as a requirement on choice. AD show the following result (reproduced as Prop. 1 of Sect. 3): Strategies surviving the Dekel-Fudenberg procedure (see D

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1 On the Epistemic Foundation of Backward Induction? Geir B. Asheim Department of Economics, University of Oslo, P.O. Box 1095 Blindern, N-0317 Oslo, Norway Abstract. I characterize backward induction in an epistemic model of perfect information games where rationality is associated with 'rational preferences' instead of 'rational choice'. In this approach, backward induction corresponds to the rationality principle that a player believes that his opponents reveal their preferences in each subgame. At an interpretative level this result resembles the one established by Aumann [4]. By instead imposing that a player believes that his opponents reveal their preferences only in the whole game, I interpret Ben-Porath's [10] support of the Dekel-Fudenberg procedure. 1 Introduction In recent years, two widely cited and inuential contributions on backward induction in nite perfect information games have appeared, namely Aumann [4] and Ben-Porath [10]. These contributions both of which consider generic perfect information games (where all payos are dierent) reach opposite conclusions: While Aumann establishes that Common Knowledge of Rationality (CKR) implies that the backward induction outcome is reached, Ben-Porath shows that the backward induction outcome is not the only outcome that is consistent with Common Certainty of Rationality (CCR). The models of Aumann and Ben-Porath are dierent. One such dierence is that Aumann makes use of 'knowledge' in the sense of 'absolute certainty', while Ben-Porath's analysis is based on 'certainty' in the sense of 'belief with probability one'. The present paper shows how the conclusions of Aumann and Ben-Porath can be captured by imposing dierent principles of rationality on the players within the same general framework. Furthermore, the interpretations of the present analysis correspond closely to the intuitions that Aumann and Ben-Porath convey in their discussions. Hence, the present contribution may increase our understanding of the dierences between the analyses of Aumann and Ben-Porath, and thereby enhance our understanding of the rationality requirements underlying backward induction. The analysis follows Asheim & Dufwenberg [3] [AD] in arguing that rationality in games should be imposed as a requirement on preferences rather than? This paper builds in part on jointwork with Martin Dufwenberg, who has contributed with helpful suggestions. I also thank Stephen Morris for useful comments.

2 as a requirement on choice. AD show the following result (reproduced as Prop. 1 of Sect. 3): Strategies surviving the Dekel-Fudenberg procedure (see Dekel & Fudenberg [16]), where one round of weak elimination is followed by iterated strong elimination, can be characterized as maximal elements when it is commonly known that each player takes into account all vectors of opponent strategies ('Caution') and believes that his opponents reveal their preferences in the whole game ('Belief of preference revelation'). For generic perfect information games, Ben-Porath shows that the set of outcomes consistent with CCR at the beginning of the game corresponds to the set of outcomes that survive the Dekel-Fudenberg procedure. Hence, maximal elements when 'Caution' and 'Belief of preference revelation' are commonly known correspond to the outcomes consistent with CCR. An extensive game allows preference revelation, not only in the whole game, but in proper subgames. In perfect information games (and, more generally, in multi-stage games) the subgames constitute an exhaustive set of situations in which preferences can be revealed. Hence, in perfect information games one can argue that 'Belief of preference revelation' should be replaced by 'Belief of preference revelation in each subgame': Each player believes that his opponents reveal their preferences in each subgame. The main results of the present paper (Props. 2 and 3 of Sect. 5) show how, for generic perfect information games, common knowledge of 'Caution' and 'Belief of preference revelation in each subgame' is possible and uniquely determines the backward induction outcome. Hence, by substituting 'Belief of preference revelation in each subgame' for 'Belief of preference revelation', the present analysis provides an alternative route to Aumann's conclusion, namely that CKR implies the backward induction outcome. Section 2 presents the formal framework in which extensive games will be analyzed. AD's characterization of the Dekel-Fudenberg procedure is reviewed in Sect. 3. Section 4 applies this result to generic extensive games of perfect information and compares, by means of an example, the present analysis to that of Ben-Porath [10]. Section 5 introduces 'Belief of preference revelation in each subgame' as an alternative rationality principle and establishes the paper's main results. Section 6 interprets the analysis in view of Aumann [4] and Ben-Porath [10] as well as Battigalli's [8] concept of a 'rationality ordering'. 2 States and Knowledge The purpose of this section is to present a framework for extensive games where each player is modeled as a decision maker under uncertainty. The decisiontheoretic analysis builds on Anscombe & Aumann [1]. However, for the analysis of extensive games, continuity must be relaxed in the fashion of Blume et al. [11] to allow each player to take into account the possibility that any subgame can be reached. Moreover, by not imposing completeness, the analysis does not require subjective probabilities. The framework is summarized by the concept of a knowledge system. The Appendix contains a presentation of the decisiontheoretic terminology, notation and results that will be utilized.

