The Local Best Response Criterion: An Epistemic Approach to Equilibrium Refinement. Herbert Gintis

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1 The Local Best Response Criterion: An Epistemic Approach to Equilirium Refinement Herert Gintis May 2, 2008 Astract The standard refinement criteria for extensive form games, including sugame perfect, perfect, perfect Bayesian, sequential, and proper are too strong, ecause they reject important classes of unojectionale Nash equiliria, and too weak, ecause they accept many ojectionale Nash equiliria. This paper develops a new refinement criterion, ased on epistemic game theory, that captures the concept of a Nash equilirium that is plausile when players are rational. I call this the local est response (LBR) criterion. This criterion is conceptually simpler than the standard refinement criteria, ecause it does not depend on out-of-equilirium, counterfactual, or passage to the limit arguments. The LBR is also informationally richer, ecause it clarifies the epistemic conditions that render a Nash equilirium acceptale. 1 Introduction A Nash equilirium refinement of an extensive form game is a criterion that applies to all Nash equiliria that are deem plausile, ut fails to apply to other Nash equiliria that are deemed implausile, ased on our informal understanding of how rational individuals might play the game. A voluminous literature has developed in search of an acceptale equilirium refinement criterion. A numer of useful criteria have een proposed, including sugame perfect, perfect, perfect Bayesian, sequential, and proper equilirium (Harsanyi 1967, Myerson 1978, Selten 1980, Kreps and Wilson 1982, Kohlerg and Mertens 1986), which introduce player error, model eliefs off the path of play, and investigate the limiting Santa Fe Institute and Central European University. I would like to thank Roert Aumann, Jeffrey Ely, Larry Samuelson, and Fernando Vega-Redondo for their help, and the European Science Foundation for financial support. 1

2 ehavior of pertured systems as deviations from equilirium play go to zero. 1 I present a new refinement criterion that is more accurate is reflecting our intuitions concerning the implications of rationality, and reveals the implicit criteria we use to judge a Nash equilirium as plausile or implausile. Moreover, the criterion does not depend on counterfactual or disequilirium eliefs, tremles, or limits of neary games. I call this the local est response (LBR) criterion. The LBR criterion asserts that the choices of the players in an extensive form game must e consistent with the assumption that for each information set that is reached in some Nash equilirium, player i choosing at forms a conjecture of other players actions that consists of est responses in some Nash equilirium that reaches, and chooses an action at that is a est response to such a conjecture. If player i choosing at has several choices that lead to different information sets of the other players (we call such choices decisive), i chooses among those with the highest payoff for i. The LBR equiliria of the game are the Nash equiliria in which every move y every player conforms to this criterion. Most readers will find this informal description sufficient to egin to apply the LBR criterion to specific cases, and might want to skip directly to section 2. Formally, the LBR criterion is defined using epistemic game theory (Aumann and Brandenurger 1995). However, standard epistemic game theory applies only to normal form games, so I will define a distinct structure for extensive form games of perfect recall. Its relationship to the normal form epistemic game structure is layed out in the Appendix. We assume a finite extensive form game G of perfect recall, with players i D 1; : : : ; n and a finite pure strategy set S i for each player i, so S D S 1 : : : S n is the set of pure strategy profiles for G, with payoffs i WS!R. Let S i e the set of pure strategy profiles of players other than i, and let S i e the set of proaility distriutions over S i. Let N i e the information sets where player i chooses. For player i and 2 N i, we call 2 S i a conjecture of i at. If is a conjecture at 2 N i and j i, we write j for the marginal distriution of on j 2 N j, and we call j i s conjecture at of j s ehavioral strategy at j. We consider G an extensive form epistemic game with state space, which consists of all states of the form.; a ; /, where 2 N i for some i, a 2 A, the set of choices availale to i at, and 2 S i is a conjecture that reaches with positive proaility for some s i 2 S i. We say an action a 2 A is a est response to if there is a strategy s i 2 S i such that.s i ; / reaches, s i is a est response to, and s i./ D a. 1 Distinct categories of equilirium refinement for normal form games, not addressed in this paper, are focal point (Schelling 1960; Binmore and Samuelson 2006), and risk-dominance (Harsanyi and Selten 1988) criteria. The perfection and sequential criteria are virtually coextensive (Blume and Zame 1994), and extend the sugame perfection criterion. 2

