Bayesian coalitional rationality and rationalizability

Transcription

1 Bayesian coalitional rationality and rationalizability Xiao Luo a, Yongchuan Qiao a, Chih-Chun Yang b a Department of Economics, National University of Singapore, Singapore b Institute of Economics, Academia Sinica, Taipei 115, Taiwan August 2017 Abstract We offer an epistemic notion of Bayesian coalitional rationality i.e., Bayesian c-rationality) in strategic environments by a mode of behavior that no group of players wishes to change. In a semantic framework in which each player is endowed with a CPS belief, we characterize the game-theoretic solution concept of Bayesian coalitional rationalizability i.e., Bayesian c-rationalizability) in [31] by means of common knowledge of Bayesian c-rationality. We also formulate and show Bayesian c-rationalizability is outcome equivalent to a coalitional version of a posteriori equilibrium. Our analysis provides the epistemic foundation of the solution concept of Bayesian c-rationalizability. JEL Classification: C70, C72, D81. Keywords: Bayesian c-rationality, Bayesian c-rationalizability, common knowledge, a posteriori Bayesian c-equilibrium, conditional probability systems CPS) This paper is based on part of an earlier manuscript Bayesian Coalitional Rationalizability. We thank Adam Brandenburger, Yi-Chun Chen, Yossi Greenberg, Takashi Kunimoto, Shravan Luckraz, Xuewen Qian, Chen Qu, Yang Sun, Satoru Takahashi, Licun Xue, Junjian Yi, Shmuel Zamir, Shenghao Zhu, and seminar participants at National University of Singapore and Academia Sinica. The earlier version of the paper was presented at International Conference on Game Theory, SAET Conference on Current Trends in Economics, and China Meeting of the Econometric Society. Financial supports from National University of Singapore and National Science Council of Taiwan are gratefully acknowledged. The usual disclaimer applies. Corresponding author. Fax: ecslx@nus.edu.sg X. Luo); qiaoyongchuan@ gmail.com Y. Qiao); cyang16@econ.sinica.edu.tw C.C. Yang). 1

2 1 Introduction In their seminal papers, Bernheim [13] and Pearce [39] proposed the game-theoretic solution concept of rationalizability as the logical implication of common knowledge of Bayesian rationality cf. also Tan and Werlang [42]). The notion of Bayesian rationality requires a player to adopt a strategy that maximizes his expected utility under his subjective probabilistic belief about the other players strategies; that is, it reflects a simple idea that each individual player, in a strategic situation, makes his own optimal decision in the absence of coalition considerations. Notably, in a standard textbook, Osborne and Rubinstein [38, Section 5.4] presented an epistemic characterization of rationalizability by common knowledge of Bayesian rationality within a semantic framework. The collective and coalitional behavior has been recognized as an important issue in reallife examples such as cartels, trade blocs, political party formation, special interest groups, public goods provision, and so on. To cope with coalitional aspects of strategic interactions, Ambrus [2] first offered a well-defined solution concept of coalitional rationalizability by using an iterative procedure of restrictions, in which members of a coalition are willing to confine their play to a subset of their strategies if doing so is in their mutual interest; see, for example, Acemoglu et al. [1], Ambrus and Argenziano [4], Jullien [27], Newton [36], and Grandjean et al. [25] for interesting applications. 1 This concept is based on the crucial and novel idea that the players go through an internal reasoning procedure : The coalitional agreements players can consider in this context take the form of restrictions of the strategy space. This means that players look for agreements to avoid certain strategies, without specifying play within the set of non-excluded strategies.... A restriction is supported if every group member always for every possible expectation) expects a higher payoff if the agreement is made than if he instead chooses to play a strategy outside the agreement. Ambrus [2, p.904]) 1 Liu et al. [30] adopted the idea of coalitional) rationalizability to define a stable matching with incomplete information, which requires to attain common knowledge among a blocking pair that no profitable pairwise deviation exists. Their analysis, however, refrains from considering deviations by larger groups of agents. Kobayashi [28] made use of the similar idea to study equilibrium contracts for syndicates with differential information, by assuming an agent declares his intention to join a blocking coalition only when it is common knowledge for the coalition members that he would intend to join. 2

3 Ambrus [2] showed that, in contrast to an equilibrium approach, allowing coalitions to make set-valued restrictions agreements) in the concept of rationalizability runs into no logical inconsistencies; that is, the solution set of coalitionally rationalizable strategies is never empty in the class of finite games. Along the lines of Ambrus s [2] approach, Luo and Yang [31] took a traditional gametheoretic approach to define an alternative solution concept of Bayesian coalitional rationalizability henceforth, Bayesian c-rationalizability) from the Bayesian point of view. The alternative notion extends Ambrus s original one to situations in which, in pursuit of mutually beneficial interests, the players in a coalition i) evaluate their payoff expectations, if an agreement is made, by Bayesian updating, instead of holding fixed the marginal expectations concerning nonmembers as in Ambrus [2], and ii) contemplate various plausible deviations; that is, the validity of deviation is checked not only against restricted subsets of strategies as in Ambrus [2], but also against arbitrary sets of strategies. They showed that all the major features and properties of the conventional rationalizability, as a special case of Bayesian c-rationalizability with the restriction of singleton coalitions only, are essentially preserved. However, what exact epistemic conditions lead to the notion of Bayesian c-rationalizability is a bit subtle and unclear, especially in highly complex coalitional environments. We aim to fill that gap. In this paper, we carry out the epistemic game theory program to formally express the epistemic assumptions behind the solution concept of Bayesian c-rationalizability. More specifically, we provide some epistemic characterization for Bayesian c-rationalizability, by using epistemic states that specify what each player knows and believes about the information, knowledge, and rationality of coalitions. We put forth the notions of Bayesian coalitional rationality henceforth, Bayesian c-rationality) and a posteriori Bayesian c- equilibrium. The first notion reflects the idea of a truism among coalition members: no credible coalitional gains exist; the second notion is a coalitional version of a posteriori equilibrium in Brandenburger and Dekel [15]. The main results of this paper are as follows: 1) Bayesian c-rationalizability is the logical implication of common knowledge of Bayesian c-rationality Theorem 1); 2) Bayesian c-rationalizability is outcome equivalent to a posteriori Bayesian c-equilibrium Theorem 2). Our study reveals the concept of Bayesian c-rationalizability is epistemically a natural extension of rationalizability in the presence of coalitional reasoning. 3