3 A Finite Extensive Game of Almost Perfect Information, with n players and K, 1 stages can be described as follows. Any nite multi-stage game (i.e., a perfect information game or a nitely repeated game) ts this description. The sets of histories is determined inductively: The set of histories at the beginning of the rst stage 1 is H 1 = f;g. Let H k denote the set of histories at the beginning of stage k. At h 2 H k, let, for each player i 2 N := f1;:::;ng, i's action set be denoted A i (h), where i is inactive athif A i (h) is a singleton. Write A(h) :=A 1 (h):::a n (h). Dene the set of histories at the beginning of stage k + 1 as follows: H k+1 := (h; a) jh 2 H k and a 2 A(h). This concludes the induction. Let H := S K,1 k=1 Hk denote the set of subgames and let Z := H k denote the set of outcomes. For each player i, let the von Neumann-Morgenstern utility function i : Z! < determine i's payo. Assume that there exist z, z 0 2 Z such that i (z) > i (z 0 ). A pure strategy for player i is a function s i that assigns an action in A i (h) to any h 2 H. Let S i denote player i's nite set of pure strategies, and write S := S 1 ::: S n and S,i := j6=i S j.write p, r, and s (2 S) for pure strategy vectors. For any s 2 S, let z(s) denote the outcome reached when s is used. Dene u i : S!<by u i (s) = i (z(s)). Refer to G =(S i ;u i ) i2n as the normal form of the extensive game,. Since u i is a von Neumann-Morgenstern utility function, we may extend u i to the set of objective randomizations on S. For any h 2 H [ Z, let S(h) =S 1 (h) :::S n (h) denote the set of strategy vectors that are consistent with h being reached. If h 0 is the predecessor of h, then S(h 0 ) S(h). If s i 2 S i and h 2 H, let s i j h denote the strategy in S i (h) satisfying s i j h (h 0 ) = s i (h 0 ) at any h 0 2 H except at h 0 with S(h 0 ) S(h) where s i j h (h 0 ) is dictated by s i j h being consistent with h. States and Types. When a normal form game G is turned into a decision problem for each player (see Tan & Werlang [23]), the uncertainty faced by a player is the strategy choices of his opponents, the beliefs of his opponents about the strategy choices of their opponents, and so on. A type of a player corresponds to a belief about the strategy choices of his opponents, a belief about the beliefs of his opponents about the strategy choices of their opponents, and so on. Given an assumption of coherency, models of such innite hierarchies of beliefs (Boge & Eisele [12], Mertens & Zamir [20], Brandenburger & Dekel [15], Epstein & Wang [18]) yield S T as the universal state space, where S is the underlying space of uncertainty and where T = T 1 ::: T n is the set of all feasible type vectors. Furthermore, for each i, there is a homeomorphism between T i and the set of beliefs on S, where := j6=i T j. In a decision-theoretic sense, the set of beliefs on S corresponds to the set of "regular" binary relations on the set of acts on S, where an (Ascombe-Aumann) act on S is a function that to any element ofs assigns an objective randomization on S. In the references above, with the notable exception of Epstein & Wang [18], a binary relation is "regular" if and only if it is represented by some subjective probability distribution in (S ). In conformity with the literature on innite hierarchies of beliefs, let the set of states of the world (or simply states) be := S T, and let each type t i of

4 any player i correspond to a binary relation ti on the set of acts on S,i. However, like AD, I do not construct a universal state space by explicitly modeling innite hierarchies of beliefs. For tractability I instead directly consider an implicit model with a nite type set T i for each player i from which innite hierarchies of beliefs can be constructed. Moreover, since completeness and continuity are not imposed, the "regularity" conditions on ti consist of reexivity, transitivity and objective independence only, meaning that ti is not necessarily represented by some subjective probability distribution in (S,i ). Knowledge. The construction is summarized by the following denition. Denition 1. A knowledge system for a normal form game G = (S i ;u i ) i2n consists of for each player i, a nite set of types T i, for each type t i of any player i, a binary relation ti (t i 's preferences) on the set of acts on S,i, where ti is reexive and transitive and satises objective independence. For each player i, i's knowledge can be derived from the knowledge system by following AD in their adaptation of Brandenburger & Dekel's ([15], p. 197) knowledge denition. For each player i and each state! 2, let t i (!) denote the projection of! on T i. Associate 'knowledge' of an event with the property that the complement of the event issavage-null: IfE, write n K i E :=! 2 (s; t) 2 ne ) t i 6= t i (!) or o t i = t i (!) and (s,i ;t,i ) is Savage-null acc. to ti(!) : Say that i knows the event E given! if! 2 K i E.Write KE := K 1 E \:::\ K n E.Say that the event E is mutually known given! if! 2 KE.Write CKE := KE \ KKE \ KKKE \ :::.Say that the event E is commonly known given! if! 2 CKE. Theory and Preferences over Strategies. Let ti denote the marginal of ti on. Since ti expresses t i 's belief concerning the type vector of i's opponents, I follow the terminology of Aumann & Brandenburger ([6], p. 1164) and refer to ti as t i 's theory. Likewise, let ti S,i denote the marginal of ti on S,i. A pure strategy s i 2 S,i can be viewed as an act x S,i on S,i that assigns (s i ;s,i )toany s,i 2 S,i.Amixed strategy x i 2 (S i ) corresponds to an act x S,i on S,i that assigns (x i ;s,i ) to any s,i 2 S,i. Hence, ti S,i is a binary relation also on the subset of acts on S,i that correspond to i's mixed strategies. Thus, ti S,i can be referred to as t i 's preferences over i's mixed strategies. The set of mixed strategies is the set of acts that are at t i 's actual disposal. Since ti is reexive and transitive and satises objective independence, ti S,i shares these properties, and, for any h 2 H, C ti i (h) :=fs i 2 S i (h)js i is maximal w.r.t. ti S,i(h) in (S i(h))g

