David Ettinger

Philippe Jehiel
email: jehiel@enpc.fr

K Binmore

D Fudenberg

D Laibson

A Newman

A Rubinstein

A Theory of Deception

This paper proposes an equilibrium approach to belief manipulation and deception in which agents only have coarse knowledge of their opponent's strategy. Equilibrium requires the coarse knowledge available to agents to be correct, and the inferences and optimizations to be made on the basis of the simplest theories compatible with the available knowledge. The approach can be viewed as formalizing into a game theoretic setting a well documented bias in social psychology, the Fundamental Attribution Error. It is applied to a bargaining problem, thereby revealing a deceptive tactic that is hard to explain in the full rationality paradigm.

Deception and belief manipulation are key aspects of many strategic interactions, including bargaining, poker games, military operations, politics and investment banking. Anecdotal evidences of belief manipulation and deception are very numerous, and Michael Lewis's (1990) best-seller "Liar's Poker" reports colorful illustrations of such strategic behaviors in the world of investment banking in the late 1980s. For example, Lewis explains how "he spent most of his working life inventing logical lies" that worked amazingly well (thanks to the logical appearance, see Lewis (1990) page 186).

belief manipulation and deception are delicate to capture because traditional equilibrium approaches assume that players fully understand the strategy of their opponents. [START_REF] Compte | Repeated Relationships with Limits on Information Processing[END_REF] 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 manipulate beliefs so as to take advantage of the erroneous inferences. [START_REF] Crawford | Lying for Strategic Advantage: Rational and Boundedly Rational Misrepresentation of Intentions[END_REF] To illustrate the phenomenon of deception, we will consider and formalize the following bargaining situation. The owner of a house, Mrs A, wishes to sell her good at some price considered to be high (say above the market price as perceived by real estate agents). A potential buyer, Mr B, comes in. Mr B will accept paying the high price if he is afraid enough that another buyer may be interested in the house. Otherwise, he will prefer to continue bargaining in the hope of getting a lower price. The owner, Mrs A, after mentioning some slight problems with the heating system (thereby conceding a small discount in the price) tells Mr B that there is another potential buyer, and so she is not willing to discount the price any further. Mr B has no way to verify Mrs A's claims (in a reasonable amount of time). Should Mr B trust Mrs A when she says that there is another buyer, or is she blu¢ng?

In the theory to be developed below, mentioning that there are heating de…ciencies will make it more likely in Mr B's eyes that Mrs A is an honest seller always telling the truth.

As a result, Mr B will be convinced enough that there is indeed another buyer when Mrs A says so, and he will accept paying the high price (minus the small discount conceded for the reported heating de…ciencies). By mentioning that there are de…ciencies, Mrs A manipulates Mr B's belief about her true nature (whether she is an honest seller or an opportunist), and she exploits Mr B's inference error when she says that there is another buyer. Such a deceptive tactic works in our theory in so far that mentioning small de…ciencies is more representative of honest sellers than of opportunist sellers over all transaction situations (with high or low prices, say), and in forming his judgement about Mrs A's type, Mr B somehow only considers the general attitudes of the various types of sellers and does not distinguish how the various types of sellers behave in those various transaction situations with high or low price.

We will present a detailed formalization of the above deceptive bargaining tactic in Section II, pointing out that it would not work if Mr B were fully rational. [START_REF] Esponda | Behavioral Equilibrium in Economies with Adverse Selection[END_REF] Before developing that application, we present in Section I a general framework that allows us to model quite generally such inference errors as the one made by Mr B in a game theoretic equilibrium approach.

Speci…cally, the class of games considered in this paper are two-player multi-stage games with incomplete information and observable actions in which players may be of several types, past actions are assumed to be observable by everyone, and types may a¤ect the preference relations of players. A key non-standard ingredient is that players are also parameterized by how …nely they understand their opponent' s strategy. In addition to their preference and informational characteristics, players are endowed with cognitive types.

Following Jehiel (2005), 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 aggregate 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 the behaviors of the various types of his opponent. Thus, cognitive types may vary in two dimensions: a player may be more or less …ne in the partition of the decision nodes of his opponent (what we call the analogy part), and a player may or may not distinguish the behaviors of the various types of his opponent (the sophistication part). In the above bargaining story, Mr B bundles the announcement nodes of sellers into one analogy class, whether the price is high or low, and he distinguishes the behaviors of honest and opportunist sellers. Thus, Mr B uses a coarse analogy partition, but he is sophisticated in the terminology just de…ned.

Given a strategic environment that includes the speci…cation of players' cognitive types, we de…ne an equilibrium concept that we refer to as the analogy-based sequential equilibrium. In equilibrium, players have correct expectations about the aggregate behavior of their opponents in their various analogy classes -these are referred to as analogy-based expectations. 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. [START_REF] Eyster | Cursed Equilibrium[END_REF] In Section I we show that in …nite environments (…nite numbers of types, actions, and nodes), an analogy-based sequential equilibrium always exists. We also suggest how to interpret the solution concept from a learning perspective. Finally, we illustrate the working of the concept in a simple two-person two-period zero-sum game in which the payo¤ structure is commonly known to players but players may have cognitive types other than the fully rational one. The example serves to illustrate 1) why a player with non-fully rational cognitive ability cannot be viewed as a rational player who does not distinguish between some situations (a player with coarser information), 2) how, in a mixed population of rational and coarse players, a rational player always performs better, and 3) why, in our framework with incorrect inferences, there may be room for reputation building even in zero-sum games where there is no value to commitment. [START_REF] Fudenberg | Reputation and Equilibrium Selection in Games with a Patient Player[END_REF] The framework of Section I is then used in Section II to formalize the above deceptive bargaining tactic. Section III concludes. We shall start, however, by situating our work in the perspective of various literatures.

