Bayesian coalitional rationality and rationalizability∗

a a b Xiao Luo ,†Yongchuan 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. † 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, Πj J 2| | 1 possible coalitional deviations (i.e., all restrictions of strategies in S for each∈ coalition− member j J except the empty j
set). Due to the enormous complexity of coalitional reasoning, exploring∈ how to make such desirable epistemic states possible is fascinating and intriguing. Our work is closely related to Ambrus [3] and provides complementary support to his ap- proach. Ambrus [3] offered a wide range of epistemic definitions of coalitional rationalizability by rationality and common certainty of coalitional γ-rationality (where γ is a coalitional best response operator). Despite its great advantages in terms of generality and simplicity, his formalism is less comprehensive from an epistemic perspective; it does not provide a full epistemic expression of coalitional rationality. The major issue is that the shortcut operator γ is not based on assumptions about players’epistemic beliefs and behavior in the possi- ble worlds associated with coalitions; for instance, the conjectures used in the “sensible” operator γ are not linked to epistemic states.4 Nevertheless, the missing part of the ex-

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.” 3 To coordinate its interrelated actions, a group of players needs to have recourse to “common knowledge” (see, e.g., Lewis [30], Chwe [20], and Rubinstein [42]). 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 [24] and Fagin et al. [23, 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. 4 Ambrus’s [3] notion of γ-rationality, although defines the scope of “conceivable” γ-rational strategic

4 pression could be critical in the epistemic program. Examples such as backward induction and iterated elimination of weakly dominated strategies show the hidden assumptions in the causal arguments about players’beliefs and rationality at epistemic states are crucial; see, for example, Brandenburger [15], Dekel and Siniscalchi [21], and Perea [41] for discussions. To overcome the aforementioned shortcoming, this paper offers a more delicate definition of coalitional rationality in the Bayesian paradigm, which makes an epistemic expression of how a player interacts with other players within a coalition if a new agreement is made. Our epistemological notion of Bayesian c-rationality in Definition 2 is designed mainly for this purpose. We would like to point out that in the special case of singleton coalitions, our definition of Bayesian c-rationality turns out to be consistent with the definition of Bayesian rationality (Proposition 1). By comparison, there is a discrepancy between γ-rationality and individual rationality; the γ-rationality for the singleton coalition j does not necessarily imply the individual rationality of the player j, because the operator{ γ}is independent of the epistemic states under consideration. To deal with hypothetical coalitional reasoning in a Bayesian paradigm, we carry out our analysis in an epistemic framework in which each player has a “conditional probability system”(CPS) belief at a state. The CPS belief specifies the family of probabilistic beliefs about the opponents’behavior in all possible contingencies; thus, even in the case of an unex- pected or hypothetical event, the player must have a probabilistic assessment of opponents’ strategies contingent on that event. 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 exam- ple, extensive-form rationalizability in a complete CPS type structure model in Battigalli and Siniscalchi [13] and iterated weak dominance in a complete “lexicographic conditional probability system (LCPS)”type structure model in Brandenburger et al. [18].5 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. behavior, does not explicate what is exactly the γ-rational epistemic state of affairs; in particular, the notion does not provide us with the epistemic precondition for the changing process from the J-commonly-known set A to another set γ (A, J). 5 Although this kind of completeness is crucial to our epistemic analysis, an analogous concept does not appear in Ambrus’s[3] framework, because of his shortcut operator γ. In Ambrus [3], the prominent 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).