5 is non-empty and supports any maximal mixed strategy. Refer to C ti i (h) ast i's choice set in the subgame h, and refer to C ti i : H! 2 Si(h) nf;g as t i 's choice function. Note if s i is maximal in a subgame h, then s i is maximal in any later subgame that s i is consistent with: Lemma 1. If s i 2 C ti i (h), then s i 2 C ti i (h0 ) for any h 0 2 H with s i 2 S i (h 0 ) S i (h). Proof. Suppose that s i is not maximal w.r.t. ti in S,i(h 0 ) (S i(h 0 )). Then there exists x S,i such that x S,i ti S,i(h y 0 ) S,i, where x S,i assigns (x i ;s,i ) to any s,i 2 S,i with x i 2 (S i (h 0 )), and where y S,i assigns (s i ;s,i ) to any s,i 2 S,i. By Mailath et al. ([19], Defs. 2 and 3 and the if-part of Theorem 1), S(h 0 )isastrategic independence for i. Hence, x S,i can be chosen such that u i (x S,i (s,i )) = u i (y S,i (s,i )) for all s,i 2 S,i ns,i (h 0 ). By conditional representation, x S,i ti y S,i(h) S,i, which contradicts that s i is maximal w.r.t. ti S,i(h) in (S i (h)). ut Write C ti i := C ti C t,i,i (h) := j6=ic tj j (h). i (;).For any h 2 H, write Ct (h) :=C t1 1 (h) ::: Ctn n (h) and 3 Rational Preferences Usually rationality isassociated with 'rational choice'. This means that rationality is a requirement on a pair (s i ;t i ), where s i can be said to be a rational choice by t i if s i 2 C ti i. See e.g. Epstein ([17], Sect. 6) for a presentation of this approach in a general context. The present paper follows AD by letting rationality beassociated with requirements on t i only. Since t i corresponds to the preferences ti, such requirements will be imposed on ti. In support of this alternative approach which will be referred to by the term 'rational preferences' one can note the following: The approach allows... rationality to be associated with types rather that strategy-type pairs.... conventional concepts like 'rationalizable strategies' and strategies surviving the Dekel-Fudenberg procedure to be characterized under very weak and natural conditions (see Prop. 4 of Asheim [2] as well as Prop. 1 below).... caution to be imposed as a strengthening of rationality without having to change the denition of knowledge. Under 'rational choice' the notion of 'knowledge' must be weakened to accommodate caution (see the introduction of AD as well as Borgers ([13], pp. 266{267) and Epstein ([17], p. 3)). In contrast, this paper models players that take into account all vectors of opponent strategies, while still using the term 'knowledge' in the strong sense described in the above subsection on knowledge. Thus, knowledge of rationality need not be weakened to allow players to take into account the possibility that any subgame in the extensive game be reached.

