The following strategy pro…le is an Analogy-based Expectation Sequential Equilibrium. 1) Rational Row Player: Play U in period 1. Play D in period 2 if U was played in period 1, and U otherwise. 2) Coarse Row Player: Play U both in periods 1 and 2. 3) Column Player (Sophisticated Coarse): Play L in period 1. Play R in period 2 if the Row player played U in period 1. Play L in period 2 if the Row player played D or B in period 1. As in all our examples, the key aspect is about understanding the inference process of the Sophisticated Coarse Column player. The Coarse Row player always plays U , and the Rational Row player plays U and D with an equal frequency on average. These (average) behaviors of the two types of Row players de…ne the analogy-based expectations of the Column player. Given these expectations, the Column player updates her belief about the type of the Row player as follows: When action D is being played in period 1, the Column player believes that she faces the Rational player for sure. We also assume that this is her belief after action B is being played in period 1.38 When action U is being played in period 1, the Column player believes that she faces the Coarse Row player with probability 1=2 = 23 . Accordingly, the Row player plays R in period 2 because given her belief 1=2+1=2 1=2 this looks like the smartest decision (even though in reality it is not). Thus, by playing U in period 1, the Rational Row player builds a (false) reputation for being more likely to be a Coarse Row player, which he later exploits in period 2 by getting 38

Action B is never played in equilibrium; so action B could come from either type. The chosen belief can be rationalized by assuming that with some small probability the Rational Row player trembles and plays B.

22

the high payo¤ of 7. Interestingly, in period 1, the Rational Row player does not go for the highest immediate payo¤ of 11=2 he would obtain by playing B. He realizes that it is better (in fact best) to deceive the Column player …rst by playing the most typical action of the Coarse Row player, and then exploit it in period 2. Clearly, the same conclusion as the one arising in Proposition 5 would hold if the Coarse Row Player had been exogenously assumed to systematically play U in both periods (as in the crazy type approach). Here we get the extra insight that even though the underlying game is the same zero-sum game for all players, the heterogeneity in cognitive types endogenously explains the di¤erence of behaviors of the various types of Row players.39 Remark: It should be noted that if there were only one type for each player characterized by his analogy partition as in Jehiel (2005), then it would be impossible to reproduce the behavioral strategies as described in Proposition 5: For the Column player to play a di¤erent action in periods 1 and 2 she should either be indi¤erent between playing L or R (which cannot be the case here since the Row player does not play U with probability 7=9 on average) or treat separately the behavior of the Row in the two time periods, but then in period 1 she could not …nd it optimal to play L given that the Row player always plays U .

6

Deception as a bargaining tactic

We consider a wage negotiation game between a professor and the dean of a university. The professor can generate outside o¤ers but this is costly, and the dean is not ready to pay a wage rise unless he feels there is a signi…cant probability that the professor would leave otherwise. Payo¤s are speci…ed so that in the standard rationality framework the professor’s threat to leave would not be credible, and thus there would be no wage increase. For the sake of illustration, we assume that there is a single preference type both for the professor and the dean so that players’types bear only on the cognitive limitations of the players. As we will see, such cognitive limitations and the resulting deceptive tactic will ensure that a rational professor may sometimes get his wage rise in such a scenario! Speci…cally, the game tree of the wage negotiation is described as follows. At t = 1, the professor chooses between accepting the status quo (SQ) or developing contacts with another university (D) in view of an alternative faculty position in another department. Establishing these contacts costs him > 0. If he develops contacts, the professor asks for a pay rise (> 0) to the dean. At t = 2, the dean decides either to refuse (R) or to accept (A) the pay rise. If the dean accepts, the professor stays in the department and the negotiation process 39

The Coarse Row player plays U because he expects the Column player to play L and R with an equal frequency and U is a best-response to such an expectation.