5 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. 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 ≡ ωaa, ωab, ωba, ωbb with a complete information structure. At state ωab, for example, player 1{ plays strategy a}and player 2 plays strategy b; each player holds the correct belief about the opponent’susing 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 ωab, ωba, and ωbb in this model. (Intuitively, state ωaa 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 ωaa.) Because this epistemic model has a complete information structure, Bayesian c-rationality is commonly known at states ωab, ωba, and ωbb. In Theorem 1(a), we show players must play a Bayesian c-rationalizable strategy profile under common knowledge of Bayesian c-rationality, in any “null-event-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 pur- pose, we may consider an alternative model with an incomplete information structure show- ing that each player cannot distinguish the opponent’susing strategies at states in Ω. Each player believes with probability 1 that the opponent is playing b at each state. In this new model, ωaa becomes a Bayesian c-rational state because every player at this state can expect to obtain the highest payoff of 3; hence, the profile (a, a) can be attained by common knowl- edge of Bayesian c-rationality. In Theorem 1(b), we show every Bayesian c-rationalizable strategy profile can be attained by common knowledge of Bayesian c-rationality. In this pa- per, we also follow Brandenburger and Dekel [16] to establish an equivalence result between Bayesian c-rationalizability and a posteriori Bayesian c-equilibrium (Theorem 2).

6 3 Preliminaries

Consider a finite game:

(I, Si i I , ui i I ), G ≡ { } ∈ { } ∈ where I is a (nonempty) finite set of players, Si is a (nonempty) finite set of i’s pure strategies, and ui : S i I Si 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 SJ j J Sj, S J i/J Si, ⊆ ≡ × ∈ − ≡ × ∈ and S j i=jSi. For any si Si and any probability distribution µ over S i, define u (s , µ−) ≡ × 6 µ (s ) u (s , s∈ ). − i i s i S i i i i i ≡ − ∈ − − − P 3.1 Bayesian c-rationalizability: definition

We adopt the notation in Luo and Yang [32]. Let ∆∗ (S j) denote the set of all “conditional − probability systems”(CPSs) on finite state space S j faced by player j (see, e.g., Myerson [35, 36]). For nonempty subset A S, let − ⊆

∆A∗ (S j) µ ∆∗ (S j): µ S j (A j) = 1 . − ≡ | ∈ − | − −  For nonempty product subsets A, B S, we say a coalition AB is a “feasible coalition from ⊆ J A to B”if B = B AB A AB . J × −J Definition 1 (Luo and Yang [32]). A nonempty product subset S is a coalitional R ⊆ rationalizable set (CRS) if V 0 only for 0 = , where for = , we define V 0 as: a feasible coalition R , jR suchR thatR R 6 ∅ R R ∃ JRR0 ∀ ∈ JRR0 (1) [Profitability] r , µ ∆ (S ), if r / or µ = µ , then j j ∗ j j j0 j 0 j ∀ ∈ R ∀ | ∈ R − ∈ R |R− 6 |R− u (r , µ ) < u (s , µ ) for some s S , and j j j j j 0 j j j |R− |R− ∈ (2) [Credibility] r , there exists µ ∆ (S ) such that u (r , µ ) j0 j0 j ∗ j j j0 0 j ∀ ∈ R \R | ∈ R − |R− ≥ u (s , µ ) s S . j j 0 j j j |R− ∀ ∈ That is, a CRS is a (nonempty) product set of pure strategies from which no group of players would like to make a vital deviation. In particular, ri i is said to be a Bayesian ∈ R c-rationalizable strategy for player i. With the restriction of 0 = 1, Definition 1 yields a correlated version of rationalizability (see Luo and Yang [32]).|JRR |

7 3.2 Aumann’smodel of knowledge

We follow Aumann [8, 7, 9, 10] and Brandenburger and Dekel [16] to consider an epistemic model for game (where each player is allowed to hold a CPS belief): G

( ) < Ω, Pi ( ) i I , µi ( ) i I , si ( ) i I > , M G ≡ { · } ∈ { | · } ∈ { · } ∈ where Ω is the set of states with typical element ω Ω, Pi(ω) Ω is i’s partitional infor- ∈ ⊆ mation structure at ω, µi (ω) ∆∗ (S i) is the CPS belief that i holds at ω, and si(ω) Si | ∈ − ∈ is i’s using strategy at ω; cf. also Osborne and Rubinstein [39, Chapter 5] for a detailed introduction to Aumann’smodel of knowledge. Throughout this paper, we require the belief