6 Here I will focus on showing how 'rational preferences' as an approach to rationality in games can be used to shed light on the analyses of Aumann [4] and Ben-Porath [10], and thereby enhance our understanding of the rationality requirements underlying backward induction. For this purpose, it is useful to reproduce AD's characterization of the Dekel-Fudenberg procedure. Admissible Rationality. The Dekel-Fudenberg procedure is made up of one round of elimination of weakly dominated strategies followed by iterated elimination of strongly dominated strategies. AD characterize this procedure by imposing on the preferences ti the following rationality principles, where the terms 'conditional representation' is dened in the Appendix, where weak and strong dominance are used w.r.t. i's vnm utility function u i, where n o E,i := (s,i ;t,i ) s,i 2 C t,i,i and t,i is not Savage-null acc. to ti n F,i := (s,i ;t,i ) s,i 2 S,i and t,i is not Savage-null acc. to ti ; and where x and y are two arbitrary acts on S,i. o CR ti is Conditionally Represented by u i. CAU (CAUtion) If x F,i weakly dominates y F,i, then x ti y. BPR (Belief of Preference Revelation) If x E,i strongly dominates y E,i, then x ti y. These principles can be explained as follows. When it is commonly known that all players are of types that satisfy CR, then it is commonly known that the game G =(S i ;u i ) i2n is played, meaning that G is a game of complete information. CAU means that (s,i ;t,i ) is not Savage-null acc. to ti if t,i is not Savagenull acc. to ti. The principle guarantees that all vectors of opponent strategies are taken into account, implying that the marginal of ti on S,i (i.e., t,i 's preferences over i's mixed strategies ti S,i ) is admissible. BPR means that the type 'believes' that the preferences of opponent types are revealed in the whole game, in the sense that the type always prefers an act which strongly dominates another act on the event that all opponent types choose maximal elements in the whole game, regardless of what happens outside this event. that satisfy CR, CAU, and BPR can be con- An example of preferences ti structed by letting 1. ti be represented by u i and some lexicographic probability system (LPS) `ti =(m 1 ;:::;m k ) 2 L(S,i ) (see Prop. A2 and Blume et al. [11]), 2. the marginal of ti on (i.e., t i 's theory ti ) be represented by u i and some subjective probability distribution m ti 2 ( ) (see Prop. A1), 3. supp(`ti )=S,i supp(m ti ), and m 1 (r,i ;t,i ) > 0 only if r,i 2 C t,i,i.

7 In this construction, ti is complete and the marginal of ti on is continuous. While these properties do not contradict CR, CAU, and BPR, they are not implied by CR, CAU, and BPR. Note that CAU in conjunction with CR and BPR rules out that the marginal of ti on S,i is continuous. Denition 2. Type t,i is admissibly rational if ti satises CR, CAU, and BPR. Refer to A := A 1 \:::\A n as the eventofadmissible rationality, where, for each i, A i := f! 2 jt i (!) is admissibly rationalg. The Dekel-Fudenberg procedure can now be characterized as maximal elements in states where admissible rationality is commonly known. Proposition 1 (Asheim & Dufwenberg [3]). A pure strategy r i for i survives the Dekel-Fudenberg procedure if and only if there exists a knowledge system with r i 2 C ti(!) i for some! 2 CKA. 4 Generic Games of Perfect Information Within the formalism of Sect. 2, a nite extensive game is... of perfect information if, at any h 2 H, there exists at most one player that has a non-singleton set of actions.... generic if, for each i, i (z) 6= i (z 0 ) whenever z and z 0 are dierent outcomes. Generic extensive games of perfect information has a unique subgame-perfect equilibrium. Moreover, in such games the procedure of backward induction yields in any subgame the unique subgame-perfect equilibrium outcome. If p denotes the unique subgame-perfect equilibrium, then, for any subgame h, z(pj h ) is the backward induction outcome in the subgame h, and S(z(pj h )) is the set of strategy vectors consistent with the backward induction outcome in the subgame h. Both Aumann [4] and Ben-Porath [10] analyze generic extensive games of perfect information. As already pointed out, while Aumann establishes that Common Knowledge of Rationality 1 (CKR) implies that the backward induction outcome is reached, Ben-Porath shows that the backward induction outcome is not the only outcome that is consistent with Common Certainty of Rationality (CCR). The purpose of the present section is to interpret the analysis of Ben- Porath by applying Prop. 1 to the class of generic perfect information games. Ben-Porath [10] establishes through his Theorem 1 that the set of outcomes consistent with CCR at the beginning of the game corresponds to the set of outcomes that survive the Dekel-Fudenberg procedure. Hence, by Prop. 1, maximal elements when admissible rationality is commonly known correspond to the outcomes consistent with CCR. 1 Aumann's [4]) analysis is based on substantive rationality. See Aumann [4], pp. 14{16, and Aumann [5].

8 F f F 5 D d D Fig. 1. A centipede game An Example. To illustrate how common knowledge of admissible rationality is consistent with outcomes other than the unique backward induction outcome, consider the simple centipede game of Fig. 1 where backward induction implies that down is being played at any decision node. Let T 1 = ft 0 1 ;t00g and 1 T 2 = ft 0 2 ;t00 2 g. Assume that the preferences of each type of player i is represented by u i and a 2-level LPS over S,i, implying that all types satisfy CR. In Table 1. The lexicographic probability systems for each type of the two players t 0 1:, t 0 2 t 00 2 t 00 1 : t 00 2 t 00 2 d 4 5 ; 10 7, 0; 1, 10 d 3 5 ; 10 5, 0; 1, 10 f 0; 1, ; 10 1, f 0; 1, ; 10 3 t 0 2:, t 0 1 t 00 1 t 00 2 : t 0 1 t 00 1 D 1 2 ; 1, 4 0; 1, 8 D 1; 1, 2 (0; 0) FD 0; 8 1, 12 ; 1, 4 FD 0; 1, 4 (0; 0) FF 0; 8 1, 0; 1, 8 FF 0; 1 4 (0; 0) Table 1, the rst numbers in the parentheses express primary probability distributions, while the second numbers express secondary probability distributions. The strategies DD and DF are merged as their relative likelihood does not matter. Note that all types satisfy CAU. With these 2-level LPSs each type's preferences over the player's own strategies are given by t 0 1 : D FD FF t 00 1 : FD D FF t 0 2 : d f t 00 2 : f d It is easy to check that all types satisfy BPR (e.g. both t 0 2 and t 00 2 believe that t 0 1 reveals his preferences in the sense that they deem his maximal element D innitely more likely than each of his non-maximal elements FD and FF).

