Why the Ultimatum Game may not be the Ultimate Experiment?

Transcription

1 Why the Ultimatum Game may not be the Ultimate Experiment? Yoram Halevy Michael Peters y July 11, 2006 Abstract The Ultimatum Game seems to be the ideal experiment to test for the structure of preferences or the sequential rationality assumptions underlying subgame perfection. We study the theoretical implications of introducing the possibility of misconceptions - that actions may potentially a ect continuation payo s - and show that the set of Perfect Bayesian Nash Equilibria does not converge to the subgame perfect equilibrium when the possibility of misconception approaches zero. The perfect equilibria studied corresponds qualitatively to the experimental ndings of fair o ers made and unfair o ers rejected. 1 Introduction and Discussion One of the appealing and important properties of experimental economics stems from its ability to test the predictions of economic theory: as more information is collected in experiments (both laboratory and eld), economists are able to sort theories to those that are consistent with the data and those that can be refuted by it. A prominent example of this methodology has been the test of the assumption that decision makers have sel sh preferences. Many bargaining experiments have shown that in certain environments the equilibrium predictions of economic theory which are based on backwards induction and sel sh preferences are not consistent with the University of British Columbia y University of British Columbia 1

2 experimental results (see Roth [14] for an extensive review). This has been demonstrated most clearly in one of the simplest and most in uential experiments: the ultimatum game. A proposer makes an o er of x (between 0 and x). If the responder accepts the o er then the proposer receives (x x) while the responder receives x: If the responder rejects - both receive zero. This is the simplest dynamic game of complete and perfect information, for which the backwards induction argument is exactly devised. The latter (together with sel sh preferences) predicts that the responder should accept any positive o er and therefore the proposer should o er only a minimal o er. As is now well known (since Güth et al [10]), the experimental evidence refute these predictions: low o ers are rejected, and the modal o er involves equal split. Di erent researchers have drawn di erent conclusions from this experiment. One group of researchers have assumed that although the backwards induction argument is valid, the agents have other regarding preferences. They care about their relative payo (Bolton [5], Rabin [13], Levine [11], Fehr and Schmidt [7], Bolton-Ockenfels [6] and many others). Another group (Binmore, Samuelson and others [9, 3, 2, 4]) have been arguing that excessive attention has been given to the backwards induction arguments. They claim that since the set of Nash equilibria is much larger, one should try and describe the outcome of the game in terms of another process that leads to NE, but not necessarily to SPE - like an evolutionary model. The goal of the current study is to study how robust is the backwards induction outcome in the ultimatum game to perturbations - representing possible misconceptions of participants concerning the game structure. The conclusions drawn from this study may be viewed in two di ering levels of insight. First, it points out (what is almost obvious in a hindsight) that the equilibrium predictions of game theory are based not only on assumptions about the preferences, but also on common knowledge of the game structure. But we show much more than that: even when the game structure subjects have in mind is in nitesimally close to the structure the experimenter intended to induce in the experiment, but allows for the possibility that the other party holds di erent information, or does not understand the experiment (the mapping from actions to payo s) in the way the experimenter intended, the theoretical equilibrium behavior may be very di erent from the equilibrium outcome in the game the experimenter had in mind and corresponds much better to the experimental results. Why is it natural and important to study the implications of misconcep- 2

3 tions in experimental games? We argue that participants arrive at the experiment with very poor information about the payo s associated with actions they take during the experiment, and if and how these actions will a ect their future payo s (e.g. subsequent experiment). Any experimenter knows that explaining the game to subjects could be the most challenging part of the experiment. In the ultimatum game - this task is extremely di cult: subjects may nd the environment super cially similar to bargaining environment from their day to day life. Indeed, Aumann [1] argues that the rule of rejecting low o ers is a case of rule rationality : a decision making process that rewards a behavior that utilizes a rule which works well in the repeated environmentthe subjects encounter in every day life. When applying the decision rule to the one-shot environment, the behavior may be hard to rationalize. 1 The resemblance between the ultimatum game and naturally occurring repeated bargaining environment suggests that the natural focus should be that continuation payo s might depend on the actions subjects take during the experiment. A perfectly designed experiment should dispel any uncertainty about these, and leave the participants convinced that though their continuation payo s may be uncertain, they are independent of any action they take during the experiment. This is presumably accomplished by the messages the experimenter sends to the subjects when the rules and outcomes are described, the way the experiment is described when participants are solicited, the reputation of the experimenter and so on. When one allows for the possibility of misconceptions, it is easy to explain why a proposer won t demand everything in the ultimatum game: he is afraid his proposal will be rejected. It is harder to explain why the responder rejects a positive o er. The simple intuition is that she believes her continuation payo s are higher when she rejects o ers. 2 However, consideration of exactly why she rejects the o er leads to the positive conclusions about the ultimatum game. If the experiment is very well designed, but not perfect, participants will still assign positive, though small, probability to the possibility that continuation payo s are a ected by behavior in the experi- 1 One path some researchers have chosen in order to minimize misconceptions, is to allow subjects to gain experience in previous rounds. However, In light of the above view of the problematic nature of the experiment, this might not be the optimal way to pursue since it reinforces a perception that the game may be repeated. 2 There is never any uncertainty here about the fact that her payo s during the (stage) experiment are lower if she rejects the o er. 3