µi (ω) to be consistent with i’sknowledge; that is, µi (ω) ∆s∗(P (ω))(S i). Assume, as usual, | | ∈ i − that i knows his own using strategy and belief; that is, si(ω) = si(ω0) and µ (ω) = µ (ω0) i| i| ω0 Pi(ω). ∀ ∈ Let s i(ω) denote i’sopponents’strategy profile at ω Ω, and let s(ω) (si(ω), s i(ω)). − ∈ ≡ − We refer to a set of states E Ω as an event. Let s (E) s(ω): ω E . For an event E, ⊆ ≡ { ∈ } we take the following standard definitions:

KiE ω Ω: Pi(ω) E is the event that i knows E. • ≡ { ∈ ⊆ }

KE i N KiE 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 PJ be the meet of the partitional information structures of coalition J, that is, the

finest common coarsening of partitions Pj for all j J (see, e.g., Aumann [8] and Milgrom ∈ [33]). Let P PI 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”. We say a model ( ) is null-event belief complete if, for every ω Ω, non-singleton M G ∈ coalition J, j J and µ ∆∗ (S j) with µ S j = µj S j (ω), there is ω0 PJ (ω) such ∈ | ∈ − | − | − ∈

8 that µj (ω0) = µ . Note that µ S j = µj S j (ω) implies µ coincides with µj (ω) for | | | − | − | | all non-null events of strategies —that is, for every A j S j such that µ S j (A j) = 0, − ⊆ − | − − 6 µ A j = µj A j (ω). In a null-event-belief-complete model ( ), every possible CPS be- | − | − M G lief µ , which is inconsistent with µj (ω) only on some null event of strategies, can be | | 6 generated by some state ω0 in the J-commonly-known subspace PJ (ω). The null-event- belief-completeness is a fairly weak belief-richness condition that requires, contingent on null events of strategies, all plausible conditional probabilistic beliefs must reside 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 [15] and Dekel and Siniscalchi [21]).

Remark 1. Following the line of the epistemic game theory paradigm advocated by Aumann and Brandenburger [11], for the purpose of this paper, we adopt a simple epistemic framework in which each player has a CPS belief (cf. Battigalli and Siniscalchi [13] for the construction of a CPS belief universal type space). Like Aumann and Brandenburger’s [11] 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 models.7

4 Bayesian c-rationality

Consider an epistemic model ( ) for game . We next formulate the concept of Bayesian c- M G G rationality; that is, coalition members share a common knowledge that no credible profitable coalitional deviation changes the behavior of coalition members.

6 Because each player j is assumed to know his own CPS belief in the model ( ), j must hold a unique M G CPS belief across the states in Pj (ω). Hence, the belief-completeness requirement should not be made for singleton coalitions. 7 Brandenburger and Dekel [17] 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.” The “information-structure” model is well suited to our epistemic analysis in this paper. Notable examples in the literature include Aumann [7, 8, 9, 10], Brandenburger and Dekel [16], Monderer and Samet [42], and Rubinstein [42] (cf. Samuelson [43]).

9 Definition 2. For state ω Ω and coalition J I, let A(ω, J) i I si (PJ (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 ω0 PJ (ω) and j J, uj sj (ω0) , µj A j (ω,J)(ω0) < uj sj, µj B j (ω0) ∈ ∈ | − | − for some sj Sj whenever sj (ω0) / Bj or µj B j (ω0) = µj A j (ω,J)(ω0); ∈ ∈ | − 6 | −

(C2) [Credibility] for all j J and bj Bj Aj(ω, J), there exists ω0 PJ (ω) such ∈ ∈ \ ∈ that uj bj, µj B j (ω0) uj sj, µj B j (ω0) sj Sj. | − ≥ | − ∀ ∈ 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 A(ω, 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 coalitional best response set B of the coalition, to which players in the coalition would be willing to confine their play.8 (Obviously, the set B is dependent on state ω and coalition J, but we use the shorthand notation B instead of B(ω, J) for notational simplicity.) The additional credibility condition