3 Let N e the set of Nash equiliria that reach information set with positive proaility. We say a choice etween two actions at information set is decisive if they actions lead to distinct information sets for players choosing after. We say a state.; a ; / for 2 N i is admissile if (a) a is a est response to ; () for all j i, 2 N j, is the marginal on A of some 2 N ; and (c) if a choice etween a 1 ; a 2 2 A is decisive, and one Nash equilirium using a 2 has a higher payoff than all Nash equiliria using a 1, then a a 1, and the other players conjecture this. We say a Nash equilirium D. 1 ; : : : ; n / is admissile at information set 2 N i if either does not reach, or reaches with positive proaility and there is some admissile state.; a ; / such that.a / > 0. We say a Nash equilirium is and LBR equilirium if it is admissile at every information set. 2 Sugame Perfection a,5 L 1 l R r 2 0,0 2,1 Figure 1: A Simple Game with (a D 1) and without (a D 3) an Incredile Threat In Figure 1, first suppose a D 3. Using the informal definition, we see that if gets to move, he chooses r, so conjectures this. Her est response is L. All Nash equiliria.l; 2 /, where 2 is any ehavioral strategy for, are LBR equiliria ecause they conform to s choice and never get to s choice set. Formally, has four types of states: D f. 1 ; L; 2 /;. 1 ; R; 2 /;. 2 ; l; 1 /;. 2 ; r; 1 /g; where i 2 S i, and 1.R/ > 0. The states. 1 ; L; 2 / are admissile ecause all 2 are part of a Nash equilirium, and L is a est response to all 2. The states. 1 ; R; 2 / are not admissile ecause R is not a est response to any 2. The states. 2 ; l; 1 / and. 2 ; r; 1 / are not admissile ecause there is no conjecture 1 of for that is part of a Nash equilirium and satisfies 1.R/ > 0. All Nash equiliria.l; 2 / are admissile at state. 1 ; L; 2 /, and do not reach states at 2. Hence all Nash equiliria are LBR equiliria. Note that only one of these LBR equiliria is sugame perfect. Nevertheless, s ehavior is rational in all LBR equiliria. 3

4 Now, suppose a D 1. The Nash equiliria are now.r; r/ and.l; 2 /, where 2.r/ 1=2. Informally, still conjectures r for, ut now R is the only est response, so the.l; 2 / are not LBR equiliria. must conjecture R for, ecause this is the only move she makes in a Nash equilirium that reaches s information set. His est response is r. Thus.R; r/ is the unique LBR equilirium. Formally, the state space has the same four types of states. Because the only Nash equilirium to reach 2 has play r, the only admissile conjecture for at 1 is r, to which her est response is R. Thus,. 1 ; R; r/ is the only admissile state at 1. Because the only Nash equilirium that reaches 2 has play R, the only admissile state at 2 is. 2 ; r; R/. Thus, only.r; r/ is a LBR equilirium. Note that this argument does not require any out-of-equilirium elief or error analysis. Sugame perfection is assured y epistemic considerations alone; i.e., a Nash equilirium in which plays l with positive proaility is an incredile threat. One might argue that sugame perfection can e defended ecause there is, in fact, always a small proaility that will make a mistake and play R in the a D 3 case. However, why single out this possiility? There are many possile imperfections that are ignored in the passage from a real-world strategic interaction to the game depicted in Figure 1, and they may work in different directions. Singling out the possiility of an error is thus aritrary. For instance, suppose l is the default choice, ut y spending B on a decision-making procedure, can choose r when it is to his advantage to do so. The new decision tree is depicted in Figure 2. n 3,5 L d R n 3,5 B 0,0 l d r 0, B 0,1 B Figure 2: Addition Infinitesimal Decision Costs for In this new situation, does not know s choice, and if he chooses not to find out (n), the payoffs are as efore, with choosing l y default. But, if pays inspection cost B, he oserves s choice and shifts to the nondefault r when she accidentally plays R. If plays R with proaility A, it is easy to show that will choose to inspect only if A B. 4

