Rationalizability in general situations

Transcription

1 Rationalizability in general situations Yi-Chun Chen a, Xiao Luo a,y, Chen Qu b a Department of Economics, National University of Singapore, Singapore b Department of Economics, BI Norwegian Business School, N-0442 Oslo This version: January 2012 Abstract We present an analytical framework that can be used to analyze rationalizable strategic behavior in arbitrary strategic games with various modes of behavior. We show that the notion of rationalizability de- ned in general situations has nice properties similar to those in nite games. Our approach does not require any kind of technical assumptions on the structure of the game or particular behavioral assumptions on players preferences. In this paper, we also investigate the relationship between rationalizability and Nash equilibrium in general games. JEL Classi cation: C70, D81. Keywords: General games; rationalizability; Nash equilibrium; common knowledge of rationality We thank Stephen Morris for pointing out this research topic to us. We also thank Bin Miao, Ben Wang, Xiang Sun, Chih-Chun Yang, Chun-Lei Yang, Yongchao Zhang, and participants at the NUS theory workshops for helpful comments and discussions. Financial support from the National University of Singapore is gratefully acknowledged. y Corresponding author. Fax: ecsycc@nus.edu.sg (Y.-C. Chen), ecslx@nus.edu.sg (X. Luo), Chen.Qu@bi.no (C. Qu). 1

2 1 Introduction The notion of rationalizability proposed by Bernheim (1984) and Pearce (1984) is one of the most important and fundamental solution concepts in non-cooperative game theory. The basic idea behind this notion is that rational behavior must be justi ed by rational beliefs and conversely, rational beliefs must be based on rational behavior. The notion of rationalizability captures the strategic implications of the assumption of common knowledge of rationality (see Tan and Werlang (1988)), which is di erent from the assumption of commonality of beliefs or correct conjectures in an equilibrium (see Aumann and Brandenburger (1995)). In the literature, most of the studies of rationalizable strategic behavior have been restricted to nite games. 1 The main purpose of this paper is to extend the notion of rationalizability to general games that inherits the properties of conventional rationalizability. Since many important models arising in economic applications are games with in nite strategy spaces and discontinuous payo functions, e.g., models of price and spatial competition, auctions, and mechanism design, 2 it is clearly important and practically relevant to extend the notion of rationalizability to arbitrary games. In the de nition of conventional rationalizability, each player is implicitly assumed to be Bayesian rational i.e., each player maximizes the expected utility given his probabilistic belief about the opponents strategy choices. The Ellsberg Paradox and related experimental evidence demonstrate that a decision maker usually displays an aversion to uncertainty or ambiguity and, thereby, motivates generalizations of the subjective expected utility model. Epstein (1997) extended the concept of rationalizability to a variety of general preference models by characteriz- 1 Berheim (1984, Proposition 3.2) studied the properties of rationalizable strategies in compact and continuous games. There are also a few exceptional examples on in nite games such as Arieli s (2010) analysis of rationalizability in continuous games where every player s strategy set is a Polish space and the payo function of each player is bounded and continuous and Zimper s (2006) discussions on a variant of strong point-rationalizability in games with metrizable strategy sets. 2 See, e.g., Bergemann and Morris (2005a, 2005b), Bergemann et al. (2011), and Kunimoto and Serrano (2011). In particular, Bergemann et al. (2011) and Kunimoto and Serrano (2011) considered in nite mechanisms (game forms) for which trans nite rounds of deletion of never-best replies or dominated strategies are necessary. 2

3 ing rationalizability and survival of iterated deletion of never best response strategies as the (equivalent) implications of rationality and common knowledge of rationality. In particular, Epstein o ered a model of preference to allow for di erent categories of regular preferences such as subjective expected utility, probabilistically sophisticated preference, Choquet expected utility and the multi-priors model. However, from a technical point of view, Epstein s (1997) analysis relies on topological assumptions on the game structure and, in particular, most of his discussions on rationalizability are restricted to nite games. 3 Apt (2007) relaxed the nite set-up of games and studied rationalizability by an iterative procedure, but his analysis implicitly requires players preferences to have a form of expected utility. In this paper we study rationalizable strategic behavior in general situations: arbitrary games with various modes of behavior. To de ne the notion of rationalizability, we need to consider a system of preferences/beliefs for possible subgame situations. By using Harsanyi s ( ) notion of type, we introduce the simple model of type, which speci es a set of admissible and feasible types for each of players in every possible restriction of game situation. For each type of a player, the player is able to make a decision over his own strategies. Our approach is topology-free and is applicable to arbitrary games with various modes of behavior. In a related paper, Apt and Zvesper (2010) provided a broad and general approach to various forms of customary iterative solution concepts in arbitrary strategic games with a special emphasis on the role of monotonicity in rationality. Our analysis of rationalizable strategic behavior is, in this respective, harmonious with Apt and Zvesper s (2010) approach. As we have emphasized, this paper focuses on how to extend the notion of rationalizability to general games with various modes of behavior that inherits the properties of conventional rationalizability, while Apt and Zvesper s (2010) paper focuses on examining and comparing, in the context of epistemic analysis with possibility correspondences, various forms of customary iterative solution concepts in arbitrary strategic games through the monotonic property of rationality behind the iterated dominance notions. 3 See also Asheim s (2006) related discussions on rationalizability under alternative epistemological assumptions. 3