C2 requires each incremental strategy bj Bj Aj(ω, J) be justified by individual rationality. ∈ \ Our formalism of Bayesian c-rationality is explicitly and entirely based on the assumptions about coalitional members’beliefs and behavior at epistemic states, in terms of what the players in coalitions know and believe about each other’sstrategies, knowledge, and beliefs. Although the set B in Definition 2 can be interpreted as a variant of the “coalitional best response” operator γ (A, J) in Ambrus [3], the notions of Bayesian c-rationality and γ-rationality differ in some important aspects. Because the operator γ (A, J) contains only restricted subsets of strategies, the notion of γ-rationality entails an excessive amount of attention to a wide range of “initial agreements” —all product supersets of A(ω, J) that satisfy “closed under rational behavior”in Basu and Weibull [12]; that is, it requires players in J to be logically omniscient with respect to a special class of initial agreements hinging

8 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 [46] concept of coarse core in an exchange economy with asymmetric information.

10 on a “behavior-complete” framework. By contrast, the notion of Bayesian c-rationality merely requires the initial agreement under consideration to be the “exact” product set

A(ω, J) = i I si (PJ (ω)) at state ω. Without the requirement of “closed under rational × ∈ behavior,”the set A(ω, J) represents the “finest”agreement commonly known by coalition J at state ω. We note the operator associated with Definition 2 is outside the class of the “sensible”operators studied in Ambrus [3] (see Appendix+ on p.20). One important feature of Definition 2 is that the definition of Bayesian c-rationality has two coalition-common-knowledge requirements: (i) coalition members in J share a common knowledge that play is in the initial agreement A(ω, J), and (ii) coalition members in J share a common knowledge that the contemplating joint movement from the agreement A(ω, J) to a new agreement B is a vital deviation. 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 (see, e.g., Lewis [30] and Sugden [44] for more discussions). The first coalition-common-knowledge requirement (i) 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. The second coalition-common-knowledge requirement (ii) is an additional epistemic precondition for the changing process from the initial agreement to another new agreement in the concept of coalition rationality. Proposition 1 below establishes a formal relationship between the notions of Bayesian rationality and Bayesian c-rationality. That is, the classical notion of Bayesian rationality is equivalent to our epistemological notion of Bayesian c-rationality with the restriction of singleton coalitions; in the latter notion, the self-evident knowledge assumption is explicitly 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., uj sj (ω) , µj S j (ω) uj sj, µj S j (ω) sj Sj. | − ≥ | − ∀ ∈ By Proposition 1, the conventional notion of Bayesian
rationality requires
the individual player to be self-evidently aware he cannot do better by a replacement of strategies. Sub- sequently, 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.

11 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 con- sequence of common knowledge of Bayesian c-rationality. Let ∗ denote the set of Bayesian R c-rationalizable strategy profiles in game . Formally, we have G

◦ Theorem 1. (a) In every null-event-belief-complete model ( ), s CKR ∗. (b) M G ⊆ R   ◦ There is a null-event-belief-complete model ( ) such that s CKR = ∗. M G R   Theorem 1 is consistent with Ambrus’s[3, Definition 5] alternative definition of coalitional rationalizability by γ-rationality and common certainty that every coalition is γ-rational.9 Our analysis of this paper provides a detailed and comprehensive epistemic description for the rationalizable behavior in the presence of complex coalitional reasoning.

The following example shows that without imposing the null-event-belief-complete con- dition, 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

9 For the belief operator B, CKR◦ = R◦ CBR◦ (see, e.g., Dekel and Siniscalchi [21, Section 12.3.1]); thus, ∩ CBR◦ = B CKR◦ . Because the set of Bayesian c-rationalizable strategy profiles is “closed under rational   ◦ behavior,”Theorem 1(a) implies s R CBR ∗, where R is the event in Ω that every player is rational. ∩ ⊆ R As a consequence, Bayesian c-rationalizablility  can alternatively be characterized by rationality and common belief of Bayesian c-rationality.