5 The LBR criterion is thus the correct refinement criterion for this game. Standard refinements fail y rejecting non-sugame perfect equiliria whether or not there is any rational reason to do so (e.g., that the equilirium involves an incredile threat). The LBR gets to the heart of the matter, which is expressed y the argument that when there is an incredile threat, if gets to choose, he will choose the strategy that gives him a higher payoff, and knows this. Thus maximizes y choosing R, not L. If there is no incredile threat, can choose as he pleases, or simply refuse to choose, permitting the outcome to e a yproduct of other processes in which he is engaged. In the remainder of this paper, I compare the LBR criterion with traditional refinement criteria in a variety of typical game contexts. I address where and how the LBR differs from traditional refinements, and which criterion etter conforms to our intuition of rational play. I include some cases where oth perform equally well, for illustrative purposes. Mainly, however, I treat cases where the traditional criteria perform poorly and the LBR criterion performs well. I am aware of no cases where the traditional criteria perform etter than the LBR criterion. Indeed, I know of no cases where the LBR criterion, possily strengthened y other epistemic criteria, does not perform well, assuming our intuition is that a Nash equilirium will e played. My choice of examples follows Vega-Redondo (2003), which includes an extremely complete treatment of equilirium refinement theory. I have tested the LBR criterion for all of Vega-Redondo s examples, and many more, ut present only a few of the more informative examples here. The LBR shares with the traditional refinement criteria the presumption that a Nash equilirium will e played, and indeed, in every example in the paper I would expect rational players to choose an LBR equilirium (although this expectation is not acked y empirical evidence). In many games, however, such as the Rosenthal s centipede game (Rosenthal 1981), Basu s traveler s dilemma (Basu 1994), and Carlsson and van Damme s gloal games (Carlsson and van Damme 1993), oth our intuition and the ehavioral game theoretic evidence (Gintis 2009) violate the presumption that rational agents play Nash equiliria. The LBR criterion does not apply to these games. Many games have multiple LBR equiliria, only a strict suset of which would e played y rational players. Often epistemic criteria supplementary to the LBR criterion single out this suset. In this paper, I use the Principle of Insufficient Reason and what I call the Principle of Honest Communication to this end. 5

6 C B A 1,0,0 2 0,1,0 a 0,0,0 V Carole U 3 a V U 2,2,2 0,0,0 0,0,0 0,0,1 Figure 3: Only the equilirium BV is acceptale, ut there is a connected set of Nash equiliria, including the pure strategy AU, all memers of which are perfect Bayesian. 3 The LBR Rejects Unacceptale Perfect Bayesian Equiliria Figure 3 (Vega-Redondo 2003, p. 128) depicts a game in which all Nash equiliria are sugame perfect and perfect Bayesian, ut only one is plausile, and this is the only equilirium that satisfies the LBR. The game has two sets of equiliria. The first, A, chooses A with proaility 1 and B./ C.V / 1=2, which includes the pure strategy equilirium AU, where A, B and C are mixed strategies of,, and Carole, respectively. The second is the strict Nash equilirium BV. Only the latter is an acceptale equilirium in this case. Indeed, while all equiliria are sugame perfect ecause there are no proper sugames, and AU is perfect Bayesian if Carole elieves chose a with proaility at least 2/3, it is not sequential, ecause if actually gets to move, chooses with proaility 1 ecause Carole chooses V with positive proaility in the pertured game. The informal argument egins y noting that the only moves and Carole use in a Nash equilirium where they get to choose are and V, respectively. Thus, conjectures this, to which her est response is B. Therefore, only BV is an LBR equilirium. Formally, the state space for this prolem has eight types of states: Let A, B and C e the information sets for,, and Carole, respectively. The eight types are. A ; A; B ; C /,. A ; B; B ; C /, and. A ; C; B ; C /;. B ; a; A ; C /, and. B ; ; A ; C /, where A.A/ < 1; and. C ; U; A ; B /,. C ; V; A ; B /, where A.B/ > 0. The only Nash equilirium reaching C has Carole choose V, so oth and must conjecture that Carole play V in any admissile state. Similarly, the 6

