Bayesian coalitional rationality and rationalizability a a b Xiao Luo ,yYongchuan Qiao , Chih-Chun Yang aDepartment of Economics, National University of Singapore, Singapore 117570 bInstitute of Economics, Academia Sinica, Taipei 115, Taiwan October 2017 Abstract We offer an epistemic definition 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 at a state, we characterize the game-theoretic solution concept of “Bayesian coalitional rationalizability”(i.e., Bayesian c-rationalizability) in [32] 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 system (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 support from the National University of Singapore and the National Science Council of Taiwan is gratefully acknowledged. The usual disclaimer applies. y Corresponding author. Fax:+65 6775 2646. E-mail: [email protected] (X. Luo); qiaoyongchuan@ gmail.com (Y. Qiao); [email protected] (C.C. Yang). 1 1 Introduction In their seminal papers, Bernheim [14] and Pearce [40] proposed the game-theoretic solution concept of “rationalizability”as the logical implication of common knowledge of “Bayesian rationality”(cf. also Tan and Werlang [45]). Notably, in a standard textbook, Osborne and Rubinstein [39, Section 5.4] presented an epistemic characterization of rationalizability by common knowledge of Bayesian rationality within a semantic framework. The definition of “Bayesian rationality” here requires a player to adopt a strategy that maximizes his expected utility under his subjective probabilistic belief about the other players’strategies; in particular, each player, in a strategic situation, makes his own optimal decision in the absence of coalition considerations. The collective and coalitional behavior has been recognized as an important issue in real- life examples such as cartels, trade blocs, political party formation, special interest groups, public goods provision, social networks and matchings, 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 [28], Newton [37], and Grandjean et al. [26] 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 re- strictions 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 expec- tation) 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]) 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. 1 Liu et al. [31] adopted the idea of (coalitional) rationalizability to define a stable matching with in- complete 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 [29] 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 Along the lines of Ambrus’s [2] approach, Luo and Yang [32] took a traditional game- theoretic approach to define an alternative solution concept of “Bayesian coalitional ratio- nalizability”(henceforth, Bayesian c-rationalizability) from the Bayesian point of view. The alternative notion extends Ambrus’soriginal one to situations in which, in pursuit of mutu- ally beneficial interests, the players in a coalition (i) evaluate their payoff expectations by Bayesian updating, if an agreement is made, instead of holding fixed the marginal expecta- tions concerning the behavior of 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 solution con- cept 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 notion of Bayesian c-rationalizability, in terms of what each player knows and believes about the information, knowledge, and rationality of coalitions. More specifically, we put forth a novel epistemic definition of “Bayesian coalitional rationality” (henceforth, Bayesian c-rationality) that identifies a mode of behavior that no group of players wishes to change —that is, a truism among coalition members: no credible coalitional gains exist. The main results of this paper are as follows: (1) Bayesian c-rationalizability is characterized by common knowledge of Bayesian c-rationality (Theorem 1); (2) Bayesian c-rationalizability is outcome equivalent to a posteriori Bayesian c- equilibrium, a coalitional version of a posteriori equilibrium in Brandenburger and Dekel [16] (Theorem 2). Thus, our analysis provides the epistemic foundation of the solution concept of Bayesian c-rationalizability in Luo and Yang [32]. One key ingredient to our approach is the epistemic definition of Bayesian c-rationality (Definition 2), which is the coalitional counterpart of Bayesian rationality. From an epistemic perspective, Bayesian rationality is referred to an epistemic state in which a player atten- tively awares that no alternative choice of strategy can attain a higher expected payoff. The coalitional version of Bayesian rationality captures the similar idea for coalitional reason- ing: coalition members share a common knowledge that improving every coalition member’s expected payoff by coordinating their moves in a credible way is impossible. The notion 3 of coalitional rationality needs to call for the coalition-wise common-knowledge assumption that no coalitional gains exist, because each coalition member’sbehavior and beliefs will be typically affected if a new agreement is made.2 In fact, 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. That is, each player in the group has to think about what would happen in hypothetical circumstances, in one of which every player takes into account what the group of players would do and what the group would contemplate in the hypothetical circumstances.3 Thus, it is epistemically important and necessary to consider not only what a 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. Our study contributes to a better understanding of the concept of coalitional rationality in complex social interactions. Moreover, according to the notion of Bayesian c-rationality, we need to check 2n 1 feasible coalitions (i.e., all subsets of n players Sj except the empty set) and, for a coalition J, j J 2j j 1 possible coalitional deviations (i.e., all restrictions of strategies in S for each2 coalition member j J except the empty j set). Due to the enormous complexity of coalitional reasoning, exploring2 how to make such desirable epistemic states possible is fascinating and intriguing. Our work is closely related to Ambrus [3] and provides