23

is over. The professor ends up with a higher wage, and he stays in his original position. If the dean refuses, at t = 3, the professor chooses again between accepting the status quo (SQ) - staying in his department with his initial salary - or developing further contacts (D) with the other university at cost , getting from it an o¤er. If the professor chooses the second option, he goes back to the dean, exhibits his alternative o¤er and asks for a pay rise 0 (> 0).40 At t = 4, the dean decides whether to accept (A) the pay rise 0 or refuse it (R). The professor stays in the department if the dean accepts the pay rise or leaves the department and goes to the other university if the dean refuses the pay rise.41 If the professor accepts the o¤er of the other university, the original department incurs a cost X and the professor gets U 2 .42 We normalize payo¤s so that in the original situation both the dean and the professor have a pay-o¤ 0. We further assume that X < 0 and U < , and to …x ideas, we let = 3, 0 = 4, = 1, X = 72 and U = 12 . In a perfect rationality world, X < 0 implies that at t = 4, the dean prefers to let the professor go rather than accept the pay rise. Given that there is no pay rise at t = 4, U < implies that at t = 3 the professor does not …nd it useful to generate an outside o¤er of U for an extra cost . Anticipating that no further search e¤ort will be made by the professor, the dean at t = 2, …nds it optimal not to accept the pay raise. Finally, at t = 1, the professor does not develop contacts because he anticipates no pay rise will be accepted. The game is represented in Figure 5 where the …rst and second decision nodes of the professor (resp. dean) are labelled n1 and n3 (resp. n2 and n4 ).

n1

SQ (0,0)

D

n2

A (2,-3)

R

n3

D

SQ

n4

R

(-3/2,-7/2)

A

(-1,0)

(2,-4)

Fig 5. The Wage Negotiation Game

In the standard rationality paradigm, even though the professor has the possibility to go for another job, the outside option is perceived as non-credible, and there is no pay rise. 40

We have in mind that 0 > so that the new pay rise compensates at least partially for the extra search cost. 41 In the latter event, the professor could stay in his original position, but we assume then that the induced atmosphere is quite bad for the professor (colleagues are quite upset). 42 U is equal to the value of the alternative o¤er minus the costs the professor incurs leaving his university.

24

This is a stylized and simpli…ed version of the so called outside option principle (see Binmore et al. (1989)). Consider now the following cognitive environment where we identify a node with the history that leads to that node. With probability 1=2, the dean is Sophisticated Coarse, he puts in the same analogy class both decision nodes of the professor, i.e. An = ffn1 ; n3 gg, and he distinguishes between the various types of the professor, i.e. = 1. With probability 1=2, the dean is Rational, An = ffn1 g; fn2 gg and = 1. With probability 1=2, the professor is Coarse, An = ffn2 ; n4 gg, he puts in the same analogy class both decision nodes of the dean and = 0, he does not distinguish between the various types of the dean. With probability 1=2, the professor is Rational, An = ffn2 g; fn4 gg and = 1. The following proposition illustrates the possibility of search activity and pay rise in equilibrium: Proposition 6 The following strategy pro…le43 is an equilibrium: A Coarse professor establishes contacts with an alternative university whenever he has the opportunity to and a Rational professor establishes contacts in n1 and accepts the status quo in n3 . A Sophisticated Coarse dean accepts the pay rise in n2 and refuses to give a pay rise in n4 . A Rational dean always refuses to give a pay rise.

In equilibrium, the professor always makes an e¤ort to get an alternative o¤er at t = 1. With probability 21 , the dean accepts the pay rise; with probability 14 , the professor leaves his department (thus taking the outside option) and, with probability 14 , he stays in his department with his initial salary. We observe that the outside option of the professor which, in the standard rationality paradigm, would be considered as non-credible (and would thus have no e¤ect), does a¤ect the equilibrium play here. The logic of the equilibrium is as follows. First, it is readily veri…ed that the behaviors of the rational professor and the rational dean are optimal. So let us focus on the Coarse professor and the Sophisticated Coarse dean. A Coarse professor perceives that the dean accepts the pay rise with a probability equal 1=2 2 ) on average. For such a high probability of acceptance, it is worthwhile to 5 (= 1+1=4 developing further contacts at t = 3 since 25 ( 0 2 ) + 35 (U 2 ) > . It is also readily veri…ed that a Coarse professor …nds it optimal to develop outside contacts at t = 1. The Sophisticated Coarse dean perceives that a Coarse professor always chooses D and a Rational professor chooses D with probability 23 on average44 . At t = 2, he observes that 43

The Subgame Perfect Nash equilibrium strategy pro…le (described earlier) is also an equilibrium. In the …rst node, reached with probability 1, a Rational professor chooses D and in n3 , reached with probability 21 , a Rational professor chooses SQ. Therefore, a Sophisticated Coarse dean perceives that a 1+0 = 23 in the analogy class gathering n1 and n3 . Rational Professor chooses D with a probability 1+1=2 44