4 ment. Both participants will have some information about this that depends on their conversations with the experimenter, maybe previous experimental experience, etc. The structure of the ultimatum game is such that there will always be equilibria in which the actions of both players depend on this information. In a well designed experiment, the proposer should think it very unlikely that the responder has any information that would lead her to believe that she would be better o rejecting. Even so, he may think it very likely that they are playing an equilibrium in which certain proposals will be perceived to convey high informational content. If a responder thinks it is very unlikely that a proposer with normal information would demand a high share of the pie, then it might make perfect sense after appropriate prior revision, for the responder to reject low o ers. This logic can be carried to the limit of a perfectly designed experiment. In such case the subjects can imagine the possibility that their continuation payo s depend in some ways on their actions in the experiment., but they assign zero probability to this possibility. Perfect Bayesian Equilibrium then admits the following equilibrium: the proposer in the ultimatum game demands half of the pie, and the responder accepts it. Since this outcome is supposed to occir with probability 1, anything else that happens is now o the equilibrium path. So any deviation by the proposer is rejected because the responder revises her beliefs in a way that assigns high probability to continuation payo s being much higher when an o er (any o er) is rejected than when it is accepted. Sequential rationality is consistent with just about any o er, even in a perfectly designed experiment. The theoretical contribution in this paper, is simply to analyze these equilibria. All these equilibria involve various proposals being accepted and rejected with positive probability on the equilibrium path. As uncertainty about the continuation payo s shrinks, the set of equilibria of this kind converges to the set of all pure Nash equilibria of the ultimatum game. As is well known, there is a continuum of such equilibria. Beyond the methodological issues, the practical conclusion is that the ultimatum game is not the ultimate experiment to test for, say, sel sh preferences. Sel sh preferences and sequential rationality impose no restriction at all on the o er made by the proposer. This is true even in a perfectly designed experiment, unless participants simply aren t able to conceive of the possibility that their behavior in the experiment will a ect their continuation payo. 4

5 2 A Simple Model of the Ultimatum Game This example is a stylized version of the ultimatum game. 100 dollars are to be split in the usual way, except that the proposer only has two alternatives - demand 99 dollars, leaving 1 for the responder, or demand 50 dollars, leaving the other 50 for the responder. The perturbation considered here is that the proposer has some information about the joint payo s of the players in the continuation. For simplicity, assume that this joint payo in the continuation is and that it is the same for both players. The rst type of proposer has a signal 1 that tells him that these continuation payo s are independent of the responder s behavior during the experiment, and for simplicity are normalized to zero. The payo s from the experiment for both players are the entries in the following table: rr ar ra aa 99 0; 0 99; 1 0; 0 99; ; 0 0; 0 50; 50 50; 50 The rows represent the di erent demands (100 minus the proposal): 99 or 50. The columns index the various strategies for the responder. For example, ar means: accept 99 and reject 50. If the responder adopts the strategy ar, then the proposer gets 99 dollars plus his payo (zero) from the continuation if he demands 99, but gets only his continuation payo (zero) if he o ers 50. The other entries are interpreted similarly. There are multiple Nash equilibria in this game. The (unique) sub-game perfect equilibrium, of course, has the proposer demanding 99 and the responder accepting both o ers. There is another Nash equilibrium in which the proposer demands 50, while the responder accepts 50 and rejects 99. Another equilibrium exists where the proposer demands 50 and the responder randomizes over ra and aa, playing the rst with probability The second type of proposer has a signal that suggests that continuation payo s depend on the behavior of the responder. In particular, this proposer s information suggests that both players will be strictly better o in the continuation if the responder rejects an o er. Call the signal of the proposer in this case 2, and suppose that such a signal suggests the following payo matrix: rr ar ra aa 99 x; x 99; 1 x; x 99; 1 50 x; x x; x 50; 50 50; 50 5 ( 1 ) ( 2 )

6 To be concrete, suppose the proposer in this case knows that if the pair disagree - they will be involved in a subsequent experiment, but if they agree - they will be sent home. The expected payo from the second experiment is the entry x that appears in the matrix. We are interested in what happens when 1 < x < 50. The responder doesn t know the signal the proposer has received. The responder in the example believes that the proposer has received signal 1 with probability q which could be (and in fact should be, if the experiment is well designed) very close to one. This belief is common knowledge.