7 ! only Nash equilirium reaching B has play, so the only admissile states have and Carole conjecture that play. The only admissile states at C are thus. C ; V; A ; / where A.B/ > 0. The only admissile state at A is then. A ; B; ; V /, since B is the only est response to.; V /. Thus the only admissile Nash equilirium is BV, which thus the only LBR equilirium. 4 LBR Picks Out the Sequential Equilirium Figure 4 depicts another example where the LBR criterion rules out unacceptale equiliria that pass the sugame perfection and perfect Bayesian criteria, ut sequentiality and LBR are equally successful in this case. In addition to the Nash equilirium Ba, there is a set A of equiliria in which plays A with proaility 1 and plays with proaility 2=3. The set A are not sequential, ut Ba is sequential. The LBR criterion specifies that play a if he gets to choose, and thus that conjecture that play a. Hence, knows that she earns 2 y playing B, which is greater than the payoff to playing A. thus chooses B, and Ba is the only LBR equilirium. Formally, using the aove notation, there are five sets of states:. A ; A; B /,. A ; B; B /, and. A ; C; B /; and. B ; a; A /, and. B ; ; A /, where A.A/ < 1. In the Nash equiliria that reach s information set, chooses a. Thus, only. B ; a; A / with A.A/ < 1 are admissile at s information set. All states at s information set must conjecture a and e a est response to a. Thus, the only admissile state at s information set is. A ; B; a/, so the only Nash equilirium admissile at s information set is Ba, which is thus the only LBR equilirium. A 0,2 C B a a 2,0 1,1 1, 1 2,1 Figure 4: A Generalization of Sugame Perfection 7

8 5 Selten s Horse: Sequentiality and LBR Disagree Selten s Horse is depicted in Figure 5. This game shows that sequentiality is neither strictly stronger than, nor strictly weaker than the LBR criterion, since the two criteria pick out distinct equiliria in this case. There is a connected component M of Nash equiliria given y M D f.a; a; p C.1 p //j0 p 1=3g; where p is the proaility that Carole chooses, all of which of course have the same payoff (3,3,0). There is also a connected component N of Nash equiliria given y N D f.d; p a a C.1 p a /d; /j1=2 p a 1g; where p a is the proaility that chooses a, all of which have the same payoff (4,4,4). $ A % a (3,3,0) 3l D Carole " # (4,4,4) (1,1,1) (5,5,0) (2,2,2) Figure 5: Selten s Horse The M equiliria are sequential, ut the N equiliria are not even perfect Bayesian, since if were given a choice, his est response would e d, not a. Thus the standard refinement criteria select the M equiliria as acceptale. By contrast, only the pure strategy Da in the N component satisfies the LBR criterion, ecause the only Nash equilirium in which Carole gets to choose is in the set N, where she plays with proaility 1. LBR also specifies that choose a, ecause it is the only move that is part of a Nash equilirium that reaches his choice node. LBR stipulates therefore that conjecture that play a and Carole play, so her est response is D. This generates the equilirium Da. Selten s horse is thus a case where the LBR criterion chooses an equilirium that is acceptale even though it is not even perfect Bayesian, while the standard refinement criteria choose an unacceptale equilirium in M. The M equiliria are unacceptale ecause if did get to choose, he would conjecture that Carole play, ecause that is her only move in a Nash equilirium where she gets to d 3r 8

9 / ( '. & 3 -, + 4 move, and hence will violate the LBR condition of choosing an action that is part of a Nash equilirium that reaches his choice node. However, if he is rational, he will violate the LBR stricture and play d, leading to the payoff (5,5,0). If conjectures that will play this way, she will play a, and the outcome will e the non-nash equilirium Aa. Of course, Carole is capale of following this train of thought, and she might conjecture that and will play non-nash strategies, in which case, she could e etter off playing the non-nash herself. But of course, oth and might realize that Carole might reason in this manner. And so on. In short, we have here a case where the sequential equiliria are all implausile, ut there are non-nash choices that are as plausile as the Nash equilirium singled out y the LBR. 6 The Spence Signaling Model: LBR Correctly Rejects a Sequential Pooling Equilirium 2,10 8,5 14,5 20,10 U S U Y L N pd1=3 2 l N 2 r pd2=3 S Y H N ) * U S U S 12,10 18,5 14,5 20,10 Figure 6: The Unacceptale Pooling Equilirium is Rejected y the LBR Figure 6 represents the famous Spence signaling model (Spence 1973). is either a low quality worker (L) with proaility p D 1=3 or a high quality worker (H) with proaility 2/3. Only knows her own quality. is an employer who has two types of jos to offer, one for an unskilled worker (U ) and the other for a skilled worker (S). If matches the quality of a hire with the skill of the jo, his profit is 10; otherwise, his profit is 5. can invest in education (Y ) or not (N ). Education does not enhance s skill, ut if is low quality, it costs her 10 to e educated, while if she is high quality, it costs her nothing. Education is thus purely a signal, possily indicating s type. Finally, the skilled jo pays 6 more than the unskilled jo, the uneducated high quality worker earns 2 more than 9