12 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 con- { } × { } fine their play to a subset of strategies a, b a, b in which each player can guar- { } × { } antee an expected payoff of at less 1.5.) We consider an epistemic model ( ) where M G Ω = ωaa, ωab, ωac, ..., ωcc with a complete-information structure. For i = 1, 2 and typical { } state ωxy Ω, define Pi(ωxy) = ωxy , s (ωxy) = (x, y) and µi (ωxy) ∆∗x y ( a, b, c ) ∈ { } | ∈ { }×{ } { } such that µ1 a,b (ωcc) = µ2 a,b (ωcc) = 1 b. It is easy to check that Bayesian c-rationality |{ } |{ } ◦ is commonly known at ωcc, because no credible deviation exists under this circumstance.

But the using strategy profile (c, c) at ωcc is not Bayesian c-rationalizable. In this model ( ), the CPS beliefs of players are rather sparse; no rich beliefs support M G the grand coalition to make the profitable deviation a, b a, b from the initial agreement { }×{ } c c . The model ( ) fails to satisfy the null-event-belief-complete condition. This { } × { } M G weak belief-completeness in Theorem 1(a) is mainly for the credibility requirement in the original definition of Bayesian c-rationalizablility.

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) rationaliz- ability. As an immediate corollary of Theorem 1 and Proposition 1, we can obtain a simpler characterization for (correlated) rationalizability without appealing to the null-event-belief- completeness requirement. Let be the set of (correlated) rationalizable strategy profiles R in , and let R be the event in which every player is Bayesian rational in ( ); that is, G M G R = ω Ω: players are Bayesian rational at ω . { ∈ } Corollary 1: (a) In any arbitrary model ( ), s (CKR) . (b) There is a model ( ) M G ⊆ R M G such that s (CKR) = . R Remark 2. In the spirit of Aumann’s [6] notion of strong Nash equilibrium, we can obtain the stronger notions of Bayesian c-rationality and Bayesian c-rationalizablility by removing the credibility requirement. The null-event-belief-complete condition in Theorem 1(a) is no longer necessary for this “strong” Bayesian c-rationalizability. However, like strong Nash equilibrium, the “strongly”Bayesian c-rationalizability may fail to exist.

Brandenburger and Dekel [16] proposed the notion of a posteriori equilibrium, a strength- ening of Aumann’s[7] notion of subjective correlated equilibrium, and showed the equivalence

13 between rationalizability and a posteriori equilibrium (see also Epstein [22, Theorem 5.1] and Chen et al. [19, Proposition 6] for the related study under general preferences). The equivalence implies the assumption of common knowledge of rationality provides an epis- temic justification for the equilibrium notion. In the spirit of Brandenburger and Dekel [16], we extend this kind of equivalence to complex coalitional interactions. A strategy-profile function s :Ω S is said to be an a posteriori Bayesian c-equilibrium in ( ) if, for all → M G ω Ω, players are Bayesian c-rational at ω. ∈

Theorem 2. The strategy profile s∗ is Bayesian c-rationalizable in if, and only if, there G exist an epistemic model ( ) and an a posteriori Bayesian c-equilibrium s in ( ) such M G M G that s∗ = s(ω) for some ω Ω. ∈ 6 Concluding remarks

The study of how groups of players act in their mutually beneficial interest in social envi- ronments is of great importance in economics and social sciences (see, e.g., Olson [38]). 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 ad- vocated by Aumann and Brandenburger [11], in this paper, we conduct a thorough epistemic analysis of the solution concept of Bayesian c-rationalizability in Luo and Yang [32]. 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 represents 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-

14 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 [8, 7, 9, 10] and Aumann and Brandenburger [11], we carry out our epistemic analysis within an arbitrary model, including finite and infinite mod- els discussed in the literature. We have emphasized our analysis of this paper is consistent with that of Ambrus [3]. Through the lens of a symbolical operator γ that allows for all possible arrangements of strategies by coalitions, our definition of Bayesian c-rationality can be regarded as a concrete form of coalitional γ-rationality by expressing coalition members’ behavior and beliefs about what the players in coalitions know and believe about the other’s strategies, knowledge, and beliefs. In this view, our paper provides an epistemic characteriza- tion of a peculiar form of coalitional γ-rationalizability, namely, Bayesian c-rationalizability, in a partition-information model. Exploration of the epistemic foundation for the whole class of coalitional γ-rationalizability in the model is certainly an interesting topic, but we leave it to future work. 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. [27] 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. Asheim [5], Chen et al. [19], and Epstein [22] 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.