25

the professor chose D in n1 . Given this behavior, the Sophisticated Coarse dean’s updated (1=2) belief at n2 is that the Professor is Coarse with probability (1=2)+(1=2)(2=3) = 35 . Combining the perceived behaviors and beliefs, the Sophisticated Coarse dean expects the professor . According to this to make further search e¤orts in n3 with probability ( 35 )1 + ( 25 ) 23 = 13 15 45 expectation, the dean prefers to accept the pay rise in n2 . It should be noted that the dean accepts the pay rise here because he puts a su¢ ciently high probability on the professor being coarse due to this erroneous belief updating. If the dean were to keep the prior belief that the professor is Coarse with probability 21 (which would be the standard belief at node n2 if the dean were fully rational), he would choose not to accept the pay rise at n2 .46 For rational agents, the presence of boundedly rational agents has two main consequences. First, at the start of the interaction, Rational professors mimic Coarse professors and exert e¤orts towards the outside university. That way, they deceive Coarse deans, and make them believe that they are facing a Coarse professor with a high probability. Coarse deans accept the pay rise at t = 2 because they are su¢ ciently afraid that the professor would otherwise leave. Second, Rational deans fail to be identi…ed as Rational deans by Coarse professors at t = 3. Coarse professors do not perceive that there exist two types of deans. Thus, even though a Rational dean behaves di¤erently from a Coarse dean at t = 2, a Coarse professor keeps on believing, at t = 3, that the dean will concede with probability 25 at t = 4. In this case, Rational deans would prefer being identi…ed as what they are : Rational deans who never accept pay rises. Coarse professors would then choose the status quo at t = 2 and the rational dean would get 0 rather than 47 (the expected payo¤ in the equilibrium of Proposition 6). This illustrates that it may be costly for a rational agent not to be distinguished from other types due the cognitive limitations of other players.

7

Deception in a repeated expert/agent interaction

Some deceptive tactics concern situations in which information must be repeatedly transmitted by an expert to an agent. The expert may share the same preference as the agent in which case she is a friend, or she may have a di¤erent objective, in which case she is selfinterested. While a friend would always transmit the information accurately, a self-interested expert may like to fool the agent and let him think B is true when A is true. This would certainly be easier if in an early stage of interaction the self-interested expert could convey the belief that she is more likely to be a friend (relative to the prior). Then, in a later stage, the agent would be more likely to take the words of the agent at face value, which the 13 2 ( 27 ) + 15 (0). This is because 3 > 15 He would have perceived that his payo¤ obtained by not accepting the pay rise is 2 7 ( 12 1 + 12 23 ))(0)) and he would not have accepted the pay rise (since 3 )( 2 ) + (1 45 46

26

35 12 (= 35 12 >

( 12 1 + 3).

1 2

self-interested expert would then exploit. With standard approaches, it would not be possible for the self-interested expert to persuade the agent for sure that she is more likely to be a friend (relative to the prior). This is a consequence of the assumption (made in standard approaches) that the agent knows the strategy of each type of expert perfectly well. With the machinery developed here, such a deceptive tactic may occur. The game we use to illustrate this shares a number of features with that studied by Sobel (1985). The game involves a Sender and a Receiver. There are n periods of interaction k = 1; 2; :::n. At the start of each period k, the Sender (but not the Receiver) is informed of the state of the world sk prevailing in period k. We assume that sk , k = 1; :::n are independently distributed from each other and that each sk may take two values 0 or 1 with an equal probability. In contrast to Sobel (1985) we do not assume that the Receiver observes the state of the world sk at the end of period k (except for the outside option alternative, our informational assumptions resemble those made in Ely and Välimäki (2003)).47 In each period k, the Sender sends a message mk = 0 or 1 to the Receiver who then makes a decision ak 2 [0; 1]. The message mk = 0; 1 is meant to represent the state prevailing in period k, but both the Sender and the Receiver are strategic and the Receiver is aware that the message need not represent the true state of the world. We assume that the Receiver is very impatient and values only the current period k payo¤ according to the quadratic scoring rule (ak sk )2 where ak is the period k decision. That is, the Receiver would ideally like the decision ak to be as close as possible to the period k state sk and the quadratic scoring rule ensures that the Receiver will pick the action that corresponds to what he expects to be the mean value of sk given the message he received and the history of the interaction (i.e the sequence of messages sent up to the current period). The Sender is assumed to care about the stage game payo¤s she obtains in all periods k = 1; :::n. We call k the weight she assigns to period k payo¤. That is, if the stage game P payo¤ of the Sender in period k is uk , the overall payo¤ of the Sender is k=1;:::n k uk . We will focus on situations in which 1) one of the periods denoted k has a much higher weight k = 1 than the other periods k 6= k whose weights k are assumed small, and 2) from period k + 1 onwards earlier payo¤s count much more than later payo¤s, i.e. k k+1 for k = k + 1; :::n 1, so that from period k + 1 onwards it is as if the Sender were myopic.48 We assume that there are two types of Senders, the friend F who shares the same 47