10 the uneducated low quality worker in the unskilled jo, and the ase pay for a low quality, uneducated worker in an unskilled jo is 12. This gives the payoffs listed in Figure 6. This model has a separating equilirium in which gets educated only if she is high quality, and assigns educated workers to skilled jos and uneducated workers to unskilled jos. In this equilirium, s payoff is 10 and s payoff is prior to finding out whether she is of low or high quality. Low quality workers earn 12 and high quality workers earn 20. There is also a pooling equilirium in which never gets an education and assign all workers to skilled jos. Indeed, any comination of strategies S S (assign all workers to skilled jos) and S U (assign uneducated workers to skilled jos and educated workers to unskilled jos) is a est response for in the pooling equilirium. In this equilirium, earns 8.33 and earns However, uneducated workers earn 18 in this equilirium and skilled workers earn 20. Both sets of equiliria are sequential. For the pooling equilirium, consider a completely mixed strategy profile n in which chooses SS with proaility 1 1=n. For large n, s est response is not to e educated, so the approximate proaility of eing at the top node a t of s right-hand information set r is approximately 1/3. In the limit, as n! 1, the proaility distriution over t computed y Bayesian updating approaches (1/3,2/3). Whatever the limiting distriution over the left-hand information set l (note that we can always assure that such a limiting distriution exists), we get a consistent assessment in which S S is s est response. Hence the pooling equilirium is sequential. For the separating equilirium, suppose chooses a completely mixed strategy n with the proaility of US (allocate uneducated workers to unskilled jos and educated workers to skilled jos) equal to 1 1=n. s est response is N Y (only high quality gets educated), so Bayesian updating calculates proailities at r as placing almost all weight on the top node, and at s left-hand information set l, almost all weight is placed on the ottom node. In this limit, we have a consistent assessment in which elieves that only high quality workers get educated, and the separating equilirium is s est response given this elief. There are eight types of states:.h; Y; 2 l; 2 r/,.h; N; 2 l; 2 r/,.l; Y; 2 l; 2 r/, and.l; N; 2 l; 2 r/, where 2 l and 2 r are s conjectures for at l 2 and r 2, respectively; and s states.2 l ; S; 1 L; 1 H/,.l 2 ; U; 1 L; 1 H/,.r 2 ; S; 1 L; 1 H/, and.2 r ; U; 1 L; 1 H/, where 1 H and 1 L are s conjecture for at nodes L and H, respectively. Both Nash equiliria specify that choose S at 2 l, so must conjecture this, and that choose N at L, so must conjecture this, so the only admissile states are.h; Y; S; 2 r/,.h; N; S; S/,.L; N; S; 2 r/,.2 l ; S; Y; 1 H/,.r 2 ; S; N; 1 H/ and.r 2 ; U; N; 1 H/. 10

11 = 8 9 < ; 7 6 > C D A E 5 The equilirium.nn; SS/ is admissile at.h; N; S; S/ and.l; N; S; 2 r/, and.2 l ; S; Y; 1 H/ since it never reaches this information set, and at.r 2 ; S; N; 1 H/. Hence.NN; SS/ is admissile. The Nash equilirium.n Y; US/ is admissile at states.h; Y; S; u r/ and.l; N; S; 2 r/, and.l 2 ; S; Y; 1 H/ and.r 2 ; U; Y; N /. Hence.N Y; NS/ is admissile. Neither player has a decisive choice among Nash equiliria, so oth the separating and pooling equiliria are LBR equiliria. 7 Sequentiality is Sensitive, and the LBR Insensitive, to Irrelevant Node Additions T : B 2 M 2,2 I B 2 N M 1 T 2,2 L R L R L R L R 0,0 1,1 3,3 x,0 0,0 1,1 3,3 x,0 Figure 7: The only acceptale equilirium for 1 < x 2 is ML, which is sequential and satisfies the LBR criterion. However, the unacceptale equilirium TR is sequential in the left panel, ut not the right, when an irrelevant node is added. Kohlerg and Mertens (1986) use Figure 7 with 1 < x 2 to show that an irrelevant change in the game tree can alter the set of sequential equiliria. We use this game to show that the LBR criterion chooses the acceptale equilirium in oth panes, while the sequential criterion only does so if we add an irrelevant node, as in the right panel of Figure 7. The acceptale equilirium in this case is ML, which is sequential. However, TR is also sequential in the left panel. To see this, let f A.T /; A.B/; A.M/g D f1 10; ; 9g and f B.L/; B.R/g D f; 1 g. These converge to T and R, respectively, and the conditional proaility of eing at the left node of 2 is 0.9, so s mixed strategy is distance from a est response. In the right panel, however, M strictly dominates B for, so TR is no longer sequential. To apply the LBR criterion, note that the only Nash equilirium allowing to choose is ML, which gives payoff 3, as opposed to the payoff 2 from choosing T. Therefore, conjecturing this, maximizes her payoff y allowing to choose; i.e., ML is the only LBR equilirium. 11