Related literature

There have been many attempts to relax the rationality assumptions imposed on economic agents. These include relaxing the ability of agents to optimize their strategy given their beliefs (as in the Quantal Response Equilibrium, Richard McKelvey and Thomas Palfrey, 1995) or relaxing the ability of agents to form correct expectations. By maintaining the ability of agents to optimize their strategies given their beliefs, our paper contributes to the second form of departure from rationality, which we refer to as cognitive limitations.

Several routes have been pursued to model cognitive limitations either introducing explicit biases in the inference process (see Daniel Khaneman et al., 1982 for an exposition of such biases as the gambler's fallacy, the base rate neglect, the conjunction fallacy etc...) or deriving the expectations from limited introspective reasoning (as in the level k approach, Dale Stahl, 1993) or deriving the expectations and inference process from the erroneous or coarse perception held by agents about their environment (approaches based on subjective prior or the self-con…rming equilibrium and this paper, respectively). Our paper contributes to the last of these routes by further postulating that the coarse perception held by boundedly rational agents is the simplest representation -or model of others-that is consistent with their coarse statistical knowledge. Such a line of research that views bounded rationality equilibrium concepts as a result of partial learning is the common theme of the limited foresight equilibrium (Jehiel, 1995), the analogy-based expectation equilibrium (Jehiel, 2005) and the valuation equilibrium (Jehiel and Samet, 2007). [START_REF] Jackson | Social Learning in Recurring Games[END_REF] Jehiel (2005) developed the analogy-based equilibrium concept to capture bounds on rationality that accommodate coarse perception but fully rational information processing, and extended to static games of incomplete information in Jehiel and Frederic Koessler (2008). Our aim in this paper is to extend this basic structure to extensive games with incomplete information, which is necessary to analyze the evolution of beliefs over time. The extension of these concepts to dynamic games allows us to examine the basic ideas of belief manipulation and deception. Connected to the analogy-based expectation equilibrium, Erik Eyster and Matthew 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 their information. [START_REF] Jehiel | Limited Horizon Forecast in Repeated Alternate Games[END_REF] In problems with interdependent preferences, the cursed equilibrium of Eyster and Rabin gives rise to erroneous equilibrium beliefs (as the analogy-based expectation equilibrium does) about the relation between the strategy and the signal of the opponent. Yet, by the very static nature of the games considered by Eyster and Rabin, no belief manipulation can be captured by their approach, which constitutes a key di¤erence from the present framework.

Even though the starting point of our approach is about modeling the consequences of the coarse perception of agents with cognitive limitations as just explained, it turns out that our paper can also be viewed as formalizing a well studied bias in social psychology, e.g., the Fundamental Attribution Error (FAE) (see Edward Jones and Keith Davis (1965), Lee Ross (1977), Ross, Teresa Amabile and Julia Steinmetz (1977)). 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" (Maureen O' Sullivan (2003)). [START_REF] Jehiel | Analogy-based Expectation Equilibrium[END_REF] In the above bargaining story, Mr B is subject to the FAE. In forming his judgement about whether he is facing an honest seller after Mrs A has reported minor heating de…ciencies, Mr B "ignores" that sellers' attitudes are not the same whether the price is high or low. Our model provides an explicit way to formalize such a neglect by Mr B.

There have been several earlier game theoretic attempts to capture the phenomenon of deception. These include the ideas of playing mixed strategy (to avoid being detected) in zero-sum interactions (John von Neuman and Oskar Morgenstern (1944)) and of playing a pooling or semi-pooling equilibrium (thereby not revealing one's own type) in signaling games (Michael Spence (1973)) or communication games (Joel Sobel (1985) and Vincent Crawford (2003)) or repeated games (Kreps and Wilson (1982b), Kreps et al. (1982), Drew Fudenberg and David Levine (1989)). Our approach to deception di¤ers from these earlier approaches in that it is based on the idea of belief manipulation (by which we mean that some players end up having erroneous beliefs based on their observation), which cannot arise in the standard rationality paradigm considered in these earlier approaches. In our theory, deception can be viewed as the exploitation by rational players of the FAE made by other 8 Ross et al. (1977) report a striking example in support of the FAE. In a pool of Stanford students from various …elds, subjects were divided between questioners and answerers. The "questioners" were requested to ask the answerers di¢cult questions. Every questioner was matched to a single answerer who was almost always from a di¤erent …eld. After the quizz (answerers and questioners then knew how many correct answers were given in their match), it was observed that answerers consistently thought they were worse than questioners, thereby ignoring the fact that the pool of questions on which they performed relatively poorly was not generated at random but drawn from the esoteric knowledge of the questioner. Note that answerers were explicitly told before the quizz that questioners could freely choose the questions they liked best.

players, where FAE allows for belief manipulation.

Finally, it should be mentioned that our setup can be used to formalize a model of persuasion in the vein of the one developed independently of this paper by Sendhil Mullainathan et al. (2008), in which a persuader …nds it advantageous to send (costly) messages even when they are not informative. 9

I. A General Framework

A. The class of games and the cognitive environment

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 p i = (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 plays at the same set H i of histories, whatever his type i . [START_REF] Jehiel | Valuation Equilibrium[END_REF] Moreover, the action space of player i at history h 2 H i is common to all types i , and is denoted by A i (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 i2I(h) A i (h) is the action pro…le played by the players who must move at 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 u i and it may depend on the types of [START_REF] Jehiel | Revisiting Games of Incomplete Information with Analogy-based Expectations[END_REF] In their model, such an application requires nature in state s = 1 (or 2) to be identi…ed with the strategic persuader in state s = 0. It also requires to assume that the listener pools the message moves in state s = 1 (or 2) and s = 0 into one analogy class (while distinguishing the persuader's behavior according to her private information). The analogy-based sequential equilibrium thus obtained corresponds to the more "Bayesian" approach they present in appendix II, thereby providing a learning justi…cation to that approach rather than to the simpler one pursued in the body of their paper. [START_REF] Jehiel | Valuation Equilibrium[END_REF] A history 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. players i and j together with the terminal history. That is, u i (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.