9 Hence, preferences consistent with common knowledge of CR, CAU, and BPR need not reect backward induction since FDand f are maximal elements. Note that type t 0 2, conditional on his decision node being reached (i.e. 1 choosing FD or FF), updates his beliefs about the type of player 1 and assigns probability one to 1 being of type t Consequently, the conditional belief of type t 0 2 about 1's strategy choice assigns probability one to FD. Type t 00 2, on the other hand, does not admit the possibility that 1 is of another type than t 0 1. Since the choice of F at 1's rst decision node means that t 0 1 has not revealed his preferences in the whole game, there is no restriction concerning the conditional belief of type t 00 2 about the choice at 1's second decision node. In the terminology of Ben-Porath, a "surprise" has occurred. Subsequent to such a surprise, a type need not believe that the opponent type reveals his preferences. 5 Belief of Preference Revelation in Each Subgame A simultaneous game oers only one choice situation in which preferences can be revealed. Hence, for a game in this class, it seems reasonable that any type of a player believes that the preferences of opponent types can be revealed in the whole game only, as formalized by the rationality principle BPR. An extensive game with a non-trivial dynamic structure, however, oers such choice situations, not only in the whole game but also in proper subgames. This implies that, for such an extensive game, preferences can be revealed in any subgame. Moreover, for extensive games of almost perfect information, the subgames constitute an exhaustive set of situations in which preferences can be revealed. This motivates applying the following rationality principle in this class of games, where n E,i (h) := (s,i ;t,i ) 2 S,i (h) o s,i 2 C t,i,i (h) and t,i is not Savage-null acc. to ti : BPRS (Belief of Preference Revelation in each Subgame) For each h 2 H, x E,i(h) strongly dominates y E,i(h), then x ti S,i(h) y. This principle can be interpreted as follows. 2 BPRS means that the type 'believes' that the preferences of opponenttypes are revealed in the each subgame, in the sense that the type always prefers an act which strongly dominates another act on the event that all opponent types choose maximal elements in the subgame, regardless of what happens outside this event. Denition 3. Type t i is admissibly subgame rational if ti satises CR, CAU, and BPRS. 2 BPRS is a non-inductive analogue to forward knowledge of rationality; see Balkenborg & Winter [7].

10 Refer to A := A 1 \:::\A n as the eventofadmissible subgame rationality, where, for each i, A i := f! 2 jt i(!) is admissibly subgame rationalg. Denition 3 can be applied to any nite extensive game of almost perfect information. However, to relate to Aumann's [4] Theorems A and B, the following analysis is concerned with generic perfect information games. The Example Revisited. In the knowledge system of Table 1, type t 00 2 does not satisfy BPRS. ByBPRS any type of player 2 must believe that t 0 1 reveals his preferences in the subgame dened by 2's decision node in the sense that the type deems the maximal element oft 0 1 in the subgame, FD, innitely more likely than the non-maximal element, FF. This means that anytype of player 2 prefers d to f, implying that no type of player 1 satisfying BPRS can prefer FD to D. Thus, common knowledge of CR, CAU, and BPRS entails that any types of players 1 and 2have the preferences D FD FF and d f, respectively, meaning that if any type of a player reveals his preferences in a subgame, then his choice is made in accordance with backward induction. Demonstrating that this conclusion holds in general is the main result of the present paper. Main results. In analogy with Aumann's [4] Theorems A and B, it is established that... any vector of maximal elements in a subgame of a generic perfect information game, in a state where admissible subgame rationality is commonly known, leads to the backward induction outcome in the subgame (Prop. 2). Hence, by substituting BPRS for BPR, the present analysis yields support to Aumann's conclusion, namely that if an appropriate form of rationality is commonly known, then the backward induction outcome results.... for any generic perfect information game, common knowledge of admissible subgame rationality is possible (Prop. 3). Hence, Prop. 2 is not empty. Proposition 2. Consider a generic extensive game of perfect information,. If, for some knowledge system,! 2 CKA, then, for each h 2 H, C t(!) (h) S(z(pj h )), where p denote the unique subgame-perfect equilibrium. Proof. Some properties of the knowledge denition (see Sect. 2) must be established for the proof. It is easy to check that K i =, and, for any events E and F, K i E \ K i F = K i (E \ F ), K i E = K i K i E, and nk i E = K i (nk i E). Write K 0 E := E and, for each g 1, K g E := KK g,1 E. Since K i (E\F )=K i E\K i F and K i K i E = K i E, it follows 8g 2, K g E = K 1 K g,1 E \ :::\ K n K g,1 E K 1 K 1 K g,2 E \:::\K n K n K g,2 E = K 1 K g,2 E \:::\K n K g,2 E = K g,1 E. The truth axiom (K i E E) is not satised, since an event can be known even though the true state is an element of the complement of the event. However, since A := A 1 \ :::\ A n is an event that concerns the type vector, mutual knowledge of A implies that A is true: KA = K 1 A \:::\K n A K 1 A 1 \:::\K na n = A 1 \ :::\ A n = A since, for each i, K i A i = A i. Hence, it follows that (i) 8g 1, K g A K g,1 A, and (ii) 9g 0 0 such that K g A = CKA for g g 0