4 We o er a de nition of rationalizability in general situations (De nition 1). Roughly speaking, a set of strategy pro les is regarded as rationalizable if every strategy in this set can be justi ed by a type endowed with the set. We show that the union of all the rationalizable sets is the largest (w.r.t. set inclusion) rationalizable set in the product form (Proposition 1), which can be derived from an iterated elimination of never-best responses (IENBR) (Proposition 2). IENBR is an order-independent procedure (Proposition 3). In addition, we study the epistemic foundation of rationalizability in general situations: we formulate and prove that rationalizability is the strategic implication of common knowledge of rationality (Proposition 5). We show an equivalence between the notion of rationalizability and the notion of a posteriori equilibrium in general settings (Proposition 6). In this paper, we also investigate the relationship between rationalizability and Nash equilibrium. We demonstrate through an example that the IENBR procedure may generate spurious Nash equilibrium and, then, present a necessary and su cient condition for no spurious Nash equilibria (Proposition 4). In dominance-solvable games, the set of Nash equilibria can be solved by IENBR (Corollary 1). This analysis is related to Chen et al. s (2007) work on strict dominance in general games. We show that, through examples, rationalizability neither implies nor is implied by iterated strict dominance used in Chen et al. (2007) in general game situations. It is worthwhile to emphasize that one important feature of this paper is that, throughout this paper, we do not require any kind of technical assumptions on the structure of the game or particular behavioral assumptions on players preferences; in particular, we do not require the compactness, convexity, and continuity conditions on strategy sets and payo functions, and we do not even assume that players preferences have utility function representations. The rest of this paper is organized into six sections. Section 2 is the set-up. Sections 3 and 4 present the main results concerning rationalizability with IENBR and Nash equilibrium respectively. Section 5 provides the epistemic foundation for rationalizability. Section 6 discusses the relationship between rationalizability and iterated strict dominance. Section 7 o ers concluding remarks. 4

5 2 Set-up Consider a normal-form game G (N; fs i g i2n ; fu i g i2n ); where N is an (in) nite set of players, S i is an (in) nite set of player i s strategies, and u i : S! R is player i s arbitrary payo function where S i2n S i. For s 2 S let s (s i ; s i ). A strategy pro le s is a (pure) Nash equilibrium in G if for every player i, u i (s ) u i s i ; s i 8si 2 S i. The notion of type due to Harsanyi ( ) is a simple and parsimonious description of the exhaustive uncertainty facing a player, including the player s knowledge, preferences/beliefs, etc. 4 Given player i s type, the player has one corresponding preference over his own strategies, according to which the player can make his decision. We consider a model of type for game G: > G f> G i ()g i2n ; where > G i () is de ned for every (nonempty) subset S 0 S and every player i 2 N. The set > G i (S 0 ) is interpreted as player i s type space in the reduced game Gj S 0 (N; fs 0 ig i2n ; fu i j S 0g i2n ), where u i j S 0 is the payo function u i restricted on S 0. Each type t i 2 > G i (S 0 ) has a corresponding preference relation % ti over player i s strategies in S i. The indi erence relation, ti, is de ned as usual, i.e., s i ti s 0 i i s i % ti s 0 i and s 0 i % ti s i. For instance, we may consider > G i (S 0 ) as a probability space or a regular preference space de ned on S 0. The following example demonstrates that the analytical framework can be applied to nite games where the players have the standard subjective expected utility (SEU) preferences. 4 As Harsanyi ( , p.171) pointed out, we can regard the [type] as representing certain physical, social, and psychological attributes of player i himself in that it summarizes some crucial parameters of player i s own payo function U i as well as the main parameters of his beliefs about his social and physical environment. Harsanyi demonstrated that games with incomplete information can be solved by using the notion of type. This notion is also useful to analyze strategic uncertainty about the actual play of the games with complete information. 5