Each player i is assumed to know his own payo¤ structure (but not a priori that of his opponent).

The non-standard aspect of our strategic environment lies in the de…nition of the types i . Types i are made of two components i = (t i ; c i ) where t i is the preference type of player i that acts on players' preferences -this is the standard component in the type -and c i is the cognitive type of player i, de…ning how …nely player i understands the strategy of player j -this is the non-standard component in the type.

As common sense suggests, the cognitive type of players do not a¤ect players' preferences over the various terminal nodes. That is, for every terminal history h 2 Z, we have that u i (h; i ; j ) = u i (h; 0 i ; 0 j ) whenever i and 0 i have the same preference type t i , and j and 0 j have the same preference type t j .

Cognitive types c i are de…ned as follows. Each player i forms an expectation about the behavior of player j by pooling together several histories h 2 H j at which player j must move, and each such pool 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 c i of player i is characterized by (An i ; i ), where An i 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. As in Jehiel (2005), An i is de…ned as a partition of the set H j of histories at which player j must move into subsets or analogy classes i . [START_REF] Jones | From Acts to Dispositions: The Attribution Process in Person Perception[END_REF] When h and h 0 are in the same analogy class i , it is required that A j (h) = A j (h 0 ). That is, at two histories h and h 0 which player i pools together, the action space of player j should be the same, and A( i ) denotes the common action space in i .

B. Analogy-based sequential equilibrium

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 player i of type i , a probability measure over the action space A( i ) of player j. Types j of player j are distinguished or not by player i according to whether i = 1 or 0. If i = 1, i is a function of j and i , and i ( j ; i ) is 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 then player i's expectation about the average behavior of player j in class i (where the average is taken over all possible types). [START_REF]Judgement Under Uncertainty, Heuristic and Biases[END_REF] 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 s i . It is a mapping that assigns to every history h 2 H i at which player i must move a distribution over player i's action space A i (h). [START_REF] Kreps | Rational cooperation in the …nitely repeated prisoners' dilemma[END_REF] We let i denote the behavioral strategy of type i , and for every h 2 H i we let i (h) 2 A i (h) denote the distribution over A i (h) according to which player i of type i selects actions in A i (h) when at h. We let i (h)[a i ] be the corresponding probability that type i plays a i 2 A i (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 as time proceeds (and new observations become available). Formally, we let i denote the belief system of player i of type 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 p j throughout the game. We call i the belief system of player i for the various possible types i , and we let be the pro…le of belief systems for the two players i = 1; 2.

Sequential rationality:

From his analogy-based expectation i , player i of type i derives the following representation of player j's strategy: Player i perceives player j to play at every history h 2 i according to the average behavior in class i . [START_REF] Kreps | Sequential Equilibria[END_REF] The induced strategy depends on the type j of player j whenever i = 1 but not when i = 0. At every history h 2 H i where he must play, player i is assumed to play a best-response to this perceived strategy of player j as weighted by his belief i (h). Formally, we de…ne the i -perceived strategy of player j, i j , as

If i = 1 i j (h) = i ( j ; i ) for every h 2 i and j 2 j If i = 0 i j (h) = i ( i )
for every h 2 i and j 2 j Given the strategy s i of player i and given history h, we let s i j h denote the continuation strategy of player i induced by s i from history h onwards. We also let u h i (s i j h ; s j j h ; i ; j ) denote the expected payo¤ obtained by player i when history h has been realized, the types of players i and j are given by i and j respectively, and players i and j behave according to s i and s j respectively. De…nition 1 (Criterion) Player i's strategy i is a sequential best-response to ( i ; i ) if and only if for all i 2 i , for all strategies s i and all histories h 2 H i ,

X j 2 j i (h)[ j ]u h i ( i j h ; i j j h ; i ; j ) X j 2 j i (h)[ j ]u h i (s i j h ; i j j h ; i ; j ): Consistency:
In equilibrium, two notions of consistency are required. First, analogy-based expectations are required to be consistent with the strategy pro…le. That is, they must 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 each element of an analogy class must itself be consistent with the real probability of visiting this element. A learning interpretation of this consistency requirement will be suggested. Second, the belief system held by players must be consistent with their expectations, as in Sequential Equilibrium. Formally, letting P ( i ; j ; h) denote the probability that history h is reached when players i and j are of types i and j respectively, and players play according to , the consistency of the analogy-based expectations is de…ned as: De…nition 2 Player i's analogy-based expectation i is consistent with the strategy pro…le if and only if: For any ( i ; j ) 2 such that i = 1, and for all i 2 An i ,

i ( j ; i ) = P ( 0 i ;h)2 i i p 0 i P ( 0 i ; j ; h) j (h) P ( 0 i ;h)2 i i p 0 i P ( 0 i ; j ; h)
whenever there exist 0 i and h 2 i such that P ( 0 i ; j ; h) > 0:

For any i 2 such that i = 0, and for all i 2 An i ,

i ( i ) = P ( 0 i ; 0 j ;h)2 i p 0 i p 0 j P ( 0 i ; 0 j ; h) 0 j (h) P ( 0 i ; 0 j ;h)2 i p 0 i p 0 j P ( 0 i ; 0 j ; h)
whenever there exist 0 i , 0 j and h 2 i such that P ( 0 i ; 0 j ; h) > 0.