11 since is nite. In view of these properties, it is sucient to show for any g =0;:::;K, 2 that if there exists a knowledge system with! 2 K g A, then C t(!) (h) S(z(pj h )) for any h 2 H K,1,g. This is established by induction. (g = 0) Let h 2 H k,1. First, consider some j with a singleton action set at h. Then trivially C tj j (h) =S j(h) =S j (z(pj h )). Now, consider some i with a non-singleton action set at h; since, has perfect information, there is at most one such i. Let t i = t i (!) for some! 2 K 0 A = A. Then it follows from CR and CAU that C ti i (h) =S i(z(pj h )) since, is generic. (g =1;:::;K, 2) Suppose that it has been established for g 0 =0;:::;g, 1 that if there exists a knowledge system with g 2 K g0 A, then C t(!) (h 0 ) S(z(pj h 0)) for any h 2 H K,1,g0. Let h 2 H K,1,g. First, consider some j with a singleton action set at h. Let t j = t j (!) for some! 2 K g,1 A. Then, by Lemma 1 and the premise, S j (h) =S j (h; a) and C tj j (h) Ctj j (h; a) S j(z(pj (h;a) )) if a is a feasible action vector at h. This implies that C tj j (h) T a S j(z(pj (h;a) )) S j (z(pj h )). Now, consider some i with a non-singleton action set at h; since, has perfect information, there is at most one such i. Let t i = t i (!) for some! 2 K g A. The preceding argument implies that C t,i,i (h) T a S,i(z(pj (h;a) )) whenever t,i is not Savage-null acc. to ti since! 2 K g A K i K g,1 A. Let s i 2 S i (h) be a strategy that diers from p i j h by assigning a dierent action at h (i.e., z(s i ;p,i j h ) 6= z(p i j h ) and s i (h 0 )=p i j h (h 0 ) whenever S i (h) S i (h 0 )). Write x S,i for the act on S,i that p i j h can be viewed as, and write y S,i for the act on S,i that s i can be viewed as. Let x and y be the acts on S,i that satisfy x(s,i ;t,i )=x S,i (s,i ) and y(s,i ;t,i )=y S,i (s,i ) for all (s,i ;t,i ). By backward induction, x \as,i(z(pj (h;a) )) strongly dominates y \as,i(z(pj (h;a) )) since, is generic. Since C t,i,i (h) T a S,i(z(pj (h;a) )) whenever t,i is not Savagenull acc. to ti, x E,i(h) strongly dominates y E,i(h), and, by CR and BPRS, x ti S,i(h) y and x S,i ti y S,i(h) S,i. By Lemma 1 and the premise that C ti i (h; a) S i(z(pj (h;a) )) if a is a feasible action vector at h, it follows that C ti i (h) S i(z(pj h )). ut Proposition 3. For any generic extensive games of perfect information,, there exists a knowledge system with CKA 6= ;. Proof. Construct a knowledge system with only one type of each player, T = f(t 1 ;:::;t n )g, implying that (1) it is commonly known given any! 2 S T that the type vector is (t 1 ;:::;t n ). Write, 8i 2 N, 8k 2f1;:::;K, 1g, P,i k := fp,i j h jh 2 H k g and, P,i K := S,i. Let, 8i 2 N, `ti =(m 1 ;:::;m K ) 2 L(S,i ) satisfy the following requirement: 8k 2f1;:::;Kg, supp(m k )=P,i k. By letting ti be represented by u i and the LPS `ti, then (2) ti satises CR and CAU. Let `ti S,i denote the marginal of `ti on S,i, and let, 8h 2 H, `ti = S,i(h) (m 1 S,i(h); :::; mk0 S,i(h)) denote the conditional of `ti S,i given S,i (h) (see Blume et al. [11], Def. 4.2). By the properties of a subgame-perfect equilibrium, 8h 2 H, m 1 S (p,i(h),ij h )=1andp i j h 2 C ti i (h). Hence, since likewise p,ij h 2 C t,i,i (h), (3) ti satises BPRS. By (1), (2), and (3), it follows that! 2 CKA. ut