4 From an epistemic perspective, a player, in reasoning about what other players would do in a game, has to figure out what would happen in various alternative hypothetical circumstances. He anticipates other individuals reasonings, which in return may further complicate his own reasoning. This sort of hypothetical reasoning becomes more complex when a player takes into account what a group of players would do in a game and what the group would contemplate in this situation. The players in the group have to consider what the causal effects would be of alternative choices the players might make, what other players might think about the potential causal effects, and how their alternative possible actions might affect beliefs and behavior of the players in the group. 2 To deal with hypothetical coalitional reasoning in a Bayesian paradigm, we carry out our epistemic analysis in an analytical framework in which each player has a conditional probability system CPS) belief. Accordingly, a CPS belief specifies the family of probabilistic beliefs about the opponents behavior in all possible contingencies; thus, even in the case of an unexpected or hypothetical event, the player must have a probabilistic assessment of opponents strategies contingent on that event see, e.g., Myerson [34, 35]). One novel ingredient of our approach is the definition of Bayesian c-rationality Definition 2), which is the coalitional counterpart of Bayesian individual) rationality. In the context of strategic decisions, Bayesian rationality requires a player to play an optimal strategy with respect to the player s probabilistic belief about his opponents behavior. In other words, Bayesian rationality is referred to an epistemic state in which a player attentively contemplates every hypothetical circumstance and, under no circumstance, does an alternative choice of strategy attain a higher expected payoff. The coalitional version of Bayesian rationality is designed to capture the similar idea for coalitional reasoning: coalition members share a common knowledge that improving every coalition member s expected payoff by coordinating their movements in a credible way is impossible. In fact, our definition of Bayesian c-rationality extends Bayesian rationality to strategic games that incorporate coalitional reasoning about joint actions Proposition 1). We would like to emphasize that it is epistemically important and necessary to consider not only what a 2 In the classic article The Use of Knowledge in Society, Friedrich Hayek 1945, p.530) pointed out that we must show how a solution is produced by the interactions of people each of whom possesses only partial knowledge. To assume all the knowledge to be given to a single mind in the same manner in which we assume it to be given to us as the explaining economists is to assume the problem away and to disregard everything that it is important and significant in the real world. 4

5 coalition member knows about the contemplation of joint movements, but also what the player knows about what the other coalition members know and do not know. 3 The formalism of Bayesian c-rationality requires that no coalition would be willing to confine its members play to a set of strategies, which excludes some of their using strategies at state. This kind of requirement ensures no mutually beneficial deviation exists. Accordingly, it needs to check 2 n 1 feasible coalitions i.e., all subsets of n players except the empty set) and, for each coalition J, Π j J 2 S j 1 ) possible coalitional deviations i.e., all restrictions of strategies in S j for each coalition member j J except the empty set). Due to the enormous complexity of coalitional reasoning, exploring how to make such epistemic states possible in strategic environments is especially intriguing and profound. Our study contributes to a better understanding of the concept of coalitional rationality in strategic environments in this regard. Our work is closely related to Ambrus [3] and provides complementary support to his approach. Ambrus [3] suggested an alternative epistemic definition of coalitional rationalizability by rationality and common certainty that every coalition is γ-rational, where γ is a sensible best) response operator from sets of conjectures) strategies to sets of strategies. The approach is, however, less comprehensive from an epistemic perspective, because his formalism is not completely based on primitive assumptions about players beliefs and behavior; that is, the concept of γ-rationality is defined by using the obscure operator γ A, J), which is epistemically open and abstruse. By contrast, our definition of Bayesian c-rationality uncovers all primitive assumptions at epistemic states about players beliefs and behavior in the presence of complex coalitional reasoning. That is, our definition provides an expressible epistemic description for the internal coalitional reasoning procedure in the notion of coalitional rationalizability, in terms of what the players in coalitions know and believe about each other s strategies, knowledge, and beliefs. As we emphasize, this paper aims to provide some thorough and expressible epistemic characterization of the notion 3 Actions undertaken deliberately by a group of players are related to some particular epistemic state, which often interferes with the other states under consideration of players in the group. The salient interactive epistemological feature makes the important difference between intentional actions and arbitrary actions performed by individuals. To coordinate its interrelated actions, a group of players needs to have recourse to common knowledge see, e.g., Lewis [29], Chwe [19], and Rubinstein [40]). Each group member is willing to do so only if other group members are willing to do so. Members need to have knowledge of each other, knowledge of that knowledge, knowledge of the knowledge of that knowledge, and so on. See also Halpern and Moses [23] and Fagin et al. [22, Chapter 6] for extensive discussions on reasoning about the states of knowledge of the components of a distributed system. In particular, they show common knowledge plays a critical role in reaching agreement and coordinating actions in a distributed environment. 5