7 A second, more interesting, pooling equilibrium exists in which the proposer o ers 50 independently of his signal. If he demands 99 (which is now o equilibrium) the responder revises her beliefs so that she assumes that the proposer has surely received the signal 2. The responder is then certain she will be better o rejecting the o er of 1. We now prove that there is a third equilibrium, in which the type 1 proposer randomizes. First note that there cannot be an equilibrium in which both types of proposer randomize. Let and be the probabilities with which the responder accepts 99 and 50 respectively. Suppose that proposer of type 1 is indi erent between o ering 99 and 50. Then If the proposer of type 2 is also indi erent This gives 99 = 50 (2) 99 + (1 )x = 50 + (1 )x (3) 50 + (1 )x = 50 + (1 )x (4) or =. So if there is an equilibrium where both types of the proposer randomize, the responder must accept both o ers with the same probability. This is a contradiction, since this will make the proposer o er 99 for sure. So in every equilibrium, at least one proposer type uses a pure strategy. Suppose that the type 2 proposer uses a pure strategy of demanding 99. Then the type 1 proposer either uses the same pure strategy (the equilibrium we have already described), or the type 1 proposer will reveal his type with positive probability. Since 2 is demanding 99, 1 cannot reveal his type by the identical demand, so he must be the one who o ers 50. If so, 99 must be rejected with positive probability and 50 accepted (since x < 50). To make him indi erent 99 = 50 (5) or = In that case the type 2 proposer has payo which makes demanding 99 a best reply x > 50 (6)

8 Now the type 1 proposer has to demand 99 with just the right probability to make the responder indi erent between accepting and rejecting. Let be the probability with which the proposer demands 99. Let p be the posterior probability that the proposer s type is 1 (so that continuation payo s are independent of the responder s action) conditional on receiving an o er of 1 (demand of 99). Using Bayes rule, p is then: p = q q + (1 q) (7) The responder will be indi erent about whether or not to accept the demand of 99 if: (1 p)x = 1 giving 1 p = 1. So the appropriate randomization is x q q + (1 q) = 1 1 x (8) This has a unique solution provided q > 1 1 x. To summarize, this equilibrium has the proposer demanding 99 if he is of type 2 and randomizing between 99 and 50 if he is of type 1. The proposal 99 is rejected a little less than half the time, though the o er 50 is always accepted. Figure 2: Perfect Bayesian Nash Equilibrium of the Ultimatum Game q 1 q (50,50) (50,50) a a r (0,0) (x,x) r 50 θ1 θ 2 ( ) 1 q λ = x 1 q 1 λ a r (99,1) (99,1) a (0,0) (x,x) r

9 The number q measures the degree of uncertainty (possible misconception ) involved in this game. When q is close to one, the responder is almost sure about the payo matrix of the game, the proposer is almost sure the responder is sure and so on. First, note that the demand of 99 is rejected about half of the time no matter what q is. Furthermore, it is straightforward to check that the probability with which the demand 99 is made shrinks to zero as q approaches one ( = (x 1) (1 q) =q). As q goes to one, the set of Perfect Bayesian equilibria of the ultimatum game converges to the set of Nash equilibria of the perfect information game - 1. The implications of subgame perfection are not robust to possible uncertainty about the contiuation payo s of the game. 3 Comparison with Existing Literature Subjects behavior varies across experiments, but usually proposers demand only 50, and responders who are o ered only 1 often reject. As mentioned before, one way to account for these experimental results is to imagine that players preferences aren t sel sh. For example, the responder may care not only about his own monetary reward, but also about the di erence between his reward and the proposer s reward. Let x p be the proposer s monetary payo and x r be the responder s monetary payo. Suppose the proposer s utility function is u p = x p max[0; x r x p ] (9) and the responders is Then the payo matrix becomes u r = x r max[0; x p x r ] (10) rr ar ra aa 99 0,0 99,-97 0,0 99, ,0 0,0 50,50 50,50 (11) One Nash equilibrium has 99 being o ered and rejected (because 50 is also rejected), the other has 50 being o ered and accepted. Only the latter survives the subgame perfection requirement. Strictly speaking, neither equilibria has 99 being rejected on the equilibrium path, though uncertainty about the degree of aversion to payo di erences could presumably be used to support 9