12 M H I L K G J F 8 The LBR Criterion Rejects Improper Sequential Equiliria B F A a 1,1 a 2,1 1,0 1,0 2,1 Figure 8: The LBR Criterion Selects Out Proper Equiliria from a Set of Sequential Equiliria Consider the game in Figure 8. There are two Nash equiliria, F and Aa. These are also sequential equiliria, since if intends F, ut if the proaility that B is chosen y mistake is much greater than the proaility that A is chosen, then is a est response. Conversely, if the proaility of choosing A y mistake is much greater than the proaility that B is chosen y mistake, then a is a est response. Since B is the more costly mistake for, the proper equilirium concept assumes it occurs very infrequently compared to the A mistake, will play a when he gets to move, so should chose A. Therefore, Aa is the only proper equilirium, according to the Myerson (1978) criterion. To see that Aa is the only LBR equilirium of the game, note that the only Nash equilirium that reaches s information set is Aa. The LBR criterion therefore stipulates that choose a at his information set, and that conjecture this. With this conjecture, A has a higher payoff than F for, who hence plays A. The reader will note how simple and clear this justification is of Aa y comparison with the properness criterion, which requires an order of magnitude assumption concerning the rate at which tremles go to zero. 9 The LBR Criterion Includes Second-order Forward Induction Figure 9 depicts the famous Money Burning game analyzed y Ben-Porath and Dekel (1992), illustrating second order forward induction. By urning money, (to the amount of 1) can ensure a payoff of 2, so if she does not urn money, she must expect a payoff of greater than 2. This induces her attle of the sexes partner to act favoraly to, giving her a payoff of 3. The set of Nash equiliria can e descried as follows. First, there is YB in which plays Y (don t urn money, choose favoring ), and chooses 12

13 N R Q O P S T Y \ W X [ Z V U X Y 2l 2r s S B 1l s s S 1r B s 2,2 1,0 0,1 1,3 3,2 0,0 1,1 2,3 Figure 9: Money Burning any mixture of BB (play B no matter what did) and SB (play S if played X and play B if played Y ). This set represents the Pareto-optimal payoffs favoring. The second is YS, in which plays Ys (don t urn money, choose favoring ), and chooses any mixture of BS (play B against X and S against Y ) and SS (play S no matter what) This set represents the Pareto-optimal payoffs favoring. The third is X S, in which plays X s and chooses any mixed strategy comination of SB and S S in which the former is played with proaility 1=2. This is the money-urning Pareto-inferior attle of the sexes equilirium favoring. The fourth set is the set M in which plays Y and then.1=4/ C.3=4/s and chooses any mixed strategy that leads to the ehavioral strategy.3=4/b C.1=4/S. Second order forward induction selects out YS. All equiliria are sequential, and result from to different orders in eliminating weakly dominated strategies. The only Nash equilirium where chooses at 2l involves choosing S there. Thus, conjectures this, and knows that her est response compatile with 2l is Xs, which gives her payoff 2. There are three sets of Nash equiliria where chooses Y. In one, she chooses Y and chooses B, giving her payoff 2. In the second, she chooses Ys and chooses S, giving her payoff 3. In the third, and play the attle of the sexes mixed strategy equilirium, with payoff of 3/2 for. Each of these is compatile with a conjecture of that plays a Nash strategy. Hence, her highest payoff is with Ys. Because Y is decisive and includes a Nash equilirium, where plays Ys, with higher payoff for than any Nash equilirium using X, when moves at 2r, the LBR stipulates that he conjectures this, and hence his est response is S. Thus, must conjecture that plays S when she plays Y, to which Y s is the only est response. Thus, Y S s is the only LBR equilirium. 13