6 of Bayesian c-rationalizability, which is a coalitional counterpart of epistemic characterization for rationalizability in a semantic framework see, e.g., Osborne and Rubinstein [38]). Luo and Yang [31, Proposition 1] showed the set of Bayesian c-rationalizable strategies can be derived from an iterative procedure of restrictions to c-best response strategies. Therefore, Bayesian c-rationalizability can be viewed as a specific form of γ-rationalizablility defined by Ambrus [3] from this perspective. In this paper, the concept of Bayesian c- rationality is defined by taking into account all possible arrangements of strategies by coalitions. Consequently, this paper offers an alternative and complementary epistemic characterization of the fixed-point version of coalitional rationalizability by common knowledge of coalitional rationality. Our approach also provides additional epistemic rationale for defining coalitional rationalizability directly by an iterative procedure in Ambrus [2, 3]. To obtain Theorem 1, we need to consider a weakly belief-complete framework, which contains, in a certain sense, all plausible CPS beliefs in an epistemic model. This kind of richness condition for beliefs in the analytical framework is commonly used for the epistemic analysis of game-theoretic solution concepts, for example, extensive-form rationalizability in a complete CPS type structure model in Battigalli and Siniscalchi [12] and iterated weak dominance in a complete lexicographic conditional probability system LCPS) type structure model in Brandenburger et al. [17] cf. also Brandenburger [14] and Dekel and Siniscalchi [20] for extensive discussions). 4 The rest of this paper is organized as follows. Section 2 provides an example to informally illustrate the main idea and results in this paper. Section 3 introduces the preliminary notation and definitions. Section 4 defines the concept of Bayesian c-rationality. Section 5 presents main results. Section 6 offers concluding remarks. To facilitate reading, all the proofs are relegated to the Appendix. 4 Ambrus s [3] framework is different from the belief-based epistemic model we adopt here. Although this kind of completeness is crucial to the belief-based approach, an analogous concept does not appear to be presented in Ambrus s framework because the operator γ A, J) is belief free). As Ambrus [3, pp ] pointed out, Although coalitional response operators are defined as operators from sets of strategies... one should interpret them as operators from sets of belief profiles. In particular, the interpretation of γ A, J)... is that it is the set of strategies that players in J want to restrict play to if it is common certainty among J that play is inside A. Therefore, set of strategies A in the above definition should be thought of as a shortcut for the set of belief profiles for which all players in J think that it is common certainty among J that play is in A. In the leading example of a coalitional response operator in Ambrus [3], the operator γ implicitly requires a sort of belief-richness condition: a player must consider all possible conjectures, each of which supports a best response strategy excluded from the restricted set of strategies). 6

7 2 An illustrative example Example 1. Consider the following two-person symmetric game where the first player picks a row and the second player picks a column): a b c a 2, 2 3, 2 0, 0 b 2, 3 3, 3 0, 0 c 0, 0 0, 0 1, 1 Intuitively, confining the players play to a subset of strategies {a, b} {a, b} which is also called an agreement ) is in their mutual interest. That is, each player always expects a higher payoff if this kind of nonbinding agreement is made. The notion of Bayesian c-rationalizability yields the outcome set {a, b} {a, b}. Next, we provide an epistemic model for this game. Consider a four-state space Ω {ω aa, ω ab, ω ba, ω bb } with a complete information structure. For example, at state ω ab, player 1 plays strategy a and player 2 plays strategy b; each player holds the correct belief about the opponent s using strategy. A state is said to be Bayesian c-rational if no credible) mutually beneficial agreement can exclude using strategies at the state. There are three Bayesian c-rational states ω ab, ω ba, and ω bb in this model. Intuitively, state ω aa is not Bayesian c-rational, because the two players would be willing to jointly play the strategy profile b, b) instead of a, a) at ω aa.) Because this epistemic model has a complete information structure, Bayesian c-rationality is commonly known at states ω ab, ω ba, and ω bb. In Theorem 1a), we show players must play a Bayesian c-rationalizable strategy profile under common knowledge of Bayesian c-rationality, in any weakly belief-complete model. The Bayesian c-rationalizable strategy profile a, a) cannot be attained under common knowledge of Bayesian c-rationality in the above epistemic model. We can, however, find an epistemic model with a different information structure such that common knowledge of Bayesian c-rationality attains all the Bayesian c-rationalizable strategy profiles. For this purpose, we may consider an alternative model with an incomplete information structure showing that each player cannot distinguish the opponent s using strategies at states in Ω. Each player believes with probability 1 that the opponent is playing b. In this new model, ω aa becomes a Bayesian c-rational state because every player at this state can expect to obtain the highest payoff of 3; hence, the strategy profile a, a) can be attained by common knowledge of Bayesian c-rationality. In Theorem 1b), we show every Bayesian c-rationalizable strategy profile can be attained by common knowledge of. 7

8 Bayesian c-rationality. In this paper, we also follow Brandenburger and Dekel [15] to establish an equivalence result between Bayesian c-rationalizability and a posteriori Bayesian c-equilibrium Theorem 2). 3 Preliminaries Consider a finite game: G I, {S i } i I, {u i } i I ), where I is a nonempty) finite set of players, S i is a nonempty) finite set of i s pure strategies, and u i : S i I S i R is i s payoff function. We say J is a coalition if J is a nonempty subset of I. For coalition J I, let S J j J S j, S J i/ J S i, and S i j S i. For any s i S i and any probability distribution µ over S i, define u i s i, µ) s i S i µ s i ) u i s i, s i ). 3.1 Bayesian c-rationalizability: definition We adopt the notation in Luo and Yang [31]. Let S ) denote the set of all conditional probability systems CPS) on finite state space S faced by player j see, e.g., Myerson [34, 35]). For nonempty subset A S, let A S ) { µ S ) : µ S A ) = 1 }. For nonempty product subsets A, B S, we say a coalition J AB is a feasible coalition from A to B if B = B JAB A JAB. Definition 1 Luo and Yang [31]). A nonempty product subset R S is a coalitional rationalizable set CRS) if R R only for R = R, where for R, we define R R as: a feasible coalition J RR, j J RR such that 1) [Profitability] r j R j, µ R S ), if r j / R j or µ R µ R, then u j r j, µ R ) < u j s j, µ R ) for some s j S j, and 2) [Credibility] r j R j\r j, there exists µ R S ) such that u j r j, µ R ) u j s j, µ R ) s j S j. 8