10 such rejection on the equilibrium path associated with a Bayesian equilibrium. Fudenberg, Kreps and Levine [11] (henceforth FKL) study the robustness of equilibrium re nements to possible payo perturbations. They consider an elaboration of an extensive form game, in which one of several di erent versions of a game is chosen at random by nature. Versions di er only in the players payo s, which might be di erent from the original payo s in the game. Generally, players may hold private information concerning the version of the game. FKL s interest lies in small perturbations, for which they de ne a convergence criterion for a sequence of elaborations: the probability that a version with payo s asymptotically equal to the original game is chosen, approaches one (in addition to requiring that the number of versions and payo s are bounded). A strategy in the original game is near strict under general elaborations if there exists a sequence of elaborations that converges to the original game, and a sequence of strict equilibria for each elaboration that converges (in a well de ned way) to the strategy in the original game. FKL prove (Proposition 3) that a pure strategy pro le in the original game is near strict under general elaborations if and only if it is a Nash equilibrium in the original game. That is, any Nash equilibrium can be supported if one allows arbitrary doubts about each other s payo s. The example presented in Section 2 is a speci c FKL elaboration of the original ultimatum game. This particular elaboration represents a speci c type of possible misconception about the original game. The general model presented in the following section studies more general elaborations. Di erently from FKL, our interest lies in understanding the Perfect Bayesian Equilibrium in an elaboration, and not necessarily strict Nash equilibrium in an arbitrary elaboration. The payo to this modeling strategy is that we are able to capture behavior (as rejections of low o ers) that is not consistent with Nash equilibrium in the original game. Furthermore, while introducing a new state (even if it is assigned a prior probability zero) - we are able to disentangle the relation between common knowledge of the game structure and common certainty of it - giving rise to a PBNE in which high o ers are made and accepted, while low o ers (if made) - are rejected with probability 1. In light of FKL s result, the convergence of the PBE to the set of Nash equilibria as q approaches 1 - is not surprising, and will not be pursued further in the general model. As alluded in the introduction, an alternative explanation for the experimental results is a failure of the common knowledge assumptions that play a 10

11 key role in the Nash and subgame perfect Nash prediction. Even if a player understands the instructions given by the experimenter, and fully accepts the claims by the experimenter about payo s, the player can never be sure that the other player does. The other player might misinterpret the experimenter s instructions, or make unwarranted inferences about payo s from statements made by the experimenter. Indeed, even if the other player does none of these things, the other player might believe that the original player has misunderstood, etc. As is now known, variations in these higher order beliefs can be used to support a wide range of behavior as Bayesian Nash equilibria (Yildiz and Weinstein [15]). Their theorems provide bounds on the sets of actions that can be supported on the equilibrium path of Bayesian games with higher order uncertainty. Though our example uses the similar reasoning, it goes further by showing that rejection of an o er is consistent with Perfect Bayesian equilibrium. Technically, the assumption that the payo s in the game are common knowledge is relaxed. That is, although the game structure (including payo s) is common knowledge, the responder does not know her payo (as well as the proposer s) when she decides to accept or reject an o er. An important distinction to be made here is between common knowledge and common certainty - which is achived in the example in the case of q = 1: The existence of PBNE in this case relies on the existence of type 2 (although he recieves probability zero ex-ante). Within the experimental economics literature, Plott and Zeiler [12] have recently demonstrated the role of possible misconception in a decision theoretic experiment. They studies the observed di erence between willingness to pay (WTP) and willingness to accept (WTA.) Many previous experiments (though not all) have attributed this di erence to preferences ( endowment e ect ). Plott and Zeiler showed that this conclusion was premature. They presented convincing evidence that when misconceptions are minimized the di erence disappears. What is the source of these misconceptions? How do they in uence the results of other experiments in which decision makers interact strategically with each other? Plott and Zeiler leave these questions unanswered, and this study attempts to give some preliminary answers. It shows that in an experimental test of interactive decision making the problem of misconception is exacerbated since even if one subject has no misconception (whatever that may mean) (s)he can never be absolutely sure that the other participant(s) have no misconceptions. 11