14 d f c ` _ e a 10 Beer and Quiche without the Intuitive Criterion 1/10 Nature g 9/10 B Q B Q D N D N D N D N h i k j ] ^ 0,1 2,0 1,1 3,0 1, 1 3,0 0, 1 2,0 Figure 10: The Beer and Quiche Game This is perhaps the most famous example arguing the importance of eliefs in equilirium refinement, the Cho and Kreps (1987) Real Men Don t Eat Quiche game. It also illustrates the well-known Intuitive Criterion, which is complementary to sequentiality. However, the LBR criterion singles out the acceptale equilirium without recourse to an additional criterion. This game has two sets of (pooling) Nash equiliria. The first is Q, in which plays QQ (Q if whimp, Q if strong) and uses any mixture of DN (play D against B and play N against Q) and NN (play N no matter what) that places at least weight 1/2 on DN. The payoff to is 21/10. The second is B, in which plays BB (B if whimp, B if strong) and uses any mixture of ND (play N against B and play D against Q) and NN that places at least weight 1/2 on ND. The payoff to is 29/10. The Cho and Kreps (1987) Intuitive Criterion notes that y playing QQ, earns 21/10, while y playing BB, she earns 29/10. Therefore, a rational will choose BB. To find the LBR equilirium, we note that the only admissile Nash equiliria are BB and QQ, and BB is decisive with respect to QQ and has higher payoff for. The only admissile states at s choice set, given BB, include s conjectures of B for, to which his est response is N. Thus, the LBR equiliria includ BB for and ND with proaility 1=2 and NN otherwise, for. 14

15 p w t { o s x n v m y r z l u q 11 The LBR Criterion Selects Acceptale Equiliria in Cases where Sequentiality Fails D 1=2 Nature D 1=2 L R l Carole 2,2,0 U D U D r 0,1,0 3,0,2 0,0,3 0,2,2 0,0,0 Figure 11: An Example where Perfection is Not Sensile, ut the LBR criterion is The game in Figure 11, taken from McLennan (1985), has a strict Nash equilirium at RlU and an interval of sequential equiliria L of the form Lr and D with proaility at least 3/4. The L equiliria are unintuitive for many reasons. Perhaps the simplest is forward induction. If Carole gets to move, either played R or played l. In the former case, must have expected Carole to play U, so that her payoff would e 3 instead of 2. If moved, he must have moved l, in which case he also must expect Carole to move U, so that his payoff would e 2 instead of 1. Thus, U is the only plausile move for Carole. For the LBR criterion, note that L is ruled out, since the only Nash equilirium reaching Carole s information set uses U with proaility one. 12 The Principle of Insufficient Reason A 0,0 C B a a } ~ 2, 2 1,1 1,1 2, 2 Figure 12: Using the Principle of Insufficient Reason 15

16 ƒ Figure 12, due to Jeffery Ely (private communication) depicts a game whose only acceptale Nash equiliria form a connected component A in which plays A and plays a with proaility B.a/ 2 Œ1=3; 2=3, guaranteeing a payoff of 0. There are two additional equiliria C a and B, and there are admissile states at s information set that are consistent with each of these equiliria. At s choice information set, all conjectures A for with A.A/ < 1 are admissile, so all states are admissile at this information set. Thus, all Nash equiliria are LBR equiliria. However, y the Principle of Insufficient Reason and the symmetry of the prolem, if gets to move, each of his moves is equally likely. Thus, it is only reasonale for to assume B.a/ D B./ D 1=2, to which A is the only est response. 13 The Principle of Honest Communication L R l r l r 4,4 0,0 0,0 1,1 Figure 13: Using the Principle of Honest Communication It is easy to check that the coordination game in Figure 13 has three Nash equiliria, each of which satisfies all traditional refinement criteria, and all are LBR equiliria as well. Clearly, however, only Ll is a plausile equilirium for rational players. One justification of this equilirium is that if we add a preliminary round of communication, and each player communicates a promise to make a particular move, and if each player elieves the Principle of Honest Communication, according to which players keep their promises unless they can enefit y violating these promises and eing elieved, then each will promise to play L or l, and they will keep their promises. 14 Discussion The LBR criterion is a simplification and an improvement of refinement criteria, meant to render all other standard refinement criteria otiose. However, the LBR is not an impediment to standard game-theoretic reasoning. If a game has a unique Nash equilirium, this will e oth a sugame perfect and an LBR equilirium. 16