9 That is, a CRS is a nonempty) product set of pure strategies from which no group of players would like to make a credible deviation. In particular, r i R i is said to be a Bayesian c-rationalizable strategy for player i. With the restriction of J RR = 1, Definition 1 yields a correlated version of rationalizability see Luo and Yang [31]). 3.2 Aumann s model of knowledge We follow Aumann [7, 6, 8, 9] and Brandenburger and Dekel [15] to consider an epistemic model for game G where each player is allowed to hold a CPS belief): M G) < Ω, {P i )} i I, {µ i )} i I, {s i )} i I >, where Ω is the set of states with typical element ω Ω, P i ω) Ω is i s partitional information structure at ω, µ i ω) S i ) is the CPS belief that i holds at ω, and s i ω) S i is i s using strategy at ω; cf. also Osborne and Rubinstein [38, Chapter 5] for a detailed introduction to Aumann s model of knowledge. Throughout this paper, we require the belief µ i ω) to be consistent with i s knowledge; that is, µ i ω) sp i ω)) S i). Assume, as usual, that i knows his own using strategy and belief; that is, s i ω) = s i ω ) and µ i ω) = µ i ω ) ω P i ω). Let s i ω) denote i s opponents strategy profile at ω Ω, and let sω) s i ω), s i ω)). We refer to a set of states E Ω as an event. Let s E) {sω) : ω E}. For an event E, we take the following standard definitions: K i E {ω Ω : P i ω) E} is the event that i knows E. KE i N K i E is the event that everyone knows E. CKE KE KKE KKKE is the event that E is commonly known. That is, an event is common knowledge if everyone knows it, and everyone knows that everyone knows it, and everyone knows that everyone knows that everyone knows it, and so on ad infinitum. Let P J be the meet of the partitional information structures of coalition J, that is, the finest common coarsening of partitions P j for all j J see, e.g., Aumann [7] and Milgrom [32]). In particular, let P P I denote the meet of all players information structures. It is by now well known that E is commonly known at ω if, and only if, the element of P that contains ω is included in E. That is, ω CKE P ω) E. This equivalent relationship can be easily extended to common knowledge among the players in coalition J. 9

10 We say a model M G) is weakly belief-complete if, for all ω Ω, j J where J is a non-singleton coalition) and µ sp j ω)) S ), there is ω P J ω) such that µ j ω ) = µ. That is, every possible CPS belief µ, which is consistent with player j s knowledge P j ω), can be generated by some state ω in the J-commonly-known subspace P J ω). 5 The weak belief-completeness is a kind of belief-richness condition that requires to contain, in a certain sense, all plausible CPS beliefs in an epistemic model. The completeness requirement for beliefs in an analytical framework is commonly used in the epistemic game theory literature see, e.g., Brandenburger [14] and Dekel and Siniscalchi [20]). Remark 1. Following the line of the epistemic game theory paradigm advocated by Aumann and Brandenburger [10], for the purpose of this paper, we adopt a simple epistemic framework in which each player has a CPS belief cf. Battigalli and Siniscalchi [12] for the construction of a CPS belief universal type space). Like Aumann and Brandenburger s [10] approach, our paper does not deal with the existence of a universal type space; thus, our framework requires no technical assumptions such as topological and measure-theoretic assumptions) on the state space Ω. The framework is rather flexible and applicable to various information structures; it allows for finite and infinite epistemic models discussed in the literature. 6 4 Bayesian c-rationality Consider an epistemic model M G) for game G. We next formulate the concept of Bayesian c-rationality; that is, coalition members share a common knowledge that no credible profitable coalitional deviation changes the behavior of coalition members. 5 Because each player j is assumed to know his own belief in the model M G), j must hold a unique belief across the states in P j ω). Hence, the weak) belief-completeness requirement should not be made for singleton coalitions. 6 Brandenburger and Dekel [16] showed how to transform the types model into a standard information-structure model. They articulated on p.195) that the standard model is in fact no less general than the types model. From this, it follows that the standard model, which is, of course, a simpler construct, can be employed whenever doing so is more convenient. Notable examples by using the model include Aumann s [7] agreement theorem, Aumann s [6, 8] definition of correlated equilibrium, Aumann s [9] epistemic analysis of backward induction, Monderer and Samet s [40] notion of common p-belief, and Rubinstein s [40] game model cf. also Samuelson [41] for more discussions). The informationstructure model is well suited to our epistemic analysis in this paper. 10