6 Example 1. Consider a nite game G. Player i s belief about the strategies that the opponents play in the reduced game Gj S 0 is de ned as a probability distribution i over S 0 i, i.e., i 2 S i 0 where S 0 i is the set of probability distributions over S 0 i fs i j (s i ; s i ) 2 S 0 g. For any i, the expected payo of s i can be calculated by U i (s i ; i ) = X i (s i ) u i (s i ; s i ) s i 2S 0 i where i (s i ) is the probability assigned by i to s i. That is, i generates an SEU preference over S i. For our purpose we may de ne a model of type (on G) as follows: > G = f> G i ()g i2n ; where, for every player i 2 N, > G i (S 0 ) = S i 0 for every (nonempty) subset S 0 S. Note that the beliefs are correlated in the sense that a belief is represented by a joint probability distribution over the opponents strategies. The model of type allows to represent player s beliefs as product probability distributions over the opponents strategies. 5 Throughout this paper, we impose the following two conditions, C1 and C2, for the model of type > G. C1 (Monotonicity) 8i, > G i (S 0 ) > G i (S 00 ) if S 0 S 00. The monotonicity condition requires that when one player faces a greater degree of strategic uncertainty, the player possesses more types available for resolving uncertainty. In the literature on information economics, a type of a player is interpreted as the initial private information, about all the uncertainty regarding the state of nature in a game situation, that player has. From this point of view, it is natural to assume that there are more types available if there is more strategic 5 In the standard game-theory literature, players are typically assumed to be Bayesian rational; that is, each player forms a prior over the space of states of the world and maximizes the expected value of some xed vnm index on outcomes. The model of type also allows to represent player s beliefs as other forms of subjective expected utility preferences such as Borgers s (1993) ordinal expected utility and the state-dependent utility preferences discussed in Morris and Takahashi (2011). 6

7 uncertainty about the choices of the players. Under C1, > i > G i (S) can be viewed as the universal type space of player i in game G. For s 2 S, player i s Dirac type i (s) is a type with the property: 8s 0 i; s 00 i 2 S i ; u i (s 0 i; s i ) u i (s 00 i ; s i ) i s 0 i % i (s) s 00 i : A Dirac type i (s) is a degenerated type with which player i behaves as if he faces a certain play s i of the opponents. In probabilistic models, a Dirac type is a point mass that represents a point belief about the opponents choices. The following condition requires that the only possible type for a deterministic case i.e. a singleton of a certain play of the opponents is a Dirac type. This condition is a rather natural requirement when strategic uncertainty is reduced to a special deterministic case of certainty. C2 (Diracability) 8i, > G i (fsg) = f i (s)g if s 2 S. In Example 1, it is easy to see that C1 and C2 are satis ed for the standard SEU preference model. C1 and C2 imply that i (s) 2 > G i (S 0 ) if s 2 S 0, i.e., the type space on S 0 contains all the possible Dirac types on S 0. A strategy s i 2 S i is a best response to t i 2 > G i (S 0 ) if s i % ti s 0 i for any s 0 i 2 S i. Notice that even if a reduced game Gj S 0 is concerned, any strategy of player i in the original game G can be a candidate for the best response. Observe that s is a Nash equilibrium i, for every player i, s i is a best response to i (s ). Next we provide the formal de nition of rationalizability in general games. The spirit of this de nition is that for every strategy in a rationalizable set, the player can always nd some type in the type space de ned over this set to support his choice of strategy. Let BR (t i ) denote the set of best responses to t i. De nition 1. A subset R S is rationalizable in > G if 8i and 8s 2 R, there exists some t i 2 > G i (R) such that s i 2 BR(t i ). We call a strategy pro le rationalizable in > G if this pro le lies in a rationalizable set in > G. The following Proposition 1 asserts that there is the largest (w.r.t. set 7

8 inclusion) rationalizable set in product form that contains all the rationalizable strategy pro les in > G. Proposition 1. R [ R is rationalizable in > GR is the largest rationalizable set in > G. Moreover, R is in the Cartesian-product form. Proof: It su ces to show that R is a rationalizable set in > G. Let s 2 R. Then, there exists a rationalizable set R in > G such that s 2 R. Thus, for every player i, there exists some t i 2 > G i (R) such that s i 2 BR(t i ). Since R R, by C1, t i 2 > G i (R ) and s i 2 BR(t i ). Let s 2 i2n R i where R i fs i j s 2 R g. Then, for every player i, there exists t i 2 > G i (R ) such that s i 2 BR(t i ). Since R i2n R i, by C1, t i 2 > G i ( i2n R i ) and s i 2 BR(t i ). That is, i2n R i is a rationalizable set in > G. Since R [ R is rationalizable in > GR, R = i2n R i. Remark. For Z S let '(Z) = i2n si j s i 2 BR(t i ) for some t i 2 > G i (Z). Then R is the largest xed point of ' : 2 S 7! 2 S. See Apt and Zvesper (2010) and Luo (2001, Sec. 4.1) for a general approach to rationalizable-like solution concepts by using Tarski s xed point theorem on complete lattices; cf. also Brandenburger et al. (2011) for related discussions. 3 IENBR and rationalizability In the literature, rationalizability can also be de ned as the outcome of an iterated elimination of never-best responses. We employ a trans nite elimination process that can be used for any arbitrary game. 6 Let 0 denote the rst element in an ordinal, and let + 1 denote the successor to in. For S 00 S 0 S, S 0 is said to be reduced to S 00 (denoted by S 0! S 00 ) if, 8s 2 S 0 ns 00, there exists some player i such that s i =2 BR (t i ) for any t i 2 > G i (S 0 ). 6 Lipman (1994) demonstrated that, in in nite games, we may need the trans nite induction to analyze the strategic implication of common knowledge of rationality. See also Chen et al. s (2007) Example 1 for the reason why we need a trans nite process for iterated deletion of strictly dominated strategies in general games. 8