This assumption is not critical. If the Receiver could observe state sk at the end of period k, then we should consider an alternative strategy for Sender SI: Sender SI should tell the truth from period 1 to period k and consistently sends message m = 1 from period k on irrespective of the state. In the standard approach, SI would get approximately 2 whereas he could get a payo¤ close to 0 when n is su¢ ciently large in our boundedly rational framework (see below for further speci…cations of the model). 48 We suspect that the same analysis carries over even if the weight of periods k 6= k are the same (and equally low), but this requires a more subtle analysis for the incentives of SI Sender’s communication strategy from period k + 1 onwards. If k are su¢ ciently increasing from period k + 1 on, we suspect that no signi…cant belief manipulation can arise in equilibrium, even if all k , k 6= k are very small.

27

stage game preferences as the Receiver and the self-interested Sender SI who would like the Receiver’s decision ak to be as close as possible to 1 in every period whatever the state of the world. To be more speci…c, we assume that SI’s stage game payo¤ in period k is given by the linear scoring rule (1 ak ) where ak is the period k decision made by the Receiver.49 Finally, we assume that the ex ante (or prior) probability that the Sender has type SI is . We will focus on strategy pro…les in which F always tells the truth and SI’s message is uninformative (in every period k, Sender SI adopts the same communication strategy whether sk = 0 or 1). (This sounds like a natural assumption, since Sender F would really like as much information as possible to be transmitted to the Receiver, and Sender SI does not care about the state and the state is never observed by the Receiver.) In a world with perfectly rational agents, Sender SI would get an expected payo¤ approximately equal to 2 (as all k , k 6= k approach 0) in equilibrium. This corresponds to a situation in which in period k the Receiver would (on average) perceive that the Sender has type SI with probability (the prior), the Receiver would understand that F tells the truth and that SI’s message is uninformative while Sender SI would send message mk = 1 in period k .50 This expected payo¤ can be achieved if SI sends message mk = 0 or 1 with an equal probability in all periods k = 1; :::k 1 (so as to match the behavior of F as observed by the Receiver - remember that the two states s = 0; 1 are equally likely and their realization is not observed by the Receiver). The reason why in a standard rationality paradigm SI cannot achieve a payo¤ larger than 2 is simple. Perfect understanding of the strategies of SI and F implies that the Receiver’s belief k about the Sender being self-interested in period k must rise on average over those histories that are induced by the e¤ective strategy of type SI. (This property is called the submartingale property.) Hence, whatever the equilibrium strategy followed by SI, conditional on the Sender being SI, the Receiver in period k will believe that he is facing type SI with a probability whose expected value is at least . Given the Receiver’s (correct) understanding that F tells the truth while SI’s message is uninformative, it follows that SI cannot achieve more than 2 (as all k , k 6= k approach 0). It is worth pointing out that this expected payo¤ depends neither on the number of periods that precedes k nor on the total number n of periods of the interaction. Consider now an environment in which the Receiver is boundedly rational (while both types of Sender are rational). More precisely, the Receiver is assumed to understand only for each type of Sender the average communication strategy over the n periods as a function of the current period signal. This game falls outside the class of games de…ned in Section 3 because the Sender is getting private information after the start of the game. Yet, 49

The linearity of the Sender’s stage game payo¤ simpli…es the writing of the expected payo¤ as it ensures that this expected payo¤ only depends on the expected belief of the Receiver at that stage. 50 When receiving message mk = 1 in period k , The Receiver would then pick an action in period k whose expected value would be 12 + (1 )1, hence the expected payo¤ of SI.