12 4 The Model More generally, let P be a nite collection of feasible o ers for the proposer. These are normalized to lie between 0 and 1. Suppose these o ers are indexed in such a way that 0 = p 1 < p 2 < : : : < p n = 1. The lowest demand p 1 is assumed to give all the surplus in the experiment to the responder. The highest demand p n is assumed to give all the surplus from the experiment to the proposer. The proposer receives a signal s 2 [s; s] S that a ects the payo of both players. The distribution of signals is given by F;and is assumed to be continuous with full support. Let 2 f0; 1g denote the action of the responder, = 1 meaning that she accepts the proposal. The payo s are treated asymmetrically. The payo to the proposer is given by u p (p; ; s) where s is his signal. Condition 1 Let p 0 > p. Then 1. u p (p 0 ; 1; s) > u p (p; 1; s) and u p (p 0 ; 0; s) u p (p; 0; s) for all s 2 S; 2. if u p (p; 1; s) > u p (p; 0; s) for some s 2 S then u p (p; 1; s) > u p (p 0 ; 0; s). The rst part of the condition states that independetly of whether an o er is accepted or rejected, the proposer is always better o with a higher demand (lower o er). The second part states that if the proposer receives signal s and prefers the o er p to be accepted rather than being rejected, then he must also prefer that the o er p be accepted to a rejection of a higher o er. The combination of the two parts imply that if u p (p; 1; s) > u p (p; 0; s) for some s 2 S then u p (p; 1; s) > u p (p 0 ; 0; s) > u p (p; 0; s) : The following condition is also used repeatedly. Condition 2 Let s 0 and s be such that u p (p; 0; s) u p (p; 1; s) < u p (p; 0; s 0 ) u p (p; 1; s 0 ) and suppose that for some p j > p k and q j < q k, q j u p (p j ; 1; s) + (1 q j )u p (p j ; 0; s) q k u p (p k ; 1; s) + (1 q k )u p (p k ; 0; s) Then the same inequality holds strictly for type s 0. This single-crossing condition states that if the proposer s utility loss as a result of being rejected at s 0 is lower than at s, and if the proposer s expected utility from a high demand (with a certain probability of acceptance) is at least as high as the expected utility of a lower demand (with higher 12

13 probability of acceptance) at s, then he strictly prefers the high demand at s 0 : The payo to the responder also depends on the proposer s signal. This payo function is given by u r (p; ; s): We assume that: Condition 3 The function u r (p; 0; s) u r (p; 1; s) is monotonically increasing in s and super modular in p and s. Furthermore u r (p; 0; s) u r (p; 1; s) 0 with strict inequality holding at each demand except p 1 = 0. u r (p; 0; s) u r (p; 1; s) < 0 for all p 2 P. The assumption is made on the responder s marginal utility of rejection, if she knows the proposer s signal s: It is assumed that the marginal utility of rejection is increasing in the signal, and supermodular in the proposer demand and the signal. The payo s for the responder are not too extreme: a responder who believes that the proposer s signal is s will reject all o ers except (amybe) the one that gives her the entire surplus from the experiment. By monotonicity and continuity a responder will accept the demand p 1 (which gives her all the surplus) no matter what her beliefs about the signal. This isn t true for other o ers. However, we assume: Condition 4 Z s s fu r (p; 0; s) u r (p; 1; s)g df (s) < 0 for every p < 1 This assumption implies that a responder using her prior beliefs would accept every o er except possibly the o er p = 1 (where the proposer gets the entire surplus from the experiment). This game has many equilibrium outcomes. The nature of these outcomes depends on the function u p (p; 0; s) u p (p; 1; s). We begin with the case where this function is strictly increasing. Then, the higher the signal the proposer receives, he believes that both he and the responder will gain more in the coninuation if the responder rejects the o er than if she accepts. Theorem 1 Suppose that u p (p; 0; s) u p (p; 1; s) is strictly increasing and that p < 1 is a demand such that u p (p; 1; s) > u p (p; 0; s). Then there is a Perfect Bayesian Equilibrium in which all proposers make the o er p and this is accepted. 13

14 Proof. By Condition 4, the responder wants to accept every o er except 1 given his or her prior beliefs. Let p 0 > p, and suppose that responders believe that such a proposal is made by a proposer whose type lies in the interval (s ; s] where s is chosen such that Z s s fu r (p 0 ; 0; s) u r (p 0 ; 1; s)g df (s) = 0 By Condition 3 and continuity of F, such an s always exists. Then a responder who receives the o er p 0 will be just indi erent about whether or not to accept it. Choose q 0 such that q 0 u p (p 0 ; 1; s) + (1 q 0 )u p (p 0 ; 0; s) = u p (p; 1; s) q 0 is chosen such that a proposer who received a signal s is indi erent between demanding p and p 0. By Condition 1 and the assumption that u p (p; 1; s) > u p (p; 0; s), u p (p 0 ; 1; s) > u p (p; 1; s) > u p (p 0 ; 0; s), so a unique q 0 satisfying this condition exists. Then if any proposer whose type is less than s strictly prefers to o er p 0, then by Condition 2, the highest type proposer must also. This contradiction proves that all proposer types at least weakly prefer the o er p to p 0. A similar argument is used to make downward deviations unpro table. It is possible to support equilibrium with multiple o ers in this model, but in a restricted way. Theorem 2 Suppose that u p (p; 0; s) u p (p; 1; s) is strictly increasing in s. Let p j > p k be a pair of demands and s the solution to Z s s fu r (p j ; 0; s) u r (p j ; 1; s)g df (s) = 0: If u p (p j ; 1; s) > u p (p j ; 0; s) and u p (p k ; 1; s ) > u p (p k ; 0; s ), then there is a perfect Bayesian Nash equilibrium in which proposers whose signal is in the interval [s ; s] demand p j, while proposers whose signal is below s demand p k. Responders accept p k for sure and reject p j with positive probability. Proof. By Conditions 3 and 4, there is a unique signal s satisfying the property R s fu s r (p j ; 0; s) u r (p j ; 1; s)g df (s) = 0. If the responder believes that the o er p j comes from a proposer whose type is in the interval [s ; s], 14