15 7 Appendix: Proofs

Proof of Proposition 1. " ": Assume, in negation, that player j is not Bayesian rational ⇒ at ω. Then, there exists a strategy sj∗ Sj such that uj sj∗, µj S j (ω) > uj sj (ω) , µj S j (ω) ∈ | − | − and uj sj∗, µj S j (ω) uj sj, µj S j (ω) sj Sj. Define B sj∗ ( i=jsi (Pj (ω))). | − ≥ | − ∀ ∈ ≡ {
} ×× 6
Clearly, B j = A j(ω, J) = i=jsi (Pj (ω)) where J = j . Because µj (ω) ∆s∗(P (ω)) (S j) − −
× 6
{ } | ∈ j − and s j (Pj (ω)) = A j(ω, J) = B j S j, µj A j (ω,J) (ω) = µj B j (ω) = µj S j (ω). − − − ⊆ − | − | − | − Therefore, uj sj∗, µj B j (ω) > uj sj (ω) , µj A j (ω,J) (ω) and uj sj∗, µj B j (ω) uj sj, µj B j (ω) | − | − | − ≥ | − sj Sj. Since coalition J is Bayesian rational at ω, by Definition 2, s (ω) B. Thus, ∀ ∈


∈
sj (ω) = sj∗, 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), A(ω, J) i I si (PJ (ω)) − − ≡ × ∈ and Definition 2(C1-2) but s (ω) / B. Since s j (ω) A J (ω, J), sj (ω) / Bj. By De- ∈ − ∈ − ∈ finition 2(C1), uj sj (ω) , µj A j (ω,J) (ω) < uj sj, µj B j (ω) for some sj Sj. Since | − | − ∈ µj A j (ω,J) (ω) = µj B j (ω) = µj S j (ω), uj sj (ω) , µj S j (ω) < uj sj, µj S j (ω) for | − | − | −
| −
| − some sj Sj, 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 proba- − bility distributions on Z i S i. For µ ∆ (Z i), let − ⊆ − ∈ −

BR (µ) = s∗ Si : ui (s∗, µ) ui (si, µ) si Si . { i ∈ i ≥ ∀ ∈ }

Let BR (Z i) = µ ∆(Z i)BR (µ). − ∪ ∈ −

◦ Proof of Theorem 1. (a) Assume, in negation, that s CKR * ∗. By Proposition R 1 in Luo and Yang [32], there exists a reduction (product-set)  sequence τ such that τ 0 τ τ+1 {D } ∗ = ∞ with = S and for all τ 0. Hence, there is t such that R τ=0 D D D V D ≥ ◦ ◦ s CKTR t and s CKR * t+1, where t+1 t and t t+1 via J. Therefore, ⊆ D D D ⊆ D D V D     t+1 ◦ ◦ s (ω) / for some ω CKR. Apparently, PJ (ω) P (ω) CKR and A(ω, J) ∈ D t ∈0 t+1 ⊆ ⊆ ≡ i I si (PJ (ω)) . Let J = j J : Aj(ω, J) j = . We distinguish three cases. × ∈ ⊆ D ∈ ∩ D ∅ 0  t t+1 t+1 1. J = 0: Since A J (ω, J) J = J , A J (ω, J) = (A(ω, J) ) J . Define | | − ⊆ D− D− − ∩ D − B A(ω, J) t+1 = . We proceed to show B satisfies Definition 2(C1-2) at ≡ ∩ D 6 ∅ 16 ω for coalition J. (i) Let j J and ω0 PJ (ω). Since µj (ω0) ∆A∗ (ω,J) (S j), ∈ ∈ | ∈ − µ (ω ) = µ t (ω ) and µ (ω ) = µ t+1 (ω ) = µ t+1 (ω ). j A j (ω,J) 0 j 0 j B j 0 j (A(ω,J) ) j 0 j 0 − D j − ∩D j | t | − t+1 | | − |D− Since A(ω, J) V via J, by Definition 1(1), if sj (ω0) / Bj or µj A j (ω,J) (ω0) = ⊆ D D ∈ | − 6 µj B j (ω0), then uj(sj (ω0) , µj A j (ω,J) (ω0)) < uj(sj, µj B j (ω0)) for some sj Sj. | − | − | − ∈ (ii) Definition 2(C2) is void because B A(ω, J). But, since ω CKR◦ R◦ , J ⊆ ∈ ⊆ is Bayesian rational at ω. By Definition 2, s (ω) B = A(ω, J) t+1. This is a ∈ ∩ D contradiction.