11 Definition 2. For state ω Ω and coalition J I, let Aω, J) i I s i P J ω)). Coalition J is Bayesian rational at ω if s ω) B holds true for every product subset B S satisfying B J = A J ω, J) and two conditions C1 and C2 as follows: C1) [Profitability] for all ω P J ω) and j J, u j sj ω ), µ j A ω,j)ω ) ) < u j sj, µ j B ω ) ) for some s j S j whenever s j ω ) / B j or µ j B ω ) µ j A ω,j)ω ); C2) [Credibility] for all j J and b j B j \A j ω, J), there exists ω P J ω) such that u j bj, µ j B ω ) ) u j sj, µ j B ω ) ) s j S j. We call players are Bayesian c-rational at ω if every coalition J is Bayesian rational at ω. That is, coalition J is Bayesian rational at ω requires that coalition members in J share a common knowledge that no credible profitable coalitional deviation, from the initial agreement Aω, J), precludes jointly playing strategies of coalition members at state ω. In other words, a coalition is Bayesian rational at a state if it is common knowledge among coalition members that they play strategies within each response set B of the coalition, to which players in the coalition would be willing to confine their play. 7 Although the set B in Definition 2 can be viewed as a variant of supported restriction by coalition J as in Ambrus s [3, Definition 4] γ-rationality, Bayesian c-rationality and γ-rationality differ in some important aspects. Because the operator γ A, J) contains merely restricted subsets of strategies, the notion of γ-rationality entails an excessive amount of attention from coalition J to a wide range of initial agreements all product supersets of Aω, J) that satisfy closed under rational behavior in Basu and Weibull [11]; that is, it requires players in J to be logically omniscient with respect to a special class of initial agreements hinging on a behavior-complete framework. By contrast, the notion of Bayesian c-rationality simply requires the initial agreement under consideration to be the exact product set Aω, J) = i I s i P J ω)) at state ω. Without the ad hoc requirement of closed under rational behavior, the set Aω, J) represents the finest agreement commonly known by coalition J at state ω. The additional credibility condition C2 requires each incremental strategy b j B j \A j ω, J) be justified by individual rationality. Our formalism of Bayesian c-rationality is explicitly and entirely based on primitive assumptions about players beliefs and behavior at epistemic states e.g., it provides an expressible epistemic description for 7 In this paper, coalition members are not allowed to pool their private information in the noncooperative framework. The epistemic prerequisite for a vital coalitional deviation is that the coalitional members share a common knowledge that the deviation is mutually profitable. This idea has the same spirit of Wilson s [43] concept of coarse core in an exchange economy with asymmetric information. 11

12 the internal coalitional reasoning procedure used in the concept of rationalizability, in terms of what the players in coalitions know and believe about each other s strategies, knowledge, and beliefs). One important feature of the concept of c-rationality is the epistemic precondition of common knowledge among the players of coalition. Interactive knowledge plays a crucial role in all accounts of joint actions by rational players in coalitions; it allows players to explore the mutually beneficial opportunity by joint movements. In Definition 2, the requirement of common knowledge among coalition members is harmonious with the prerequisite for Ambrus s [3] γ-rationality: there is common certainty among J that play is in the initial agreement A. This requirement is also a natural extension of the implicit common-knowledge assumption of standard Bayesian rationality in the context of strategic games involving exclusively individual reasoning. Proposition 1 below establishes a formal relationship between the notions of Bayesian rationality and Bayesian c-rationality, where the former rationality involves exclusively individual reasoning and the latter involves coalitional reasoning. That is, the classical notion of Bayesian rationality is equivalent to the epistemological notion of Bayesian c- rationality with the restriction of singleton coalitions, in the latter notion detailed epistemic assumptions are stated. Proposition 1. With the restriction of J = {j}, J is Bayesian rational at ω if and only if j is Bayesian rational at ω i.e., u j sj ω), µ j S ω) ) u j sj, µ j S ω) ) s j S j. By Proposition 1, the conventional notion of Bayesian rationality actually requires the individual player be self-evidently aware he cannot do better by a replacement of strategies. Subsequently, Bayesian rationality can be alternatively viewed as a special form of Bayesian c-rationality for singleton coalitions. This alternative definition delineates the nature of Bayesian rationality in terms of detailed epistemic assumptions of awareness, common) knowledge, and introspection. 8 8 Intuitively, an individual player j s belief µ j S ω) about the opponents behavior will not be affected when he changes over from his own strategy s j ω) to s j ; thereby, a Bayesian rational player is self-evidently aware he will never do better by using alternative strategies. Nevertheless, the concept of Bayesian c- rationality virtually needs to appeal to the coalition-wise common-knowledge assumption that no credible coalitional gains exist. For nonsingleton coalitions, a coalitional member j s initial belief µ j Aω,J) about the opponents behavior will be typically affected if a new agreement B is made; hence, j s expected payoff is determined by the updated belief µ j B in light of this new information. Because j s belief µ j is private information and not accessible to other coalitional members, the effectiveness of the new agreement relies crucially on the aforementioned common-knowledge assumption. 12

13 5 Epistemic foundation of Bayesian c-rationalizability In this section, we provide suffi cient/necessary epistemic conditions for the solution concept of Bayesian c-rationalizability. Let R {ω Ω : players are Bayesian c-rational at ω}. The following theorem shows Bayesian c-rationalizablity can be regarded as the logical consequence of common knowledge of Bayesian c-rationality. Let R denote the set of Bayesian c-rationalizable strategy profiles in G. Formally, we have ) Theorem 1. a) In every weakly belief-complete model M G), s CKR R. b) ) There is a weakly belief-complete model M G) such that s CKR = R. The following example shows that without imposing the weakly belief-complete condition, common knowledge of Bayesian c-rationality may generate a strategy profile that is not Bayesian c-rationalizable. Example 2. a b c a 3,0 0,3 0,0 b 0,3 3,0 0,0 c 0,0 0,0 1,1 In this two-person game, the notion of Bayesian c-rationalizability results in the outcome set: {a, b} {a, b}. Intuitively, it is in their mutual interest for the two players to confine their play to a subset of strategies {a, b} {a, b} in which each player can guarantee an expected payoff of at less 1.5.) We consider an epistemic model M G) where Ω = {ω aa, ω ab, ω ac,..., ω cc } with a complete-information structure. For i = 1, 2 and typical state ω xy Ω, define P i ω xy ) {ω xy }, s ω xy ) x, y) and µ i ω xy ) {x} {y} {a, b, c}) such that µ 1 {a,b} ω cc ) = µ 2 {a,b} ω cc ) = 1 b. It is easy to check that Bayesian c-rationality is commonly known at ω cc, because the only credible deviation is {b} {a} under this circumstance. But the using strategy profile c, c) at ω cc is not Bayesian c-rationalizable. 13