12 6 Discussion Consider a generic perfect information game. Say that a type's preferences are in accordance with backward induction if, in any subgame, a strategy is a maximal element only if it is consistent with the backward induction outcome. Using this terminology, Prop. 2 can be restated as follows: Under common knowledge of admissible subgame rationality in a generic perfect information game, players are of types with preferences that are in accordance with backward induction. Furthermore, common knowledge of admissible subgame rationality implies that players cannot admit the possibility that opponents are of types with preferences not in accordance with backward induction. This reects in spirit the conclusion that can be drawn from Aumann's analysis. However, since admissible subgame rationality is imposed on preferences, reaching 2's decision node and 1's second decision node in the centipede game of Fig. 1 does not contradict common knowledge of admissible subgame rationality. Of course, these decision nodes will not be reached if players actually reveal their preferences. But that players are of types satisfying BPRS does not imply that they will actually reveal their preferences; rather, it means that they 'believe' (in a sense that Morris [22] shows corresponds to Brandenburger's [14] 'rstorder knowledge') that their opponents will reveal their preferences wherever the extensive form allows them to. Knowledge vs. Certainty. The term 'knowledge' (cf. the subsection on knowledge in Sect. 2) signies, in the present paper, that the complement of a known event is not taken into account. This notion of 'knowledge' is strong; in fact, it is the strongest form considered by Morris [22]. Each of the notions 'approximate knowledge' (Monderer & Samet [21], Borgers [13]), and 'rst-order knowledge' (Brandenburger [14]) represents an eective weakening of 'knowledge' exactly because the complement of a known event is allowed to be taken into account. It is stronger than Ben-Porath's [10] term 'certainty', which allows a player, while being certain that his opponents choose rationally, to take into account the complement of this event (i.e. the possibility that they will choose strategy vectors that are not consistent with rational choice). 'Knowledge', in the sense that the complement of a known event is not taken into account, can be used here since knowledge of rationality concerns preferences and not choice. When there is common knowledge of admissible (subgame) rationality (cf. Def. 2 (3)), then each player knows (in the sense of not taking into account the complement) that each opponent is of a type with preferences that satisfy CR, CAU, and BPR(S), each player knows (in the sense of not taking into account the complement) that each opponent knows (in the sense of not taking into account the complement) that any of his opponents is of a type with preferences that satisfy CR, CAU, and BPR(S), and so on. By CAU it is not the case that any player knows that opponents will not play any particular strategy vector. On the contrary, CAU imposes that a player takes into account all vectors of opponent strategies, implying that he deems possible that

13 any subgame in the extensive game be reached. By BPR(S) the type of any player merely 'believes' that (in any subgame) opponents will choose a vector consisting of maximal elements only, in the sense that the type's preferences respect strong dominance on the set of opponent vectors of maximal elements. Rationality Orderings. Consider a two-player game. The constructive proof of Prop. 3 shows how common knowledge of admissible subgame rationality may lead a type t i of player i to have preferences over i's strategies that are represented by an LPS `ti S j =(m 1 S j ; :::; m K S j ) with more than two levels of subjective probability distributions (i.e. K>2). E.g., in the centipede game of Fig. 1, common knowledge of admissible subgame rationality implies that anytype t 2 of player 2 has preferences that can be represented by `t2 S 1 =(m 1 S 1 ;m 2 S 1 ;m 3 S 1 ) where supp(m 1 S 1 )=fdg, supp(m 2 S 1 )=fd; F Dg, and supp(m 3 S 1 )=S 1. Within the 'rational choice' approach one may interpret supp(m 1 S j ) to consist of strategies for j that are "most rational", supp(m K S j )n S k 0 <K supp(mk0 j ) to consist of strategies for j that are "completely irrational", and supp(m k S j )n S k 0 <k supp(mk0 j ), for k =2;:::;K, 1, to consist of strategies for j that are at "intermediate degrees of rationality". Furthermore, for any k = 2;:::;K, t i deems any strategy in Sk 0 <k supp(mk0 j ) innitely more likely than any strategy not having this property. This illustrates that (supp(m 1 S j );:::;supp(m K S j )n S k 0 <K supp(mk0 j )) corresponds closely to what Battigalli [8] calls a rationality ordering for j. The present construction of such a rationality ordering diers from the one proposed by Battigalli along two dimensions: 1. Battigalli considers maximal elements of players in the whole game only (see his Def. 2.1), while here also maximal elements in subgames are considered (cf. BPRS). 2. Battigalli considers maximal elements given preferences where opponent strategies that are less than "most rational" are given positive primary probability, while here only maximal elements of (admissibly subgame) rational types can be given positive primary probability. This means that although Battigalli's construction of a rationality ordering also promotes the backward induction outcome in any generic perfect information game, his proof (see Battigalli [9]) is not as directly tied to the procedure of backward induction. Furthermore, Battigalli's construction of a rationality ordering promotes the forward induction outcome in an extended version of the \Battle-of-the-Sexes\ (BoS) game where the BoS game is preceded by 1 being oered an outside option that is preferred by 1 to 2's most preferred outcome in the BoS game. This conclusion is not reached in the present analysis since there is no choice situation in which 1 under all circumstances can reveal a particular preference between his BoS strategies. Appendix: The Decision-Theoretic Framework The purpose of this appendix is to present the decision-theoretic terminology, notation and results utilized and referred to in the main text. Let F be a nite set of states, and let S be a nite set of outcomes. Consider a decision maker which is uncertain about what state in F will be realized. The decision