9 De nition 2. An iterated elimination of never-best responses (IENBR) is a nite, countably in nite, or uncountably in nite family fr g 2 such that R 0 = S, R! R +1 (and R = \ 0 <R 0 for a limit ordinal ), and R 1 \ 2 R! R 0 only for R 0 = R 1. The following Proposition 2 states that De nitions 1 and 2 are equivalent. Proposition 2. R 1 = R. Proof. By De nition 2, 8s 2 R 1, every player i has some t i 2 > G i (R 1 ) such that s i 2 BR(t i ). So R 1 is a rationalizable set and, hence, R 1 R. By Proposition 1, R is a rationalizable set in > G and, by C1, survives every round of elimination in De nition 2. So R R 1. The de nition of IENBR procedure does not require the elimination of all neverbest response strategies in each round of elimination. This exibility raises a question whether any IENBR procedure results a unique set of outcomes. Proposition 2 implies that, if IENBR is well-de ned, the resulting set of outcomes must be unique. The following Proposition 3 asserts that IENBR is indeed a well-de ned order-independent procedure in any general game situation. Proposition 3. R 1 exists and is unique in > G. Moreover, R 1 is nonempty if G has a Nash equilibrium. Proof. First, we show that R 1 exists. For any S 0 S, we de ne the next elimination operation r by r [S 0 ] s 2 S 0 j 9i s.t. s i =2 BR (t i ) for any t i 2 > G i (S 0 ). By the well-ordering principle, the power set of S can be well ordered by a linear order; cf., e.g., Aliprantis and Border (2006, Chapter 1). The existence of a maximal reduction using IENBR is assured by the following prominent fast IENBR: R 1 \ 2 R satisfying R +1 = R nr R (and R = \ 0 <R 0 for a limit ordinal ), where is an ordinal that is order-isomorphic to S. Note that r R =? implies r R 0 =? for all 0 >. By using the fact that a set is never isomorphic to its 9

10 power set (cf., e.g., Suppes, 1972, Theorem 23), it is easily veri ed that R 1! R 0 only for R 1 = R 0. Second, by Proposition 2, R 1 = R for any IENBR procedure fr g 2. Thus, R 1 is unique for any arbitrary model > G. Let s be a Nash equilibrium in G. Since s i is a best response to i (s ) for every player i, by C2, fs g is a rationalizable set in > G. By Proposition 2, s 2 R 1. 4 Nash equilibrium and rationalizability Proposition 3 shows that every Nash equilibrium survives IENBR and, hence, every Nash equilibrium is a rationalizable strategy pro le. However, the following example taken from Chen et al. (2007) demonstrates that a Nash equilibrium in the reduced game after an IENBR procedure may be a spurious Nash equilibrium, i.e., it is not a Nash equilibrium in the original game. Example 2. Consider a two-person symmetric game: G N; fs i g i2n ; fu i g i2n, where N = f1; 2g, S 1 = S 2 = [0; 1], and for all s i ; s j 2 [0; 1], i; j = 1; 2, and i 6= j 8 < 1, if s i 2 [1=2; 1] and s j 2 [1=2; 1], u i (s i ; s j ) = 1 + s i, if s i 2 [0; 1=2) and s j 2 (2=3; 5=6), : s i, otherwise. It is easily veri ed that R 1 = [1=2; 1] [1=2; 1] since every strategy s i 2 [0; 1=2) is strictly dominated and hence never a best response. That is, IENBR leaves the reduced game Gj R 1 N; fri 1 g i2n ; fu i j R 1g i2n that cannot be further reduced. Clearly, R 1 is the set of Nash equilibria in the reduced game Gj R 1, but the set of Nash equilibria in game G is fs 2 R 1 j s 1 ; s 2 =2 (2=3; 5=6)g. Thus, IENBR generates spurious Nash equilibria s 2 R 1 where some s i 2 (2=3; 5=6). The game of this example is in the class of Reny s (1999) better-reply secure games. Observe that in this game, u i (:; s j ) has a maximizer for s j =2 (2=3; 5=6), but u i (:; s j ) has no maximizer for s j 2 (2=3; 5=6). That is, while each player has a best response in a non spurious Nash equilibrium, some player has no best response in a spurious Nash equilibrium. For any subset S 0 S, we say that G (N; fs i g i2n ; fu i g i2n ) has well-de ned best responses on S 0 if 8i and 8s 2 S 0, BR ( i (s)) 6=?. Let NE denote the 10