14 In this model M G), the CPS beliefs of players are rather sparse; no rich beliefs support the grand coalition to make the profitable deviation {a, b} {a, b} from the initial agreement {c} {c}. That is, the model M G) fails to satisfy the weakly belief-complete condition. This weak belief-completeness in Theorem 1a) is mainly for the credibility requirement in the definition of Bayesian c-rationalizablility. Remark 2. Ambrus [3, Definition 5] offered an epistemic definition of coalitional rationalizability by γ-rationality and common certainty that every coalition is γ-rational. Theorem 1 is consistent with his analysis. 9 Ambrus defined the concept of γ-rationality by using the belief-free operator γ although this shortcut operator can be used for one from of sets of conjectures to sets of strategies, for example, the supported-restriction operator γ in Ambrus [3, Definition 3]). In this paper, we define the notion of Bayesian c-rationality in terms of primitive assumptions at epistemic states about what the players in coalitions know and believe about each other s strategies, knowledge, and beliefs. Our analysis of this paper provides a detailed and comprehensive epistemic description for the rationalizable behavior in the presence of complex coalitional reasoning. Observe that in a finite game, a player is playing a best-reply strategy if, and only if, the player has no better-reply strategy instead of his using strategy. With the restriction of singleton coalitions, Definition 1 yields the conventional definition of correlated) rationalizability. As an immediate corollary of Theorem 1 and Proposition 1, we can obtain a simpler characterization for correlated) rationalizability without appealing to the weak belief-completeness. Let R be the set of correlated) rationalizable strategy profiles in G, and let R be the event in which every player is Bayesian rational in M G); that is, R = {ω Ω : players are Bayesian rational at ω}. Corollary 1: a) In any arbitrary model M G), s CKR) R. b) There is a model M G) such that s CKR) = R. 9 For the belief operator B, CKR = R CBR ) see, e.g., Dekel and Siniscalchi [20, Section ]); thus, CBR = B CKR. Because the set of Bayesian c-rationalizable strategy profiles is closed under ) rational behavior, Theorem 1a) implies s R CBR R, where R is the event in Ω that every player is rational. As a consequence, Bayesian c-rationalizablility can alternatively be characterized by rationality and common belief of Bayesian c-rationality. Example 1 in Section 2 shows s ω aa ) R and ω aa R CB R, but ω aa / R CB R. 14

15 Remark 3. In the spirit of Aumann s [5] notion of strong Nash equilibrium, we can obtain a stronger notion of Bayesian c-rationalizablility by removing the credibility requirement in Definition 1. Similarly, we have a stronger notion of Bayesian c-rationality by removing the condition C2 in Definition 2. The weakly belief-complete condition in Theorem 1a) is no longer necessary for this alternative strong Bayesian c-rationalizability. However, like strong Nash equilibrium, the strongly Bayesian c-rationalizable profile may fail to exist. Brandenburger and Dekel [15] proposed the notion of a posteriori equilibrium, a strengthening of Aumann s [6] notion of subjective correlated equilibrium, and showed the equivalence between rationalizability and a posteriori equilibrium see also Epstein [21, Theorem 5.1] and Chen et al. [18, Proposition 6] for the related study under general preferences). The equivalence implies the assumption of common knowledge of rationality provides an epistemic justification for the equilibrium notion. In the spirit of Brandenburger and Dekel [15], we extend this kind of equivalence to complex coalitional interactions. A strategyprofile function s : Ω S is said to be an a posteriori Bayesian c-equilibrium in M G) if, for all ω Ω, players are Bayesian c-rational at ω. Theorem 2. The strategy profile s is Bayesian c-rationalizable in G if, and only if, there exist an epistemic model M G) and an a posteriori Bayesian c-equilibrium s in M G) such that s = sω) for some ω Ω. 6 Concluding remarks The study of how groups of players act in their mutually beneficial interest in social environments is of great importance in economics and social sciences see, e.g., Olson [37]). The analysis of coalitional reasoning is fundamental and profound in game theory and economic theory. From an epistemic perspective, exploring how to make rational states possible in strategic environments involving the distinctive mode of coalitional reasoning is theoretically and conceptually important. Such an epistemic analysis can help in understanding when a particular solution concept is applicable in practical circumstances. Ambrus [3] made an attempt to present an epistemic definition of γ-rationalizablility by rationality and common certainty of coalitional rationality. Along the line of the epistemic game theory program advocated by Aumann and Brandenburger [10], in this paper, we go 15