The consistency of the belief system is de…ned as:

De…nition 3 Player i's belief system i is consistent with the analogy-based expectation i if and only if for any ( i ; j ) 2 such that i = 1 i ( j )(;) = p j :

And for all histories h, ha

i (ha)[ j ] = i (h)[ j ] whenever h = 2 H j i ( j )(ha) = i (h)[ j ] i j (h)[a j ] P 0 j 2 j i (h)[ 0 j ] i 0 j (h)[a j ]
whenever h 2 H j , there exists 0 j s.t. i 0 j (h)[a j ] > 0 and player j plays a j at h: While the consistency of the analogy-based expectations (de…nition 2) should be thought of as the limiting outcome of a learning process, the consistency of the belief system i (de…nition 3) should be thought of as an expression of player i 's inference process. Based on his representation of the strategy of the various types of his opponent, player i makes inferences using Bayes' law as to the likelihood of the various possible types he is facing.

The learning process we have in mind to justify the correctness of the analogy-based expectations involves populations of players i and j in which there is a constant share p i of players of type i . In each round, players i and j are randomly matched. At the end of a round, the behaviors of the matched players and their types are revealed. These players exit the population, and they are replaced by new players with the same type. [START_REF] Kreps | Reputation and Imperfect Information[END_REF] All pieces of information are gathered in a general data set, and players have di¤erent access to this data set depending on their types. [START_REF] Lewis | Liar's Poker[END_REF] At each round of the learning process, players choose their strategy as a best-response to the feedback they received (and the system of belief that derives from it), which in turn generates new data for the next round. If the pattern of behaviors adopted by the players stabilizes to some strategy pro…le , every player's analogybased expectations should eventually converge to the ones that are consistent with given his cognitive type, [START_REF] Mullainathan | Coarse Thinking and Persuasion[END_REF] which motivates the solution concept de…ned below. [START_REF] Kreps | Reputation and Imperfect Information[END_REF] The replacement scenario is reminiscent of the recurring game framework studied by Matthew Jackson and Ehud Kalai (1997), who assume that each individual player only plays once. This is to be contrasted with a recent paper by Ignacio Esponda (2008), who, in static games of incomplete information, elaborates on Eyster-Rabin's fully cursed equilibrium by assuming that players i have access both to the empirical distribution of actions of players j (but not to how these actions are related to j's private information) and to i's own distribution of payo¤s. [START_REF] Lewis | Liar's Poker[END_REF] A player i with cognitive type c i = (An i ; i ) such that i = 0 has access to the average empirical distribution of behavior in every analogy class i 2 An i where the average is taken over all histories h 2 i and over the entire population of players j. A player with cognitive type c i = (An i ; i ) such that i = 1 has access to the average empirical distribution of behavior in every i 2 An i for each subpopulation of types j of players j. [START_REF] Mullainathan | Coarse Thinking and Persuasion[END_REF] Observe that the average in the expression of i ( j ; i ) is taken over all possible realizations of player Equilibrium:

In equilibrium, both the analogy-based expectations and the belief systems are consistent, and players play best-responses to their analogy-based expectations at every history. In line with the Sequential Equilibrium (Kreps and Wilson (1982a)), 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 a i 2 A i (h) at every history h 2 H i .

This in turn puts additional structure on the expectations and beliefs at histories that belong to analogy classes that are never reached in equilibrium. [START_REF] Mckelvey | Quantal Response Equilibrium[END_REF] 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 ,

( k i ) k , ( k i ) 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. k i is consistent with k and 3. k i is consistent with k i .
Compared to the sequential equilibrium, the main novelty lies in the introduction of cognitive types who may only know partial aspects of the strategy of their opponent. Compared to the analogy-based expectation equilibrium (Jehiel (2005)), the main novelty lies in the introduction of players' uncertainty about the type of their opponent and the possibility that a cognitive type may distinguish the behaviors of the various types of his opponent. It is the combination of these features that allows us to speak of deception as the exploitation of the FAE. More precisely, such a deception requires the presence of players who are both uncertain about their opponent's type (so that there is room for inference processes) and i's types 0 i , hence the summation over 0 i . That is, we are assuming that player i of type i is informed of j 's behaviors whatever the type of player i they are matched with. The weight p 0 i P ( 0 i ; j ; h) on j (h) simply re ‡ects the relative frequency with which j (h) contributes to the aggregate behavior. [START_REF] Mckelvey | Quantal Response Equilibrium[END_REF] For those readers who dislike trembles, one can o¤er a weaker notion of equilibrium without trembles, similar in spirit to the self-con…rming equilibrium (see Drew Fudenberg and David Levine (1998)). Note, however, that trembles have less bite in our setup than in the standard framework 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. are partially knowledgeable of the strategy of their opponent, so that the inferences may be erroneous.

C. Basic properties

We note that in …nite environments, an equilibrium always exists, no matter how cognitive types are speci…ed and distributed.

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 a i 2 A i (h) at any history h 2 H i with a probability no less than ", and then showing that the limit as " tends to 0 of such strategy pro…les is an Analogy-based Sequential Equilibrium. Q. E. D.

We next observe that if every player i is rational (in the sense that for all types i = (t i ; c i ) of player i, the cognitive type c i = (An i ; i ) is such that An i is the …nest analogy partition S h2H j fhg, and player i distinguishes between player j's types, i = 1), then an analogy-based sequential equilibrium coincides with a sequential equilibrium of the game in which every type i = (t i ; c i ) of player i is identi…ed with her preference type t i . Thus, our framework can be viewed as providing a generalization of the sequential equilibrium that allows us to cope with situations in which the cognitive abilities of players need not be perfect.

D. A simple illustration

In this part, we construct an analogy-based sequential equilibrium in a simple two-person two-period zero-sum game. This example serves to illustrate the working of the concept in a simple scenario. Speci…cally, consider the two-period-repetition of the following zero-sum stage game G. In stage game G the Row player chooses an action U or D, the Column player chooses an action L or R, and stage game payo¤s are as represented in Figure 4. The overall payo¤ obtained by the players is the sum of the payo¤s obtained in the two periods.