28

the Analogy-based Sequential Equilibrium concept can easily be extended to cover such an application as we now explain. We will assume that k < n2 and focus on the following strategy of the SI sender (Sender F is assumed to be telling the truth in all periods). In periods k = 1; ::: k 1, Sender SI sends message mk = 0 (whatever the state sk ). In periods k = k ; :::n, Sender SI sends message mk = 1 (whatever the state sk ). Given the Sender’s strategy, the Receiver rightly perceives that F always tells the truth and that SI’s message is uninformative. But, the Receiver now has an incorrect perception of the timing of SI’s communication strategy. His (coarse) perception is that SI sends message m = 0 on average (across all periods) with probability = k n 1 and message m = 1 on average with probability 1 (independently of the current period state).51 Observe that k < n2 ensures that < 12 so that message m = 0 is more typical of F than of SI’s (average) communication strategy. Given such a perception, after observing mk = 0 in periods k = 1; :::k 1 and mk = 1 in period k , the Receiver believes that he is facing Sender SI with probability where k

1

=

(1

(1 ) k )(1=2) +1

(2)

and = k n 1 . This follows from the (iterated) application of Bayes’law given the Receiver’s perception of the Sender’s communication strategy. Given such a belief, the Receiver picks 1 action + (1 )1 in period k when receiving message mk = 1 resulting in a stage 2 game payo¤ of 2 for SI in period k . Thus, the SI sender gets an (equilibrium) payo¤ approximately equal to as all k , k 6= k approach 0. Checking that Sender SI’s 2 communication strategy is part of an equilibrium is now an easy exercise. Initially (from periods k = 1 to k 1), Sender SI strictly prefers sending mk = 0 to mk = 1 so as to increase the Receiver’s belief that the Sender is more likely to be F than SI in the crucial period k . In period k , Sender SI exploits her deceptive tactic and sends mk = 1 at a point where the Receiver is most con…dent to be facing type F . From period k + 1 onwards, SI consistently sends m = 1 because she is myopic and message m = 0 would result in a strictly lower immediate payo¤ (the Receiver keeps assigning a positive (though vanishing) weight on the Sender being type F so that the message has some impact on the decision made by the Receiver). To summarize:52 51 In line with our general motivation, one should have in mind that the Receiver has not himself played the game many times, but that he has only been informed of such aggregate statistics about the behaviors of the two types of Senders. 52 We are assuming that k < n2 . If k > n2 , the equilibrium would require that the average communication strategy of SI is slightly below 12 , that SI mixes between m = 0 and m = 1 from periods 1 to k 1 (though not with probability 1=2) and that SI sends m = 1 from period k onwards. We conjecture that in such an equilibrium, SI would get approximately 2 (for low values of k , k 6= k ), as in the standard rationality

29

Proposition 7 Assuming that all k , k 6= 0 are close to 0, Sender SI can get an equilibrium payo¤ as high as 2 where is given by (2) in our environment with boundedly rational Receivers. In our equilibrium, observe that conditional on the Sender being type SI, the history of messages up to period k is mk = 0 for k = 1; :::k 1 and mk = 1. Accordingly, conditional on the Sender being SI, the Receiver believes in period k that she is facing type SI with probability , which is strictly smaller than the prior (k < n2 ensures that < 21 and < , see above). Such a belief manipulation could not arise in a standard rationality world in which the submartingale property must hold. The di¤erence provides a measure of the extra gain made by SI due to the belief manipulation of the 2 boundedly rational Receiver. Observe that in contrast with the standard rationality case, the payo¤ obtained by SI depends (through and ) both on the number of periods that precedes the critical period k and on the total number n of periods. The most favorable case to SI is when n is large relative to k so that is small (and/or k is not too small so that there is time for belief manipulation). In the extreme case in which n is very large relative to k and k > 1, SI may lead the Receiver to believe in period k that she is almost surely type F . Investigating experimentally whether there is an e¤ect of the total duration of the interaction on the Receiver’s belief in period k (as can be inferred from his choice of action) sounds like a natural test for the present theory of deception.

8

Conclusion

The paper can be viewed as proposing a bridge between the literature on psychology, especially that related to the Fundamental Attribution Error, and the game theory literature, especially that related to bounded rationality and reputation.53 The combination of the two strands of literature has led to an equilibrium theory of deception and belief manipulation that, we think, is useful to understand a number of economic problems in a new way. There are obviously many other approaches that mix psychology and economics. The following lines give a very incomplete (and somewhat arbitrary) account of some of these approaches and how our own approach …ts into these. Following the lead of Simon (1956) many researchers have emphasized the role of behavioral heuristics in decision making (see Gigerenzer et al. (1989) or Gigerenzer and Selten case. 53 It shares also some similarities with idea of bounded awareness as developed in Bazerman (2005) (chapter 11). From this perspective, our theory assumes some form of bounded awareness at the learning stage (agents pay attention only to a limited number of regularities) whereas Bazerman’s discussion of bounded awareness is more about the understanding of the rules of the game. Bazerman’s book also discusses various self-serving biases in decision making and attribute them to the fact that people are "imperfect information processors" (chapter 1). This paper completely agrees with the latter view, and it illustrates how imperfections in the information processing may lead to the possibility of deception.