11 set of Nash equilibria in G, and NEj R 1 the set of Nash equilibria in the reduced game Gj R 1 (N; fri 1 g i2n ; fu i j R 1g i2n ). A su cient and necessary condition under which rationalizability generates no spurious Nash equilibria is provided as follows. Proposition 4. NE = NEj R 1 i G has well-de ned best responses on NEj R 1. Proof. ( Only if part.) Let s 2 NEj R 1. Since NEj R 1 = NE, s i 2 BR ( i (s )) 8i. Thus, BR ( i (s )) 6=? for all i. ( If part.) (i) Let s 2 NE. By Proposition 3, s 2 R 1 and, hence, s 2 NEj R 1. So NE NEj R 1: (ii) Let s 2 NEj R 1. Since G has well-de ned best responses on NEj R 1, for every player i there exists some ^s i 2 S i such that ^s i 2 BR ( i (s )), which implies that ^s i % i (s ) s i and (^s i ; s i) 2 R 1. Since s 2 NEj R 1, s i % i (s ) ^s i. Therefore, s i i (s ) ^s i and, hence, s i 2 BR ( i (s )). That is, s 2 NE. So NEj R 1 NE. This su cient and necessary condition in Proposition 4 does not involve any topological assumption on the original or the reduced games. In Chen et al. s (2007) Corollary 4, some classes of games with special topological structures were proved to preserve Nash equilibria for the iterated elimination of strictly dominated strategies. These results are also applicable to the IENBR procedure de ned in this paper. If a game is solvable by an IENBR procedure, the following corollary asserts that the unique strategy pro le surviving the procedure is the only Nash equilibrium. Corollary 1. NE = R 1 if jr 1 j = 1. Proof. Let R 1 = fs g. By C2, s i is a best response to i (s ) for every player i. So s 2 NE and hence R 1 NE. By Proposition 5, NE R 1. 5 Epistemic conditions of rationalizability In this section we provide epistemic conditions for rationalizability in general games. In doing epistemic analysis, we need to extend the model of type in Section 2 to the space of states. Consider a space of states, with typical element! 2. A subset 11

12 E is referred to as an event. A model of type on is given by > f> i ()g i2n ; where > i () is de ned over (nonempty) subsets E. The set > i (E) is player i s type space for given event E, which can be interpreted as player i s type space conditional on event E; each type t i 2 > i (E) has a preference relation % ti de ned on player i s strategies in S i under which the complement of E is regarded as impossible. For example, if > i (E) is applied to the case of the probability measure space, > i (E) can be considered as the space of probability measures conditional on subset E of. The model of type on can also be viewed as a type structure used in epistemic game theory to model interactive beliefs in which a type of a player is a joint belief about the states of nature and the types of the other players (see, e.g., Brandenburger (2007)). An epistemic model for game G is de ned by M (G) (; >; fs i g i2n ; ft i g i2n ) ; where is the space of states, > is a model of type on, s i (!) 2 S i is player i s strategy at state!, and t i (!) 2 > i () is player i s type at state!; cf., e.g., Aumann (1999) and Osborne and Rubinstein (1994, Chapter 5). 7 Denote by s (!) the strategy pro le at! and let S E fs (!) j! 2 Eg. For our purpose, we need the following two conditions for the model of type on. C3 (Continuity) For any sequence of events E l 1 l=1 \1 l=1 > i E l > i \ 1 l=1 El 8i. 7 We take a point of view that an epistemic model is a pragmatic and convenient framework to be used for doing epistemic analysis; cf. Aumann and Brandenburger (1995, Sec. 7(a)) for related discussions. Mertens and Zamir (1985) constructed a well-de ned compact Hausdor space of types in a probabilistic setting, and Epstein and Wang (1996) constructed a well-de ned compact Hausdor space of types in a more general setting of regular preferences; see also Epstein (1997) for various models of preference. In this paper, we are mainly concerned with the analysis of the game-theoretic solution concept of rationalizability in general situations. In particular, we do not assume that preferences have utility function representations. 12

13 C4 (Consistency) For any event E, > i (E) = > G i (S E ) 8i. The continuity condition C3 requires that the intersection of type spaces on a sequence of events is included in the type space on the intersection of the sequence of events. The condition is related to the property of information and knowledge structure termed limit closure in Fagin et al. (1999), which re ects the idea that what happens at nite levels determines what happens at the limit. This continuity condition C3 allows us to consider only a countably in nite number of levels of knowledge. 8 The consistency condition C4 requires that the type space on an event be consistent with the type space on the strategies projected from the event and, thus, each player behaves in a natural way with respect to the marginalization. This consistency requirement is much in the same spirit of the notion of coherence imposed in the analysis of hierarchy of beliefs and preferences (see, e.g., Brandenburger and Dekel (1993) and Epstein and Wang (1996)). These conditions are satis ed by many of type models discussed in the literature, e.g., (countably additive) probability measure spaces and regular preference models. We say player i knows an event E at! if t i (!) 2 > i (E) since the complement of E is regarded as impossible under t i (!) 2 > i (E). 9 Let K i E f! 2 j i knows E at!g. For simplicity, we assume the knowledge operator satis es the axiom of knowledge, i.e., K i E E. An event E is called a self-evident event, if E = K E. De ne the event E is mutual knowledge as: KE \ i2n K i E; and the event E is common knowledge as: CKE \ 1 l=1k l E 8 See, e.g., Fagin et al. (1999) and Heifetz and Samet (1998) for extensive discussions. 9 This formalism can be easily applied to Aumann s de nition of knowledge by the possibility correspondence in a semantic framework and the notion of belief with probability one in a probabilistic model, for instance. 13