That is, there is no discount between period 1 and period 2 payo¤s. There is one type for the Column player. The Column player is 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 only knows the average behavior of this type over the two time periods, i.e. he bundles all histories into one analogy class. In equilibrium, (U; L) is played in period 1 and then (D; R) in period 2 whenever the Row player is rational, and (U; L) is played in period 1 and then (U; R) in period 2 whenever the Row player is coarse. The Column player gets an expected payo¤ of 10 that is less than her value 70=9. The Rational Row player gets an overall payo¤ of 5 + 7 = 12 and the Coarse Row player gets an overall payo¤ of 5 + 3 = 8.

A key aspect of this equilibrium involves 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 Row player for sure. When action U is being played in period 1, the Column player believes that she faces the Coarse Row player with probability 1=2 1=2+1=2 1=2 = 2 3 . Accordingly, the Column player plays R in period 2 because given her belief, 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 the high payo¤ of 7. [START_REF] O'sullivan | The Fundamental Attribution Error in Detecting Deception: The Boy-Who-Cried-Wolf E¤ect[END_REF] We make several comments about the equilibrium shown in Proposition 2.

First, the Column player gets an expected payo¤ that is less than her value, 70=9, even though, by the very property of the value, the Column player could very well guarantee 70=9 -no matter what the Row player does -by playing the maximin strategy (i.e., play L with probability 4=9 and R with probability 5=9 in both periods). The Column player chooses not to follow the maximin strategy because she thinks that she can do better, given her understanding of the strategy of Row players. Such a feature would, of course, not arise in a standard rationality framework in which the Column player should obtain, in equilibrium, at least what she can secure irrespective of other players' strategies. This helps to clarify the di¤erence from Vincent Crawford (2003), who assumes in a zero-sum pre-play communication game that those agents whose behaviors are not exogenously speci…ed are fully rational and are thus bound to get at least their value in equilibrium. [START_REF] Ross | The Intuitive Psychologists and his Shortcomings[END_REF] It also helps to explain why it is not possible to interpret the analogy-based sequential equilibrium as a sequential equilibrium that would obtain in the full rationality paradigm under alternative informational assumptions. 21 [START_REF] O'sullivan | The Fundamental Attribution Error in Detecting Deception: The Boy-Who-Cried-Wolf E¤ect[END_REF] The rest of the argument to establish Proposition 2 goes as follows. It is readily veri…ed that the Rational Row player plays a best-response to the Column player's 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, and he would obviously get a lower payo¤ by playing U in period 2.) The Coarse Row player …nds it optimal to play U whenever he has to move, because he perceives the Column Player to play L and R with an equal frequency on average over the two time periods, and 1 2 (5 + 3) > 1 2 (0 + 7). [START_REF] Ross | The Intuitive Psychologists and his Shortcomings[END_REF] Vincent Crawford (2003) captures the idea of lying for strategic advantage in a zero-sum pre-play communication game that is populated by su¢ciently many mechanical types. But in Crawford's model, the belief of rational players cannot be manipulated, as equilibrium requires that rational players are not mistaken about either the distribution of types or about their strategies. This is a key di¤erence from our approach. [START_REF] Ross | Social Roles. Social Control and Biases in Social-Perception Processes[END_REF] Even if the Column player were assumed not to remember whether she is in stage 1 or 2, she could still Second, in the equilibrium of Proposition 2, the Rational Row player obtains a larger payo¤ than the Coarse Row player. This is no coincidence, as the Rational Row player always has the option to mimic other types' strategies and Rational players assess correctly the payo¤ attached to any strategy.

Finally, it should be noted that it would be impossible to reproduce the behavioral strategies described in Proposition 2 if there were only one type for each player, who would be characterized solely by his analogy partition as in Jehiel (2005). [START_REF] Schelling | The Strategy of Con ‡ict[END_REF]

II. Deception as a Bargaining Tactic

A. The basic setup

The owner of a house, Mrs A, wishes to sell her good. The initial price has already been publicly announced. It is either p or p where p > p and p may be thought of as being the "market price" of the house as perceived real estate agents.

A potential buyer, Mr B, comes in, and the following interaction between Mrs A and Mr B takes place. Mrs A tells Mr B whether or not some small repairs (say for heating de…ciencies) are needed in the house. [START_REF] Sobel | A theory of Credibility[END_REF] If minor de…ciencies are announced, the price drops by an amount . That is, the new price is p where p was the originally announced price ( should be thought of as being small relative to p p). Then Mrs A tells Mr B whether or not there is another buyer who has expressed interest in the house. When the initial price was p = p and Mrs A says that another buyer has expressed interest, the price increases by a very small amount, say ". No such price increase occurs when the initial price is p = p. [START_REF] Spence | Job Market Signaling[END_REF] Only Mrs A knows whether indeed there are small repairs needed and whether there is another potential buyer. After the announcements are made, Mr B has to decide whether or not to accept the o¤er (before he can verify the correctness of Mrs A's announcements).

secure the value, given that the maxmin strategy does not require any recall (it is stationary). See Jehiel (2005) and Jehiel and Koessler (2008) for further examples illustrating why the analogy-based expectation equilibrium cannot be interpreted as a standard equilibrium of a di¤erent game with modi…ed information structure. [START_REF] Schelling | The Strategy of Con ‡ict[END_REF] 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 .

If Mr B says yes, the transaction takes place at the agreed price (i.e., p if no de…ciencies were announced and p if de…ciencies were announced). We let V yes B denote Mr B' s payo¤ when the original price was p and de…ciencies were announced (so that the …nal price is p ).

If Mr B says no, there are several cases. When the original price was the "market price" p, no transaction takes place between Mrs A and Mr B, as we assume that Mrs A expects to sell her house at a price close to p and Mr B expects to buy a similar house at a price close to p (both Mrs A and Mr B would be slightly better o¤ making the transaction now even at prices p , p + ", respectively, due to extra delays imposed by the transaction not being made now).