15 then she is indi erent between accepting and rejecting the o er. By Condition 4, the responder will strictly prefer to accept the o er p k if she believes that it is made by proposers whose type is in the interval [s; s ). Since u p (p k ; 1; s ) > u p (p k ; 0; s ), by Condition 1, u p (p j ; 1; s ) > u p (p k ; 1; s ) > u p (p j ; 0; s ). Thus there is a unqiue q j satisfying u p (p k ; 1; s ) = q j u p (p j ; 1; s ) + (1 q j )u p (p j ; 0; s ) Then if the responder accepts the demand p j with probability q j, Condition 2 says that since proposer type s weakly prefers the higher price lower probability o er p j, every higher proposer type strictly prefers to o er p j. In a similar fashion, for each o equilibrium o er p 0, nd x 0 such that the responder is indi erent between accepting and rejecting p 0 if she thinks that the o er was made by a proposer whose type is in the interval [x 0 ; s]. The acceptance probability now depends on the o er. If p 0 > p j, then repeating the arguments above and using the assumption that the highest proposer type prefers to have p j accepted, it is possible to choose the acceptance probability such that a proposer of the highest type is indi erent between the o ers p j and p 0. Similarly, if p j > p 0 > p k, then the acceptance probability is set so that a proposer of type x is indi erent between the proposal p j and the proposal p 0. If p 0 < p k, then the acceptance probability is chosen such that a proposer of type s (who prefers all o ers to be accepted) at least weakly prefers p k to p 0. This makes it possible to check deviations to o equilibrium prices. There are three cases, p 0 > p j, p j > p 0 > p k, and p 0 < p k. In the rst case, the highest type proposer s weakly prefers the o er p j to the o er p 0 by the choice of q 0. If there is a proposer type s < s who strictly prefers p 0 to p j, then the same must be true for a proposer of type s by Condition2. Thus the deviation must be unpro table for all proposer types. If p 0 < p k, then since the proposer with the lowest type is indi erent, all proposer types must at least weakly prefer p k by Condition 2. If p j > p 0 > p k, then all proposer types above x prefer p j by the single crossing condition. If there is a proposer type less than x who strictly prefers the proposal p 0 to p k, then type x must also have this strict preference if Condition 2 holds. This contradiction shows that types below x prefer p k. The conditions required for these two theorems are easy to check. For example, one might assume that every type of proposer does strictly better when every non-zero o er is accepted. In this case, every non-zero price 15

16 can be supported as a pooling equilibrium, and any pair of prices can be supported as a separating equilibrium. This is not the most interesting case. It seems more reasonable to study the possibility that proposers who have very high signals would strictly prefer that some very low demands that they might make would be rejected. One standard property of experimental results is that higher demands are less likely to be accepted. The model described so far is ill suited to explain the experimental results in a way that is consistent with a single equilibrium being played. The following theorem illustrates. Theorem 3 Suppose that u p (p; 0; s) u p (p; 1; s) is strictly increasing. Then all perfect Bayesian equilibrium outcomes in which higher demands are accepted with lower probability involve at most two demands that are accepted with strictly positive probability. Proof. Suppose more than two demands are accepted with positive probability in some perfect Bayesian equilibrium. Let p k and p j be the lowest and next lowest demands made with strictly positive probability in this equilibrium, and let p 0 be any other demand that is accepted with strictly positive probability. Let q k, q j and q 0 be the probabilities with which these o ers are accepted along the equilibrium path. By hypothesis, q k > q j > q 0. Let ~s be the supremum of the the set of types for the proposer that o er p k with strictly positive probability. If any lower type proposer weakly prefers to demand p j or p 0, then by Condition 2, a proposer of type ~s must strictly prefer the higher demand. Thus ~s < s, and for each type below ~s, q k u p (p k ; 1; s) + (1 q k )u p (p k ; 0; s) > q j u p (p j ; 1; s) + (1 q j )u p (p j ; 0; s) and similarly for p 0. The strict inequality is reversed for types s > ~s. Repeat the argument for the supremum of the set of types, say ^s, for the proposer who o er p j. Exactly as above, ^s < s. Then if p j is o ered on the equilibrium path, the responder must believe that the proposer s type is in the interval (~s; ^s]. If the proposer makes the o er p 0, the responder must believe the proposer s type is in the interval (^s; s]. Now since 0 < q j < q k, responders who see the o er p j must be just indi erent about whether or not they accept it. As a consequence, a responder who thought that the proposer s type was ^s, would strictly prefer to reject. Since u r (p; 0; s) u r (p; 1; s) is super modular by Condition 3, the higher demand coupled with beliefs that the higher demand is made by a proposer with 16