14 where K 1 E = KE and K l E = K K l 1 E for l 2. The following Lemma 1 shows some useful properties about the knowledge operator K and the common knowledge operator CK. It is easy to see that these properties are satis ed by the standard semantic model of knowledge with partitional information structures. Lemma 1. The operators K and CK satisfy the following properties: (1) E F ) KE KF. (2) \ 1 l=1 KEl = K \ 1 l=1 El. (3)! 2 CKE i! 2 E for some self-evident E E. Proof. (1) Let! 2 KE. Then t i (!) 2 > i (E) 8i. If E F, by C1 and C4, > i (E) > i (F ). Hence t i (!) 2 > i (F ) 8i, i.e.,! 2 KF. (2) By Lemma 1.1, it su ces to show \ 1 l=1 KEl K \ 1 l=1 El. Let! 2 \ 1 l=1 KEl. Then! 2 KE l for all l 1 i, for all i, t i (!) 2 > i E l for all l 1 i, for all i, t i (!) 2 \ 1 l=1 > i E l. By C3, t i (!) 2 > i \ 1 l=1 El for all i, i.e.,! 2 K \ 1 l=1 El. (3) Only if part: Let! 2 CKE, and let E = CKE. We show that by KE E and Lemma 1.1, K l+1 E K l E for all l 1. By Lemma 1.2, K (CKE) = K \ 1 l=1 Kl E = \ 1 l=2 Kl E = CKE. Then E is self-evident and! 2 E. If part: Let! 2 E = K E E. By Lemma 1.1, K l+1 E = K l E K l E for all l 1. So! 2 E \ 1 l=1 Kl E = CKE. Say player i is rational at! if s i (!) is a best response to t i (!). Let R i f! 2 j i is rational at!g and R \ i2n R i. That is, R is the event everyone is rational. The following Proposition 5 provides the epistemic conditions for rationalizability; it shows that rationalizability is the strategic implication of common knowledge of rationality. Proposition 5. For any epistemic model, S CKR R. Moreover, there is an epistemic model such that S CKR = R. 14

15 Proof. Let s 2 S CKR. Then there exists! 2 CKR such that s (!) = s. By Lemma 1.3,! 2 R for some self-evident event R R. Therefore, for any! 0 2 R, s i (! 0 ) 2 BR (t i (! 0 )) and t i (! 0 ) 2 > i R for all i. By C4, t i (! 0 ) 2 > G i S R!. Thus, S R is rationalizable and hence s 2 S R R. De ne R. For any! = fs i g i2n 2, for every player i de ne s i (!) = s i and t i (!) = t i 2 > G i (R ) such that s i 2 BR (t i ). Clearly, every player i is rational across states in. By C4, K. Therefore, = CKR and, hence, S CKR = R. Within the standard expected utility framework in nite games, Brandenburger and Dekel (1987) o ered the notion of a posteriori equilibrium, a strengthening of Aumann s (1974) notion of subjective correlated equilibrium, and showed an equivalence between rationalizability and a posteriori equilibrium. The equivalence implies that the assumption of common knowledge of rationality also provides a formal epistemic justi cation for this equilibrium notion. In nite models, Epstein (1997) extended this equivalence result to more general regular preferences including the subjective expected utility model. We end this section by presenting such an equivalence result for arbitrary games with various modes of behavior in the analytical framework used in this paper. A strategy-pro le speci cation function s :! S in an epistemic model M (G) for game G is said to be an a posteriori equilibrium in M (G) if for every player i 2 N, 10 i.e., s i (!) 2 BR (t i (!)). 8! 2, s i (!) % ti (!) s i 8s i 2 S i, Proposition 6. The strategy pro le s is rationalizable in > G if and only if there 10 This de nition of a posteriori equilibrium does not involve an exogenous informational partition for each player as in Bandenburger and Dekel (1987) and Epstein (1997). However, an endogenous (possibly non-partitional) informational structure for each player can be elicited from the player s preference relation at states; cf. Morris (1996) and Chen and Luo (2011b) for more discussions. 15