When the original price was p and there is e¤ectively another buyer, no transaction between Mrs A and Mr B takes place. Mrs A gets a payo¤ that is less than p, due to the risk that the other buyer does not con…rm his interest, but signi…cantly larger than p, and Mr B gets a payo¤ of V out B (corresponding to the outcome of a search for another house).

When the original price was p and there is no other buyer, [START_REF] Stahl | Evolution of Smart n players[END_REF] bargaining between Mrs A and Mr B goes on. We do not model this extra piece of bargaining explicitly, but we assume that a transaction eventually takes place at a price signi…cantly lower than p (say not too far from p). [START_REF] Neuman | Theory of Games and Economic Behavior[END_REF] We denote by V no B the payo¤ obtained by Mr B in this case. [START_REF] Vrij | Detecting Lies and Deceit[END_REF] On top of the above speci…cations, we assume that there are two categories of sellers, those who always tell the truth (whom we call honest sellers) and those who do what serves their interest best (whom we call opportunists). Mrs A can belong to either of these categories, but there is no way for Mr B to know which, except by making inferences from how she behaves (here, what she says in the announcement stage).

Finally, we describe the probabilities of the various events, which are assumed to be known to both Mrs A and Mr B. We assume that the probability of the seller being honest is = Pr(Mrs A is honest) independently of the other random variables. We assume that the probability of a de…ciency is d = Pr(de…ciency) independently of the other random variables.

For plausibility reasons, we allow the probabilities that the price is p and that there is another buyer to be (presumably negatively) correlated. We let b = Pr(other buyer) be the probability that there is another buyer, = Pr(p = p jother buyer) and = Pr(p = p jno other buyer) be the probabilities that the initial price is p conditional on having another buyer or not having one, respectively. We also let = b + (1 b ) = Pr(p = p) denote the unconditional probability that the initial price is p.

B. Cognitive environment

In this bargaining problem, a key strategic aspect is the judgement Mr B makes as to the likelihood that there is another buyer as a function of the announcements made by Mrs A.

When Mr B is told that there is another buyer, should Mr B trust Mrs A? And how is Mr B's judgement a¤ected by the announcement (or the non-announcement) of minor de…ciencies?

We wish to analyze a situation in which Mr B somehow confuses the two price scenarios p = p; p when assessing the announcement strategies of sellers such as Mrs A (whereas he distinguishes the strategies of sellers in all other respects). We will also assume that opportunist sellers are fully rational, and we will show how Mrs A, when opportunist, deceives Mr B by mentioning minor de…ciencies (whether or not there are any) so as to increase his belief that there is indeed another buyer when she says there is. From the viewpoint of social psychology, Mr B is victim of the fundamental attribution error, and when she is opportunist, Mrs A exploits this. This is the essence of the deceptive tactic that we wish to highlight here. We will also illustrate later on how deception would be unsuccessful if Mr B were assumed to be fully rational instead.

To cast the above strategic environment into the framework of Section I, we let t A be the preference type of Mrs A, where t A = ( ; d; b), = h, o indicates whether A is honest or opportunist, d = 1; 0 indicates whether there are (minor) de…ciencies or not and b = 1; 0 indicates whether there is another buyer or not. Mrs A is assumed to be fully rational. Thus, her cognitive type corresponds to the standard situation in which every decision node of Mr B constitutes a singleton analogy class (since there is only one type of Mr B -see below -no distinction of the various types of Mr B is relevant here). We will identify Mrs A's type A with her preference type t A .

Mr B can only be of one type. While his preferences have already been described, his cognitive type is described as follows. Mr B puts the "de…ciency announcement nodes" of Mrs A in the same analogy class, whether p = p or p. Similarly, he puts the "other buyer announcement nodes" of Mrs A in the same analogy class, whether p = p or p. In addition, he di¤erentiates between the behaviors of the various types of sellers (i.e., B = 1 in the language of Section I).

Finally, before the interaction between Mrs A and Mr B starts, Nature chooses the type A of Mrs A and then the initial price p = p; p according to the distribution described above.

C. Analysis

When Mrs A is honest, her strategy is imposed by the de…nition of her type. She always tells the truth. That is, she mentions the de…ciency if there is one and she mentions the existence of another buyer if there is one, whether the price is p = p or p. Moreover, when the price is p = p and Mrs A is opportunist, she never mentions any de…ciency (whether or not there is one), so as to save on the discount , and she always says that there is another buyer (whether or not there is one), so as to get the extra " in the …nal price. 28 Transaction always takes place between Mrs A and Mr B when p = p.

It only remains to determine the announcement strategy of Mrs A when she is opportunist and the price is p, and also the acceptance strategy of Mr B in this case.

We will test when the following strategies constitute an analogy-based sequential equilibrium. When she is opportunist and the price is p = p, Mrs A always reports that there are de…ciencies and that there is another buyer, no matter what the truth is. Mr B says "yes" to the o¤er after such announcements and "no" after any other announcement (i.e., when Mrs A says that there is no other buyer, or that there is another buyer and no de…ciencies in the house).

A key variable in the analysis is the belief that Mr B attaches to the existence of another buyer after Mrs A has made her announcements (and the initial price was p = p). Call such a belief. Given the above de…nitions of V out B , V no B and V yes B , Mr B would accept the deal (after d = 1 and b = 1 were announced) if V out B + (1 )V no B < V yes B , and he would reject it otherwise. Accordingly, we let

= V no B V yes B V no B V out B
denote the threshold belief such that Mr B would say "yes" if > and "no" if < .