16 a step further by doing a thorough epistemic analysis of the solution concept of Bayesian c-rationalizability in Luo and Yang [31]. In this paper, we have offered a formal definition of Bayesian c-rationality for collective deliberations in noncooperative games. The definition of Bayesian c-rationality is a purely normative concept, which reflects the idea of a truism among coalition members: no credible coalitional gains exist; that is, a mode of coalitional behavior that it is commonly known among a coalition of players that, when confronted with collective deliberations from the Bayesian point of view, the coalition does not wish to jointly change it. In restricting the size of the coalition to one, the concept of Bayesian c-rationality is consistent with Bayesian rationality. In this paper, we have provided the epistemic characterization of Bayesian c-rationalizability in terms of common knowledge of Bayesian c-rationality and a posteriori Bayesian c-equilibrium. The notion of Bayesian c-rationalizability can thus be viewed as a natural extension of rationalizability in the context of strategic games involving complex coalitional reasoning. Like Aumann [7, 6, 8, 9] and Aumann and Brandenburger [10], we carry out our epistemic analysis within an arbitrary model, including finite and infinite models discussed in the literature. We would also like to emphasize that our analysis of this paper is consistent with that of Ambrus [3]; in particular, our definition of Bayesian c-rationality can be regarded as a concrete form of γ-rationality, by formally expressing belief profiles for the players in coalition J behind the shortcut operator γ A, J). Thus, our paper provides an alternative and complementary epistemic characterization of the concept of coalitional rationalizability within a partition-information model. In closing, we mention some possible extensions. In this paper, we define Bayesian c- rationality by assuming players are subjective expected utility maximizers and coordinate their play to achieve a common gain through nonbinding agreements on joint strategies. Alternatively, we can consider the coalitional preferences as the aggregation of the preferences of coalition members; see, for example, Hara et al. [26] for a coalitional expected multi-utility theory. The extension of this paper to games with different modes of coalitional behavior is an intriguing subject for further research; cf. Epstein [21] and Chen et al. [18] for related work on rationalizability under general preferences. The exploration of the notion of extensive-form c-rationalizability in dynamic settings is also an important research topic for further study. 16

17 7 Appendix: Proofs Proof of Proposition 1. " ": Assume, in negation, that player j is not Bayesian rational at ω. Then, there exists a strategy s j S j such that u j s j, µ j S ω) ) > u j sj ω), µ j S ω) ) and u j s j, µ j S ω) ) u j sj, µ j S ω) ) s j S j. Define B {s j} i j s i P j ω))). Clearly, B = A ω, J) = i j s i P j ω)) where J = {j}. Because µ j ω) sp j ω)) S ) and s P j ω)) A ω, J) = B S, µ j A ω,j) ω) = µ j B ω) = µ j S ω). Therefore, u j s j, µ j B ω) ) > u j sj ω), µ j A ω,j) ω) ) and u j s j, µ j B ω) ) u j sj, µ j B ω) ) s j S j. Since coalition J is Bayesian rational at ω, by Definition 2, s ω) B. Thus, s j ω) = s j, which is a contradiction. " ": Assume, in negation, that coalition J = {j} is not Bayesian rational at ω. Then, there exists a product subset B satisfying B J = A J ω, J), Aω, J) i I s i P J ω)) and Definition 2C1-2) but s ω) / B. Since s ω) A J ω, J), s j ω) / B j. By Definition 2C1), u j sj ω), µ j A ω,j) ω) ) < u j sj, µ j B ω) ) for some s j S j. Since µ j A ω,j) ω) = µ j B ω) = µ j S ω), u j sj ω), µ j S ω) ) < u j sj, µ j S ω) ) for some s j S j, contradicting the supposition that j is Bayesian rational at ω. To prove Theorem 1, we introduce some notation. Let Z i ) denote the set of probability distributions on Z i S i. For µ Z i ), let Let BR Z i ) = µ Z i )BR µ). BR µ) = {s i S i : u i s i, µ) u i s i, µ) s i S i }. Proof of Theorem 1. a) Assume, in negation, that s ) CKR R. By Proposition 1 in Luo and Yang [31], there exists a reduction product-set) sequence {D τ } such that R = τ=0 Dτ ) with D 0 = S and ) D τ D τ+1 for all τ 0. Hence, there is t such that s CKR D t and s CKR D t+1, where D t+1 D t and D t D t+1 via J. Therefore, s ω) / D t+1 for some ω CKR. Apparently, P J ω) P ω) CKR and Aω, J) i I s i P J ω)) D t. Let J 0 = { j J : A j ω, J) D t+1 j = }. We distinguish three cases. 1. J 0 = 0: Since A J ω, J) D t J = Dt+1 J, A Jω, J) = Aω, J) D t+1 ) J. Define B Aω, J) D t+1. We proceed to show B satisfies Definition 2C1-2) at 17