16 exist an epistemic model M (G) and an a posteriori equilibrium s in M (G) such that s = s (!) for some! 2. Proof. if part: Let s be an a posteriori equilibrium in a model M (G). Then, for every player i and every s 2 S, s i 2 BR(t i ) for some t i 2 > i (). By C4, for every player i and every s 2 S, s i 2 BR(t i ) for some t i 2 > G i (S ). That is, the set S is a rationalizable set in > G. Thus, the pro le s is rationalizable in > G if s = s (!) for some! 2. Only if part: Let R be the largest rationalizable set in > G. Then, for every player i and every s 2 R, there is t i 2 > G i (R ) such that s i 2 BR(t i ). De ne an epistemic model for G : such that M (G) (; >; fs i g i2n ; ft i g i2n ) ; (i) = (s i ; t i ) i2n j t i 2 > G i (R ) and s i 2 BR(t i ) ; (ii) 8i, > i (E) = > G i (S E ) if E ; (iii) 8i, s i (!) = s i and t i (!) = t i if! = (s i ; t i ) i2n. Therefore, for every player i and every! = (s i ; t i ) i2n in, s i (!) 2 BR (t i (!)). That is, s is an a posteriori equilibrium in M (G). Thus, for each rationalizable pro le s in > G, we can nd an a posteriori equilibrium s in M (G) and a state! = (s i ; t i ) i2n in such that s = s (! ). 6 Rationalizability and iterated strict dominance In this section, we show that, through examples, rationalizability neither implies nor is implied by iterated strict dominance used in Chen et al. (2007) in general game situations. This is because in the general environments an undominated strategy need not be a best response in a model of type, and conversely, a best response in a model of type is not necessarily undominated Chen and Luo (2011a) show that if the sets of strategies are compact Hausdor spaces and the payo functions are continuous and satisfy a condition called concave-like, then a strategy is undominated if and only if it is a best response in the subjective expected utility preference model. This class of games covers the mixed extensions of nite games and the Cournot oligopoly. 16

17 Recall that a strategy s i 2 S i is said to be dominated given S 0 S if for some strategy s 0 i 2 S i, u i (s 0 i; s i ) > u i (s i ; s i ) for all s i 2 S 0 i. For any subsets S 0 ; S 00 S where S 00 S 0, we use the notation S 0 7! S 00 to signify that for any s 2 S 0 ns 00, some s i is dominated given S 0. Let 0 denote the rst element in an ordinal, and let + 1 denote the successor to in. De nition 3. An iterated elimination of strictly dominated strategies (IESDS o ) is de ned as a nite, countably in nite, or uncountably in nite family nd 2 such that D 0 = S (and D = \ 0 <D 0 for a limit ordinal ), D 7! D +1, and D \ 2 D 7! D 0 only for D 0 = D. The following Example 3 (due to Andrew Postlewaite), which appears in Bergemann and Morris (2005a, Footnote 8), shows that an undominated strategy need not be a best response in a model of type. 12 Example 3. Consider a two-person symmetric game G = N; fs i g i2n ; fu i g i2n, where N = f1; 2g, and for i = 1; 2, S i = f0; 1; 2; :::g and 8 < 1, if s i = 0; u i (s i ; s i ) = 2, if s i 1 and s i > s i ; : 0, if s i 1 and s i s i. Let > G be the model of type generated by expected utility preferences with (countably additive) probability measures, i.e., > G i (S 0 ) = S 0 i for all S 0 S. Clearly, s i = 0 is not strictly dominated given S, because for any s i 1, u i (0; s i ) = 1 > 0 = u i (s i ; s i ) for s i s i. But, s i = 0 is not a best response in > G. To see this, note that for any i 2 (S i ) and any s i > 0, Z u i (s i ; s i ) d i (s i ) = 2 i (fs i js i < s i g)! 2 as s i! 1. Hence, there is some s i such that R u i (s i ; s i ) d i (s i ) > 1 = R u i (0; s i ) d i (s i ). Since no strategy is strictly dominated given S, D = S. But, since s i = 0 is not a best 12 Here we follow Chen et al. (2007) in eliminating only strategies that are dominanted by pure strategies. The claims in these two examples hold even if we consider eliminating strategies which are dominated by mixed strategies. 17