Assuming Mrs A follows the above strategy and given Mr B's cognitive type, the con- and Mr B's perceived probability that ( = o and p = p,

Mrs A says d = 1, b = 1) is (1 )( ) 2 . 29
After observing that p = p and hearing that d = 1 and b = 1 from Mrs A, Mr B assesses the probability that there is another buyer based on his prior belief that Mrs A is honest and his perceived informativeness of A's messages whether = h or o. Since Mr B knows that when Mrs A is honest she sends truthful messages, and since he correctly perceives that the message of opportunist sellers is not informative, we obtain that Mr B's posterior belief that b = 1 is:

post = post + (1 post ) Pr(b = 1 j p = p)
where Pr(b = 1 j p = p) = b = . We will also de…ne the corresponding probability when the probability that Mrs A is honest coincides with the prior instead of post . That is, prior = + ( 1) Pr(b = 1 j p = p):

We have:

Proposition 3 Assume that post > > prior . Then, the above strategy pro…le is an analogy-based sequential equilibrium.

Proof: Given that post > and the above derivations, after observing that p = p and hearing from Mrs A that d = 1, b = 1, Mr B optimally says "yes" to the o¤er because he infers that the chance that there is indeed another buyer post is high enough (larger than ).

Clearly, if Mr B is told that there is no other buyer b = 0, he will infer that Mrs A is honest (because opportunist sellers never say that), and accordingly he will know for sure that there is no other buyer and will say "no" when the price is p = p. Finally, after observing that p = p and hearing from Mrs A that d = 0, b = 1, Mr B will believe according to his cognitive perception that Mrs A is honest with some probability b . 30 Because post ( = h) > (as otherwise one could not have post > prior ) and ( = h j p = p, A says d = 0 or 1, b = 0) = 1, we can infer that b < (the evolution of as perceived by Mr B should be a martingale). Given that prior < (and that the o¤er when no discount is proposed is worse, everything else being equal), we conclude that Mr B …nds it optimal to reject the o¤er after observing p = p and hearing from Mrs A that d = 0, b = 1. This also ensures that when Mrs A is opportunist and p = p, she does not …nd it desirable to say that d = 0, b = 1 because she rightly anticipates that the o¤er would be rejected in such a case. Q. E. D.

D. Discussion

Observe that the condition > prior is independent of Pr(p = p) whereas post > is automatically satis…ed when Pr(p = p) is su¢ciently small. 31 Thus, the conditions of Proposition 3 require that the prior probability that Mrs A is honest be not too large, so that > prior , and that the probability that p = p be not too large, so that post > . The e¤ect of Pr(p = p) on the equilibrium analysis is, of course, due to the erroneous inference process of Mr B, caused by his bundling of the decision nodes of Mrs A into a single analogy 30 This b is derived analogously to post and can be expressed as

b = (1 d ) b (1 d ) b + (1 )(1 ) 2
31 To illustrate this, consider the case in which the probability that p = p is independent of whether b = 0; 1 so that = = . Then post ( = h) simpli…es into ) ; which converges to 1 as approaches 0, thereby implying that post approaches 1 > when ' 0. In a di¤erent vein, it is also worth noting that if Mr B had not distinguished the di¤erent categories of sellers (i.e., B = 0), then the deceptive tactic of the opportunist Mrs A would have been pointless, since Mr B would have kept believing that he is facing an honest seller with probability whatever Mrs A's announcements, and he would have rejected the deal when p = p (given that > prior ). Thus, we see that the deceptive tactic of Mrs A requires that Mr B be not too irrational, in the sense that Mr B's cognitive type should allow him to make some inferences from what he observes.

d b d b +( 1
Finally, it is instructive to contrast the insights obtained in our cognitive environment with those that would arise had we assumed that Mr B were fully rational. An important observation is that when Mr B is rational it can never be optimal for Mrs A, when she is opportunist, to say that d = 1 with probability 1 when p = p. Indeed, if she did so, then Mr B -assumed to be rational -would infer that Mrs A is an honest seller with a probability smaller than the prior , and it is not hard to see that this cannot be bene…cial to Mrs A (given the extra cost due to the discount imposed by the announcement of d = 1). 32 As it turns out, when > prior and Mr B is rational, Mrs A, when she is opportunist and p = p,

will

III. Conclusion

What are the lessons to be drawn from our approach? Firstly, a description of the prototype of a deceptive tactic. In the above bargaining story (as in many real life situations), the …rst stage of the deception involves building a relation of con…dence with the victim (even if this has some cost, as illustrated by the announcement of the de…ciencies in the bargaining story) so as to better exploit it at a later stage. Thus, from a practical viewpoint, one is more likely to discover a deceptive tactic when one sees a party making an initial sacri…ce that subsequently turns out to be of great bene…t to this same party. Secondly, according to our theory, deception requires the presence of agents who are neither fully rational (otherwise, their beliefs could not be manipulated) nor fully irrational, in the sense of not distinguishing the various types of the opponent (otherwise, there could be no inference process as the interaction proceeds). More precisely, our theory of deception requires the presence of agents who somehow have a stereotypical understanding of others' attitudes. Our

Figure 4 .

 4 Figure 4. The stage game G We assume that there are two types of Row players, the Rational type and the Coarse type, where both types are 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 only knows the average behavioral strategy of the Column player over the two time periods (i.e., he bundles period 1 and the possible histories in period 2 into one analogy class).

Proposition 2

 2 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 in period 1.