17 If a game has a non-sugame perfect equilirium with an incredile threat, this equilirium cannot e LBR ecause in an LBR equilirium, players never conjecture incredile threats. In a repeated game, it is plausile to argue, as does the LBR criterion, that players maximize at every information set that they reach. We do not need the extra aggage of the sequential equilirium concept to justify this assumption. The LBR supplies a rigorous and insightful equilirium refinement criterion. Moreover, it clarifies the meaning of an intuitively acceptale equilirium as eing one in which players conjecture that other players use actions that are part of Nash equiliria, and choose actions themselves that maximize their payoffs suject to such conjectures. In a roader sense, this paper suggests the value of epistemic game theory in clarifying and resolving prolems in classical game theory, in the spirit of Aumann and Dreze (2008). The attempt to extend the principles of Bayesian rationality directly to strategic interaction, which occupied game theorists in the 1980 s and early 1990 s (Kohlerg and Mertens 1986, Tan and da Costa Werlang 1988, Harsanyi and Selten 1988) failed for the simple reason that the epistemic assumptions of the Savage and related axiom systems are sujective, and hence cannot model intersujectivity. The possile worlds framework corrects this weakness y assuming a common world that is imperfectly known y different agents. 15 Appendix A normal form epistemic game G consists of a normal form game with players i D 1; : : : ; n and a finite pure strategy set S i for each player i, so S D S 1 : : :S n is the set of pure strategy profiles for G, with payoffs i WS!R. In addition, G includes a finite state space, elements of which are called states and susets of which are called events. On the state space, for each player i, knowledge structure.p i ; K i ; P i / is defined, and a sujective prior p i.i!/ over the strategies S i for the other players that is a function of the state! 2. The knowledge structure.p i ; K i ; P i / has the following properties. We define P i to e a partition of. For state! 2, P i! is the unique event P 2 P i such that! 2 P. We interpret P i! as the set of states that i considers possile when the actual state is!. Because player i cannot distinguish among states in P i!, p i.i!/ D p i.i! 0 / for all! 0 2 P i!; i.e., p i.i!/ is measurale with respect to P i. We can thus write i s sujective prior as p i.i P i!/. If E is an event, we define K i E D f! 2 jp i! Eg, and we say that K i E is the event that i knows the event E; i.e., K i E is the set of states! at which i knows that! 2 E, which is equivalent to P i! E. In particular, P i! is the event that i considers! possile. 17

18 If K i E D E, so E is the union of cells in the partition P i, we say E is self-evident. The state space and knowledge structures are general features of the modal logic of knowledge (Hughes and Cresswell 1996, Aumann 1999), ut are more closely specified in epistemic game theory, in which state! 2 specifies the strategy profile s.!/ D.s 1.!/; : : :; s n.!// 2 S chosen y the players at!, as well as their conjectures! D. 1!; : : : ;! n / concerning the choices of the other players, where i! 2 S i, the space of proaility distriutions on S i. Player i s conjecture is derived from i s sujective prior y i!.s i/ D p i.s i I P i!/. A player i is rational at! if s i.!/ maximizes EΠi.s i ; i! /, which is defined y EΠi.s i ; i! / D X i!.s i / i.s i ; s i /: (1) s i 2S i We extend the epistemic game structure to extensive form games of perfect recall as follows. Every state! defines a unique path from the root node to a terminal node of the game. Let N i e the information sets at which player i chooses. We say information set is reached in state! if a node along this path is a memer of. Then, given information set, we define E to e the event that is reached. The epistemic assumptions of the extensive form game are that for each player i, and for all 2 N i, i considers all states in E, and no others, possile, so K i E D E ; i.e., E is self-evident to the player choosing at. By the perfect recall assumption, the tree property partially orders the information sets for the players, where if every node in has an ancestor in. This implies that ) E E, and the inequality is strict if E contains more than one node. We define i s sujective prior at 2 N i as the conditional proaility.s i / D p i.s i I P i!j! 2 E /, which is well defined if p i.s i I P i!/ 0 for some! 2 E, ecause E is the union of cells P i!, for! 2 E. If p i.s i I P i!/ are identially zero, we define.s i / aritrarily. We require that p! i.s ii P i!/ D 0 if s i does not reach for some s i 2 S i, so conjectures satisfy.s i / D 0 if s i does not reach for some s i 2 S i. REFERENCES Aumann, Roert J., Interactive Epistemology I: Knowledge, International Journal of Game Theory 28 (1999): and Adam Brandenurger, Epistemic Conditions for Nash Equilirium, Econometrica 65,5 (Septemer 1995): and Jacques H. Dreze, Rational Expectations in Games, American Economic Review 98,1 (2008):

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