14 maker is endowed with a binary relation over all functions that to each element off assigns an objective randomization on S. Any such function x F : F! (S) is called an act on F.Write x F and y F for acts on F. A binary relation on the set of acts on F is denoted by F, where x F F y F means that x F is preferred or indierent to y F. As usual, let F (preferred to) and F (indierent to) denote the asymmetric and symmetric parts of F. A binary relation F on the set of acts on F is said to satisfy objective independence if x 0 F F (respectively F ) x 00 F i x 0 F +(1, )y F F (respectively F ) x 00 F +(1, )y F, whenever 0 <<1 and y F is arbitrary. nontriviality if there exist x F and y F such that x F F y F. continuity if there exist 0 < << 1 such that x 0 F +(1, )x 00 F F y F F x 0 F +(1, )x 00 F whenever x 0 F F y F F x 00 F. If F = F1 F2, say that F1 is the marginal of F on F1 if, x F1 F1 y F1 i x F F y F whenever x F1 (f1) =x F (f1;f2) and y F1 (f1) =y F (f1;f2) for all (f1;f2). If E F, let x E denote the restriction of x F to E. Dene the conditional binary relation E by x 0 F E x 00 F if, for arbitrary y F,(x 0 E; y,e) F (x 00 E; y,e), where,e denotes F ne. Say that the state f 2 F is Savage-null if x F ffg y F for all acts x F and y F on F. A binary relation F is said to satisfy conditional continuity if, 8f 2 F, there exist 0 <<<1 such that x 0 F +(1, )x 00 F ffg y F ffg x 0 F +(1, )x 00 F whenever x 0 F ff g y F ffg x 00 F. non-null state independence if x F feg y F i x F ffg y F whenever e and f are not Savage-null and x F and y F satisfy x F (e) =x F (f) and y F (e) =y F (f). If e, f 2 F and feg is complete, then e is deemed innitely more likely than f (e >>f) if e is not Savage-null and x F feg y F implies (x,ff g ; x 0 ffg) fe;fg (y,ffg ; yffg) 0 for all x 0 F, yf 0. According to this denition, f may, but need not, be Savage-null if e>>f. Say that y E is maximal w.r.t. E if there is no x E such that x E E y E. Let u : S!<be a vnm utility function, where there exist s 0, s 00 2 S such that u(s 0 ) >u(s 00 ). If x 2 (S) is an objective randomization, write u(x) = P x(s)u(s). s2s Say that F is conditionally represented by u if (a) F is nontrivial and (b) x F ffg y F i u(x F (f)) u(y F (f)) whenever f is not Savage-null. If F is conditionally represented by u, then F satises conditional continuity and non-null state independence, and, for any f 2 F, ffg is complete. Say that x E strongly dominates y E if, 8f 2 E, u(x E(f)) > u(y E(f)). Say that x E weakly dominates y E if, 8f 2 E, u(x E(f)) u(y E(f)), with strict inequality for some e 2 E. Say that F is admissible w.r.t. u if x F F y F whenever x F weakly dominates y F.If F is conditionally represented by u, then F is admissible w.r.t. u i no f 2 F is Savage-null. Two representation results can now be stated. The rst follows from Anscombe & Aumann [1] and the properties of a conditionally represented binary relation. Proposition A1. If F is complete, transitive, objectively independent, continuous, and conditionally represented by u, then there is a subjective probability distribution m F 2 (F ) such that x F F y F mf (f)u(yf (f)). i P f2f mf (f)u(xf (f)) P f2f The second result, due to Blume et al. ([11], Theorem 3.1), requires the notion of a lexicographic probability system (LPS): If K 1 and, 8k 2f1;:::;Kg, m k F 2 (F ), then `F =(m 1 F ; :::; m K F ) is an LPS on F. Let L(F ) denote the set of LPSs on F. Proposition A2. If F is complete, transitive, objectively independent, and conditionally represented byu, then there is an LPS `F =(m 1 F ; :::; m K F ) 2 L(F ) such that x F F y F i ( P f2f mk F (f)u(x F (f))) K k=1 L ( P f2f mk F (f)u(y F (f))) K k=1.3 3 For two vectors v and w, v L w i whenever w k >v k, 9 k 0 <ksuch that v k 0 >w k 0.

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