0 t+1 t+1 2. J > 1: Define B 0 (A(ω, J) )J J0 A J (ω, J). By Lemma 2 in J − | | ≡ D × ∩ Dτ \ τ× Luo and Yang [32], for all τ 0, BR i i i I. Since A J (ω, J) ≥ D− ⊆ D ∀ ∈ − ⊆ t t+1 e 0 J = J , Bj BR B j j J . Let
B be the (nonempty) set of surviving D− D− ⊇ − ∀ ∈ iterated elimination of never-best  responses for all the players in J 0, starting from e e 0 0 t+1 B. Then, Bj = BR (B j) j J and because of J > 1, A j(ω, J) j = − ∀ ∈ | | − ∩ D− A j(ω, J) B j = j J. We proceed to show B satisfies Definition 2(C1-2) − − e ∩ ∅ ∀ ∈ t t+1 at ω for coalition J. (i) Since A(ω, J) via J, for all j J, aj ⊆ D V D ∈ ∈ t+1 Aj(ω, J) and µ ∆A∗ (ω,J) (S j), we have uj(aj, µ A j (ω,J)) < uj(sj, µ ) for some | ∈ − | − |D j t+1 − sj Sj. By B , for all j J and ω0 PJ (ω), uj sj (ω0) , µj A j (ω,J) (ω0) < − ∈ ⊆ D ∈ ∈ 0 | uj sj, µj B j (ω0) for some sj Sj. (ii) Let j J . Then, Bj = BR (B j) and | − ∈ ∈ −
A j(ω, J) B j = . Because the model ( ) is null-event-belief-complete, for each − ∩ −
∅ M G bj Bj, there exists ω0 PJ (ω) such that uj bj, µj B j (ω0) uj sj, µj B j (ω0) ∈ ∈ | − ≥ | − sj Sj. Since coalition J is Bayesian rational at ω, by Definition 2, s (ω) B. This ∀ ∈
∈
is a contradiction.

0 0 0 t+1 3. J = 1 (with J = j ): Let d 0 BR µ 0 t+1 (ω) . Then, d 0 0 j j (A(ω,J) ) j0 j j | | { } ∈ | ∩D − ∈ D t t+1  t+1 t+1  because of A J (ω, J) J = J and BR j0 j0 . Define B dj0 − ⊆ D− D− D− ⊆ D ≡ { } × t+1 (A(ω, J) )J J0 A J (ω, J). We proceed to show B satisfies Definition 2(C1-2) − ∩ D \ × 0 at ω for coalition J. (i) Let j J. If j = j , then A j(ω, J) B j = . Similarly to − − ∈ 6 0 ∩ ∅ Case 2(i), Definition 2(C2) is satisfied. If j = j , then µ 0 B (ω0) = µ 0 t+1 (ω0) j j0 j 0 | − |D j t t+1 − ω0 PJ (ω). Since A 0 (ω) B 0 = and via J, it follows that for all ∀ ∈ j ∩ j ∅ D V D ω P (ω), u 0 (s 0 (ω ) , µ 0 (ω )) < u 0 (s 0 , µ 0 (ω )) for some s 0 S 0 . 0 J j j 0 j A j0 (ω,J) 0 j j j B j0 0 j j ∈ | − | − ∈ (ii) Since d 0 BR µ 0 (ω) , Definition 2(C2) is satisfied. Since coalition J is j j B j0 ∈ | − Bayesian rational atω, by Definition 2, s (ω) B. This is a contradiction. ∈