30

(2002)). The cognitive types in our approach can be viewed as standing for heuristics used by the players to understand the reaction of their environment. But, note that our cognitive types are better viewed as de…ning learning heuristics rather than behavioral heuristics. This view on heuristics does not seem to have its counterpart in Gigerenzer et al.’s work. The psychology literature has discussed a number of biases other than the FAE. Many of these biases relate to the laws of probabilities and in contrast to our work are generally better understood as arising in non-repeated interactions: they include the base rate and conjunction fallacies, the law of small numbers, the gambler’s fallacy, overcon…dence....54 It might be interesting in future research to combine these biases with our framework, yielding new formulations for the updating process in the belief system.

54

While Kahneman et al. (1982) identify a number of these biases, Thaler (1991) also shows their signi…cance in experimental economics. A number of economists have also developed theories motivated by these biases (see, for example, Mullainathan (2002) or Rabin (2002)).

31

9 9.1

Appendix Proof of Proposition 4

First, a Coarse employee puts the two nodes in which the supervisor has to make a decision into the same analogy class. Therefore, he always shirks at t = 0. Second, if the supervisor chooses C, there is a unique best-response for the employee whatever his type and belief are: To work at t = 2; 3. Conversely, if the supervisor chooses D, there is a unique best-response for the employee whatever his type and belief are: To shirk at t = 2; 3. Therefore, to …nd an equilibrium, we only need to focus on the decision of a Rational employee at t = 0 and the decision of the supervisor. Choosing D after having observed that the employee shirking at t = 0 cannot be part of an equilibrium strategy since the employee’s best response would be always to shirk. Suppose now that the supervisor chooses C both if the employee works and if the employee shirks at t = 0. The employee, whatever his type is, has a unique best response: To shirk at t = 0 and to work at t = 2; 3. The supervisor perceives that an employee, whatever his type is, works with a probability 2/3. C is not a best response to such a belief and choosing C in both cases cannot be part of an equilibrium either. The only remaining possibility for the supervisor is to choose C (resp: D) when he observes that the employee shirks (resp: works) at t = 0. A Rational employee has a unique best response to such a behavior, to work at t = 0. Now, we need to establish that it exists a belief consistent with these behaviors such that the supervisor behavior is a best response to this belief. If players follow the described behaviors, a Coarse employee shirks once and works twice and a Rational employee shirks twice and works once. The supervisor perceives that a Coarse (resp: Rational) employee chooses to work with a probability 2/3 (resp: 1/3). After having observed an employee working at t = 0, her revised belief is that he is Coarse with (2=3)(2=3) probability: (2=3)(2=3)+(1=3)(1=3) = 45 and, if she observes that the employee shirks at t = 0, she (1=3)(2=3) believes that he is Coarse with probability (1=3)(2=3)+(2=3)(1=3) = 12 . Crossing beliefs and analogy-based expectations, we obtain the following. After having observed that the employee shirked (resp: worked) at t = 0, the employer prefers choosing C (resp: D). Q.E.D.

9.2

Proof of Proposition 5

In the body of the paper, we showed that the Sophisticated Coarse Column player’s strategy is an equilibrium strategy. Besides, it is readily veri…ed that the Rational Row player plays a best-response to the 32

Column’s player strategy. he gets an overall payo¤ of 5 + 7 = 12 and would only get an overall payo¤ of 0+11=2 at best if he were to play D in period 1, a payo¤ of 11=2+11=2 = 11 at best if he were to play B in period 1, and he would obviously get a lower payo¤ by playing U or B in period 2. Now, the Column Player plays L and R with an equal frequency on average over the two time periods, and this is the expectation of the Coarse Row player. Given his expectation, the Coarse Row player …nds it optimal to play U whenever he has to move. This is because 1 (5 + 3) > max[ 12 (0 + 7), 12 (11=2 + 0)] . Q.E.D. 2

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