18 ω for coalition J. i) Let j J and ω P J ω). Since µ j ω ) Aω,J) S ), µ j A ω,j) ω ) = µ j D t ω ) and µ j B ω ) = µ j Aω,J) D t+1 ) ω ) = µ j D t+1 ω ). Since Aω, J) D t D t+1 via J, by Definition 11), if s j ω ) / B j or µ j A ω,j) ω ) µ j B ω ), then u j s j ω ), µ j A ω,j) ω )) < u j s j, µ j B ω )) for some s j S j. ii) Definition 2C2) is void because B Aω, J). But, since ω CK R R, J is Bayesian rational at ω. By Definition 2, s ω) B = Aω, J) D t+1. This is a contradiction. 2. J 0 > 1: Define B D t+1 J Aω, J) D t+1 ) 0 J\J 0 A J ω, J). By Lemma 2 in Luo and Yang [31], for all τ 0, BR D i) τ D τ ) i i I. Since A J ω, J) D J t = D t+1 J, Bj BR B j J 0. Let B be the nonempty) set of surviving iterated elimination of never-best responses for all the players in J 0. Then, B j = BR B ) j J 0 and because of J 0 > 1, A ω, J) D t+1 = A ω, J) B = j J. We proceed to show B satisfies Definition 2C1-2) at ω for coalition J. i) Since Aω, J) D t D t+1 via J, for all j J, a j A j ω, J) and µ Aω,J) S ), we have u j a j, µ A ω,j)) < u j s j, µ D t+1 ) for some s j S j. By B D t+1, for all j J and ω P J ω), u j sj ω ), µ j A ω,j) ω ) ) < u j sj, µ j B ω ) ) for some s j S j. ii) Let j J 0. Then, B j = BR B ) and A ω, J) B =. Because the model M G) is weakly belief-complete, for each b j B j, there exists ω P J ω) such that u j bj, µ j B ω ) ) u j sj, µ j B ω ) ) s j S j. Since coalition J is Bayesian rational at ω, by Definition 2, s ω) B. This is a contradiction. ) 3. J 0 = 1 with J 0 = {j 0 }): Let d j 0 BR µ j 0 Aω,J) D t+1 ω) )0. Then, d j 0 D t+1 j ) 0 because of A J ω, J) D J t = Dt+1 J and BR D t+1 D t+1 0 j. Define B {d 0 j 0} Aω, J) D t+1 ) J\J 0 A J ω, J). We proceed to show B satisfies Definition 2C1-2) at ω for coalition J. i) Let j J. If j j 0, then A ω, J) B =. Similarly to Case 2i), Definition 2C2) is satisfied. If j = j 0, then µ j 0 B 0 ω ) = µ j 0 D t+1 ω ) 0 ω P J ω). Since A j 0ω) B j 0 = and D t D t+1 via J, it follows that for all ω P J ω), u j 0s j 0 ω ), µ j 0 A 0 ω,j) ω )) < u j 0s j 0, µ j 0 B 0 ω )) for some s j 0 S j 0. ) ii) Since d j 0 BR µ j 0 B 0 ω), Definition 2C2) is satisfied. Since coalition J is Bayesian rational at ω, by Definition 2, s ω) B. This is a contradiction. 18

19 b) By Luo and Yang s [31] Theorem 1, R is a CRS; in particular, BR R i) R i for all i I. Define an epistemic model for G as follow: such that Ω = M G) < Ω, {P i )} i I, {µ i )} i I, {s i )} i I >, { ) r i, µ i ) i I : r i R i and µ i R S i) such that r i BR µ i R i i I, s i ω) = r i and µ i ω) = µ i for ω = r i, µ i ) i I in Ω; i I, P i ω) = {ω Ω : s i ω ) = s i ω) and µ i ω ) = µ i ω)}. } i I ; Clearly, s Ω) = R and s i P i ω)) = R i i I. Because BR R i) R ) i i I, for any µ sp i ω)) S i) there exists r i BR µ R i I. Therefore, M G) is a weakly i belief-complete model. Let ω Ω. By the construction of Ω, every player is Bayesian rational at ω. By Proposition 1, we only need to show non-singleton coalition J is Bayesian rational at ω. Assume, in negation, that J is not Bayesian rational at ω. Then, there exists B satisfies Definition 2C1-2) and B J = A J ω, J) where Aω, J) i I s i P J ω))), but s ω) / B. By the construction of Ω, s P J ω)) = s Ω) = R. Clearly, B J = R J. We proceed to show R B via J. Since B R, Definition 12) is void. Next, we verify Definition 11) holds for all j J. Consider r j R j and µ R S ). We distinguish two cases. ) 1. If r j BR µ R, by the construction of Ω, there exists ω P J ω) = Ω such that ) ) s j ω) = r j and µ j ω) = µ. By Definition 2C1), u j r j, µ R < u j sj, µ B for some s j S j whenever r j / B j or µ B µ R. ) ) 2. If r j / BR µ R, there exists r j BR µ R such that u j r j, µ R ) u j r j, µ B ) for µ B = µ R. If µ B µ R, by the construction of Ω, there exists ω P J ω) = Ω such that s j ω) = rj and µ j ω) = µ. Again by Definition ) ) ) 2C1), u j r j, µ R < u j rj, µ R < u j sj, µ B for some sj S j. Therefore, R B. Since R is a CRS, it follows that B = R and s ω) B. This is a contradiction. ) Hence, for all ω Ω, players are Bayesian c-rational at ω. Consequently, s CKR = s Ω) = R. < 19

20 Proof of Corollary 1: a) Under the restriction of singleton coalitions, i) R = R by Proposition 1, and ii) R = R cf. Luo and Yang [31]). Because the requirement of weak belief-completeness is void for singleton coalitions, Corollary 1a) follows directly from Theorem 1a). b) Consider the model M G) by replacing R with R, in the proof of Theorem 1b). In this modified model M G), players are Bayesian rational across the state space Ω, and thus s CKR) = s Ω) = R. Proof of Theorem 2. If part: Let s be an a posteriori Bayesian c-equilibrium in an epistemic model M G). Then players are Bayesian c-rational at all ω Ω. Therefore, Ω = CKR. By Theorem 1, for all ω Ω, the strategy profile s = sω) is Bayesian c-rationalizable in G. Only if part: Let s be a Bayesian c-rationalizable strategy profile in G. Then s R. Consider the model M G) defined in the proof of Theorem 1b). Then players are Bayesian c-rational at all ω Ω. That is, s is an a posteriori Bayesian c-equilibrium in M G). By the construction of M G), we can find an a posteriori Bayesian c-equilibrium s in a weakly belief-complete model M G) such that s = sω) for some ω = s i, µ i ) i I Ω. 20

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