18 response in > G, 0 =2 R 1 i. Then, it follows that 1 =2 BR (t i ) for any t i 2 > G i (Sn (0; 0)) since given Sn (0; 0), s i = 0 always yields the payo 1 while s i = 1 always yields the payo 0. Hence, 1 =2 R 2 i. Similarly, 2 =2 R 3 i, and 3 =2 R 4 i, and so on. Thus, R 1 =?. The following Example 4, which is modi ed from Stinchcombe (1997), shows that a strictly dominated strategy can be a best response in a model of type. 13 Example 4. Consider a two-person game G = N; fs i g i2n ; fu i g i2n, where N = f1; 2g, S 1 = f0; 1g, S 2 = f0; 1; 2; :::g, and u 1 (0; s 2 ) = 0, u 1 (1; s 2 ) = 1=s 2, u 2 (s 2 ; 0) = 0, and u 2 (s 2 ; 1) = s 2 for all s 2 2 S 2. Let the algebra on S 2 be the power set of S 2. Let > G be the model of type generated by expected utility preferences generated by nitely additive probability charges, i.e., > G i (S 0 ) = ba S 0 i for all S 0 S, where ba S 0 i is the space of nitely additive probability charges on S 0 i. Clearly, s 1 = 0 is strictly dominated by s 1 = 1. However, s 1 = 0 is a best response in > G. To see this, it su ces to show that R u 1 (1; s 2 ) d (s 2 ) = R u 1 (0; s 2 ) d (s 2 ) = 0 for some 2 ba (S 2 ). To nd such, let m 2 ba (S 2 ) be the uniform distribution on f1; 2; :::; mg. By Alaoglu s Theorem (see Royden (1968, Theorem 10.17)), there are 2 ba (S 2 ) and a sequence of f m g such that lim m (E) = (E) for each E N: (1) m!1 Since u 1 (0; s 2 ) = 0 for all s 2, it follows that R u 1 (0; s 2 ) d (s 2 ) = 0. R u1 (1; s 2 ) d (s 2 ) = 0, observe that for every K 1, To see Z 0 KX u 1 (1; s 2 ) d (s 2 ) (fs 2 g) + 1 K (fs 2 : s 2 > Kg). (2) s 2 =1 Since m (fs 2 g)! 0 and m (fs 2 : s 2 > Kg)! 1, it follows from (1) that (fs 2 g) = 0 and (fs 2 : s 2 > Kg) = 1. Since (2) holds for all K 1, R u 1 (1; s 2 ) d (s 2 ) = 0. In this game, D =? (since 0 is strictly dominated given S and every s 2 is strictly dominated given f1g S 2 ) and R 1 = S (since s 1 = 0 is a best response in > G (S) and every s 2 is a best response to t 2 2 > G 2 (S) which assigns probability one to s 1 = 0). 13 A similar example can also be found in Adams (1962), Seidenfeld and Schervish (1983), and Wakker (1993). 18

19 7 Concluding remarks In this paper, we have presented a simple and useful framework for analyzing rationalizable strategic behavior in general environments i.e., arbitrary strategic games with various modes of behavior. We introduce the model of type to de ne the notion of rationalizability in games with (in) nite players, arbitrary strategy spaces, and arbitrary payo functions. We have investigated properties about rationalizability in general games and shown that the notion of rationalizability possesses nice properties similar to those in nite games. More speci cally, under mild conditions C1-4 on the analytical framework, we have shown the following results on rationalizability in general situations: 1. The union of all the rationalizable sets is the largest rationalizable set in product form (Proposition 1). 2. The largest rationalizable set can be derived by the (possibly trans - nite) iterated elimination process of IENBR (Proposition 2). IENBR is a well-de ned order-independent procedure (Proposition 3). 3. The notion of rationalizability is characterized by common knowledge of rationality (Proposition 5). 4. The notion of rationalizability is equivalent to the notion of a posteriori equilibrium (Proposition 6). Our approach in this paper is completely topology-free, and is applicable to any arbitrary strategic game. We have demonstrated that, through examples, rationalizability is in general not equivalent to iterated strict dominance in general game situations. 14 In this paper, we have also investigated the relationship between rationalizability and Nash equilibrium in general games. While every Nash equilibrium survives the IENBR procedure, a Nash equilibrium in the nal reduced game after IENBR may fail to be a Nash equilibrium in the original game. That is, the IENBR procedure 14 Chen and Luo (2011a) showed that rationalizability under general preferences can be indistinguishable from the outcome of the IESDS procedure for a class of (in) nite games. 19

20 may generate spurious Nash equilibria in in nite games. We have provided a su - cient and necessary condition to guarantee no spurious Nash equilibria (Proposition 4); in particular, in dominance-solvable games, the set of Nash equilibria can be solved by IENBR (Corollary 1). We would like to emphasize that one important feature of this paper is that the framework allows the players to have various preferences which include the subjective expected utility as a special case. Throughout this paper, we do not need to make any kind of behavioral assumptions on players preference relations in the framework; in particular, we do not assume that preferences have utility function representations. The general analysis of this paper is applicable to any strategic game with various modes of behavior. To close this paper, we would like to point out some possible extensions of this paper for future research. In the ( nite) Bayesian game model, Dekel et al. (2007) o ered the notion of interim correlated rationalizability, and Morris and Takahashi (2011) examined a variant of preference-correlated rationalizability. We note that the framework presented in this paper can be used for a general analysis of rationalizability in incomplete-information environments. For example, we can recast Dekel et al. s (2007) notion of interim correlated rationalizability by using a framework in which each type of a player is viewed as an independent agent-player and the model of type for each agent-player speci es a subjective probability measure space for each product set of the payo -relevant states, the other players types, and possible restriction of action pro les, where each probability distribution over states and actions is consistent with the type s a prior belief about the payo -relevant states and the other players types. This framework can also be used for analyzing interim independent rationalizability and preference-correlated rationalizability. The extension of this paper to general games with incomplete information is an important subject for further research. In nite strategic games, Ambrus (2006) and Luo and Yang (2009) o ered the notion of (Bayesian) coalitional rationalizability in complex social interactions. The extension of this paper to permit social and coalitional interactions in general situations is an intriguing topic worth further investigation. The exploration of the notion of extensive-form rationalizability in general dynamic games also remains to be an important research topic for further study. 20

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