2

 2 sistency of Mr B's analogy-based expectations implies that he should expect honest sellers to always report the truth, and opportunist sellers either to say that d = 1 and b = 1 with probability = Pr(p = p) or to say that d = 0 and b = 1 with probability 1 , independently of p, d, b. From such a perception, the distribution of (p; d; b), and Bayes' law, one can now compute Mr B's posterior belief denoted post ( = h) that Mrs A is honest when she says that d = 1, b = 1 and p = p. post ( = h) = This follows from noting that Mr B's perceived probability that ( = h and p = p, Mrs A says d = 1, b = 1) is d b

  class, whether p = p or p. Increasing Pr(p = p) makes the deceptive tactic of Mrs A when p = p and = o more e¤ective, as reporting that there are de…ciencies becomes more representative of the overall attitude of honest sellers than of opportunist sellers. Ironically, had we assumed that Mr B were fully rational instead, he should have inferred, when p = p and Mrs A reports de…ciencies, that Mrs A is more likely to be an opportunist, since when p = p mentioning de…ciencies is more typical of opportunists than of honest sellers. But Mr B is unaware of how the behaviors of sellers di¤er in situations p = p and p, thereby explaining his erroneous judgement.We note that in the equilibrium shown above, when p = p, and Mrs A honestly reports that d = 0 and b = 1, Mr B does not trust Mrs A and he rejects her o¤er. So we see here that when Mrs A is opportunist, her deceptive tactic imposes a cost on honest sellers, who are no longer trusted in some scenarios.It should also be mentioned that under the conditions of Proposition 3 , we could not sustain an equilibrium in which opportunist sellers always say that d = 0 and b = 1 irrespective of p = p and p. Indeed, if this were so, after hearing that d = 0 and b = 1, Mr B would believe that he is facing an honest seller with a probability smaller than the prior (since saying d = 0 and b = 1 would then be more typical of opportunists than of honest sellers: -remember that opportunists say d = 0 and b = 1 when p = p). Since > prior , Mr B would then reject Mrs A's o¤er. By contrast, if Mrs A were to say that d = 1 and b = 1 when p = p, she would convince Mr B that she is an honest seller with probability 1 and Mr B would then accept Mrs A's o¤er, thereby implying that the assumed strategy of Mrs A when p = p and = o is not optimal.

  mix between announcing d = 0 or 1 and will announce b = 1: Mr B's posterior belief that Mrs A is honest after either (p = p and Mrs A says d = 0 and b = 1) or (p = p and Mrs A says d = 1 and b = 1) must be smaller than (due to the martingale property of beliefs), thereby implying that Mr B rejects Mrs A's o¤er in one of these two cases. Due to the required indi¤erence of Mrs A between announcing (d = 0 and b = 1) or announcing (d = 1 and b = 1) we conclude that Mrs A cannot obtain that a transaction with Mr B takes place with probability 1 when p = p and > prior . This is, of course, in sharp contrast with what happens in the above cognitive environment as analyzed in Proposition 3.

As regards Lewis' deceptive tactic, it is not at all clear from a game theoretic perspective why the fact that the lie is logical (in a given instance) should increase the likelihood that it is believed. If liers always use logical lies, then logic should even heighten the listener's suspicion.

From the perspective of this paper, logic may be viewed as more typical of true statements (over all possible statements), thereby making the use of logical lies more e¤ective.

Indeed, if Mr B were fully rational, he should understand that opportunist sellers more systematically concede that there are small de…ciencies when the price is high, and thus Mr B should be even more cautious about the true presence of another buyer when told that there are heating problems.

More precisely, we assume that players adopt the simplest representation of their opponent's strategy that is consistent with their knowledge (the analogy-based expectation). That is, the opponent's behavior in the various nodes bundled into one analogy class is assumed to be the same and in equilibrium it coincides with the aggregate distribution of the opponent's behavior over the set of nodes forming the analogy class. The evolution of the belief system is then similar to that in sequential equilibrium (DavidKreps and Robert Wilson (1982a)) except that it is based on the conjecture about the opponent's strategy as just de…ned (rather than on the opponent's true strategy).

The traditional approach to reputation pioneered by Thomas[START_REF] Schelling | The Strategy of Con ‡ict[END_REF] associates the idea of successful reputation building with the successful ability to commit to a particular behavior (which is of no use in a zero-sum game, due to the minmax theorem).

Other approaches based on the idea that to facilitate learning agents do not consider the set of all possible strategies but only a subset are also available, see in particular Olivier Compte and Andrew Postlewaite (2008).

The cursed equilibrium was developed independently of the analogy-based expectation equilibrium. The fully cursed equilibrium can be viewed as a special case of the analogy-based expectation equilibrium in which players' analogy partitions coincide with their own information partitions. The partially cursed equilibrium can be viewed as an alternative approach to the idea of partial sophistication to that captured by the analogy-based expectation equilibrium (see[START_REF] Jehiel | Revisiting Games of Incomplete Information with Analogy-based Expectations[END_REF] for further discussion).

A partition of a set X is a collection of subsets x k X such that S k x k = X and x k \ x k 0 = ; for k 6 = k 0 .

We could more generally allow players to distinguish partially the types. This would lead to a partitional approach de…ning which of the types are being confused. The resulting presentation would however be more cumbersome without bringing additional insights.

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.

It is assumed that Mr B cannot verify the nature of these repairs within a reasonable amount of time.

We assume this only for plausibility. The analysis is una¤ected if we assume that there is also a price increase when p = p (this is because " is assumed to be small in comparison with p p).

Presumably Mr B gets further signals (not under Mrs A' s control) about this.

This may be because Mr B values this speci…c house more than the average buyer and everyone is aware of this at this bargaining stage.

More precisely, V no B will denote Mr B's payo¤ assuming a de…ciency has been announced.

Mrs A knows that the transaction will be approved by Mr B in any event, even if the …nal price is p + " (rather than p or p).

The latter probability requires that = o (which has probability 1 ), that p = p (which has probability ), and that the announcement "d = 1, b = 1" is picked (which is perceived to have probability ) while these three events are all perceived to be independent.

Under the condition of Proposition 3 ( > prior ), Mr B would reject Mrs A's o¤er. Even if < prior , when Mrs A is opportunist she will prefer to say that d = 0 and b = 1 so as to save on the discount .