17 a higher signal means that the responder must strictly prefer to reject the demand p 0, a contradiction. The theorems above resemble the outcomes in the example described in the introduction. They don t provide enough richness to explain existing experimental results as a single equilibrium since experimental results involve many o ers and declining acceptance probabilities. At the same time, they are all independent of the particular distribution of types F for the proposer. So they do illustrate that the strong predictions associated with the subgame perfect equilibrium are not robust to mis-speci cations of the information structure. To provide an account of experimental results using a single equilibrium, we now modify the argument slightly and assume that u p (p; 0; s) u p (p; 1; s) is monotonically decreasing. A proposer with a high signal now believes that his own continuation payo is hurt by responder rejections. Responders continue to believe that their own continuation payo s are enhanced by rejection if the proposer has a high signal and the proposer knows this. Roughly speaking, the proposer now has an incentive to try to hide his information from the responder because their interests are not aligned. The following assumption is required to construct an equilibrium: Condition 5 For any o er p let s be such that u r (p 0 ; 1; s) < u r (p 0 ; 0; s) for each p 0 > p. Then u p (p; 1; s) > u p (p; 0; s) That is, if for every p 0 > p; a responder who knows the proposer s signal prefers to reject p 0, then the proposer prefers p to be accepted. In other words: let ^s be the solution to u r (p; 0; s) = u r (p; 1; s). This is the proposer s type such that if the responder believed the proposer had that type for sure, he would be just indi erent between accepting and rejecting the demand p. Roughly the Condition above states that provided a type s isn t too much lower than ^s, the proposer of type s would strictly prefer to have the demand p accepted. Condition 6 The payo to the proposer from a rejected o er is independent of p. That is, u p (p; 0; s) = u p (p 0 ; 0; s) for each p; p 0 2 P and for each s 2 S. Theorem 4 If u p (p; 0; s) u p (p; 1; s) is strictly decreasing and Conditions 1, 2, 3, 5 and 6 hold, there exists a Perfect Bayesian Equilibrium in which 17

18 multiple o ers are made along the equilibrium path. The o er p 1 is made with strictly positive probability and accepted for sure. The o er p n is made with strictly positive probability. All o ers other than p 1 are accepted with probability strictly less than one. If q k is the probability with which an o er p k is accepted on the equilibrium path and p j > p k then q j < q k. Proof. The proof is constructive. We construct a decreasing collection of m n demands 1 = m > m 1 > : : : > 1 = 0 that will be made with positive probability on the equilibrium path. Some demands in P may not be included in this set. Begin by setting m = p n. Select an interval [s; s m ) such that Z sm s fu r ( m ; 0; s) u r ( m ; 1; s)g df (s) = 0 The existence of such an interval is guaranteed by Condition 3. Now given the sequence f( m ; s m ); ( m 1 ; s m 1 ); : : : ; ( k+1 ; s k+1 )g; let k be the highest demand less than k+1 such that if the responder knew that the proposer s type is s k+1 then she would accept that demand. u r ( k ; 0; s k+1 ) u r ( k ; 1; s k+1 ) 0 (12) This highest price always exists because by Condition 3, a responder at least weakly prefers to accept the o er p 1 no matter what her beliefs about the proposer s. Now select s k such that Z sk s k+1 fu r ( k+1 ; 0; s) u r ( k+1 ; 1; s)g df (s) = 0 Again, the existence of such an interval for each strictly positive demand k follows from Condition 3. Proceed in this way until k = p 1. In this case set s k = s. Now re-index the demands and cuto s such that m is the number of demands in the sequence. Let s m+1 s (see Figure 3). We now construct the acceptance probabilities that support the equilibrium. Observe rst that by construction, for each o er k that is made with positive probability in this equilibrium, the set of types [s k+1 ; s k ) has the property that k is the highest demand less than k+1 that a responder would accept if he believed the proposer s type were exactly s k+1. Then by Condition 5, u p ( k ; 1; s k+1 ) > u p ( k ; 0; s k+1 ). The second part of Condition 1 then implies that u p ( k+1 ; 1; s k+1 ) > u p ( k+1 ; 0; s k+1 ). The rst part of Condition 1 then gives that u p ( k ; 1; s k+1 ) > u p ( k+1 ; 0; s k+1 ). 18

19 Figure 3: Construction of demands in Equilibrium 1 = π m π m 1 π k π k 1 π 2 π = 0 1 s s s s k s 2 s = s m m 1s m 2 = s k Begin with the o er 1 which is accepted for sure, i.e., q 1 = 1. Now for each k > 0, select q k+1 such that q k+1 u p ( k+1 ; 1; s k+1 )+(1 q k+1 )u p ( k+1 ; 0; s k+1 ) = q k u p ( k ; 1; s k+1 )+(1 q k )u p ( k ; 0; s k+1 ) (13) For k = 1, the existence of a q 2 satisfying (13) is guaranteed by the inequality u p ( 2 ; 1; s 2 ) > u p ( 1 ; 1; s 2 ) > u p ( 2 ; 0; s 2 ) just proved. For each subsequent k, the existence of a solution to (13) is guaranteed by the inequality u p ( k+1 ; 1; s k+1 ) > u p ( k ; 1; s k+1 ) > u p ( k+1 ; 0; s k+1 ) = u p ( k ; 0; s k+1 ), where the equality follows from Condition 6. From this construction, a proposer whose signal is s k+1 is just indi erent between demanding k and k+1. By the single crossing Condition 2, proposers whose signals are below s k+1 strictly prefer the demand k+1 to the demand k. On the other hand, if a proposer whose signal exceeds s k+1 strictly prefers to make the demand k+1, then a proposer whose signal is s k+1 must also. Applying this argument at each value of k, it follows that the best equilibrium path o er for a proposer whose type is in the interval (s k ; s k+1 ] is the o er k. To deal with o equilibrium o ers, observe that every proposer type has an equilibrium path o er that makes them better o than they are when an o er is rejected. If p 0 is an o er that doesn t appear in the equilibrium sequence 1 ; : : : ; m, then by construction, there is an s k in the equilibrium sequence s 1 ; : : : ; s m such that u r (p 0 ; 0; s k ) u r (p 0 ; 1; s k ) > 0 19

20 If that is the case, then assume that whenever the o er p 0 is made o the equilibrium path, then the responder believes that it must have been made by a proposer whose type is s k, so that the o er is rejected for sure. Then each proposer type can improve upon this payo by selecting some equilibrium path o er. One property of particular interest is the expected payo to proposers associated with di erent o ers. In a collection of experimental results, for example, one might check empirically how often an o er p k is accepted, then compute the product of p k and the observed acceptance probability in order to compute an expected payo. To check this here specialize the payo function for the proposer as follows and u p (p; 1; s) = p u p (p; 0; s) = (s) where, in this application, (s) is a decreasing function. This formulation satis es Condition 6, and can be made to satisfy Condition 5 by appropriate choice of. From (13) in the proof of Theorem 4 or q k+1 p k+1 + (1 q k+1 )(s k+1 ) = q k p k + (1 q k )(s k+1 ) q k+1 p k+1 q k p k = (q k+1 q k )(s k+1 ) By Theorem 4, q k+1 < q k. The proposers who make the low demands are those whose signals are very high. These are the proposers who think that a responder rejection will lower their continuation payo, i.e., proposers who would prefer that the responder accept demands very close to 0. The condition u p (p 1 ; 0; s) u p (p 1 ; 1; s) < 0 in this special case reduces to (s) < 0. Thus for very low values of k, we conclude that q k+1 p k+1 > q k p k. At some point (s k ) becomes positive, and thereafter q k+1 p k+1 < q k p k. The implication is that the apparent gains from the experiment are single peaked in this equilibrium. References [1] Aumann, Robert J. (1997): Rationality and Bounded Rationality, Games and Economic Behavior, 21,

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22 [13] Rabin, Matthew (1993): Endogeneous Preferences in Games, American Economic Review, 83, [14] Roth, Alvin E. (1995): Bargaining Experiments, in: The Handbook of Experimental Economics, Ed. John H. Kagel and Alvin E. Roth. Princeton, NJ: Princeton University Press. [15] Yildiz, Muhamet and Jonathan Weinstein (2006): Finite Order Implications of Any Equilibrium (mimeo). 22