17 (b) By Luo and Yang’s[32] Theorem 1, ∗ is a CRS; in particular, BR ∗ i = i∗ for R R− R all i I. Define an epistemic model for as follow: ∈ G

( ) < Ω, Pi ( ) i I , µi ( ) i I , si ( ) i I >, M G ≡ { · } ∈ { | · } ∈ { · } ∈ such that

Ω = (r , µ ) : r ∗ and µ ∆∗ (S ) such that r BR µ i I ; i i i I i i i ∗ i i i ∗ i | ∈ ∈ R | ∈ R − ∈ |R− ∀ ∈ i nI, si (ω) = ri and µi (ω) = µi for ω = (ri, µi )i I in Ω;   o ∀ ∈ | | | ∈ i I, Pi (ω) = ω0 Ω: si (ω0) = si (ω) and µ (ω0) = µ (ω) for ω Ω. ∀ ∈ { ∈ i| i| } ∈

Clearly, s (Ω) = ∗; ω Ω, s i (Pi (ω)) = ∗ i i I and PJ (ω) = Ω for non- R ∀ ∈ − R− ∀ ∈ singleton coalition J. Let (ri, µi )i I Ω. Suppose µ ∆∗ (S i) and µ S i = µi S i . | ∈ ∈ | ∈ − | − | − Because µ ∆∗ (S ) and r ∗ BR µ , it follows that µ ∆∗ (S ) and i ∗ i i i i ∗ i ∗ i | ∈ R − ∈ R ∩ |R− | ∈ R − r BR µ . Thus, (r , µ ) , r , µ  P (ω) for non-singleton coalition i i∗ ∗ i i j j j I i J ∈ R ∩ |R− | | ∈ \{ } ∈ J. Therefore,  ( )is a null-event-belief-complete model. M G
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 2(C1-2) and B J = A J (ω, J) (where A(ω, J) i I si (PJ (ω))), but s (ω) / B. − − ≡ × ∈ ∈ By the construction of Ω, s (PJ (ω)) = s (Ω) = ∗. Clearly, B J = ∗ J . We proceed to R − R− show ∗ B via J. Since B ∗, Definition 1(2) is void. Next, we verify Definition 1(1) R V ⊆ R holds for all j J. Consider rj j∗ and µ ∆∗ (S j). We distinguish two cases. ∈ ∈ R | ∈ R∗ − 1. If r BR µ , by the construction of Ω, there exists ω P (ω) = Ω such that j ∗ j J ∈ |R− ∈ s (ω) = r and µ(ω) = µ . By Definition 2(C1), u r , µ < u s , µ for j j j j j ∗ j j j B j | | |R− | − some s S whenever r / B or µ = µ . j j j j B j ∗ j   ∈ ∈ | − 6 |R−
2. If r / BR µ , there exists r BR µ such that u r , µ < j ∗ j j∗ ∗ j j j ∗ j ∈ |R− ∈ |R− |R− u r , µ . If µ  = µ , u r , µ < u r, µ . If µ  = µ , by j j∗ ∗ j B j ∗ j j j ∗ j j j∗ B j B j ∗ j |R− | − |R− |R− | − | − 6 |R− the construction of Ω, there exists ω PJ (ω) = Ω such that sj (ω) = r∗ and µ (ω) = ∈
j j| µ . Again by Definition 2(C1), u r , µ < u r , µ < u s , µ for j j ∗ j j j∗ ∗ j j j B j | |R− |R− | − some sj Sj.     ∈

18 Therefore, ∗ B. Since ∗ is a CRS, B = ∗ and s (ω) B. This is a contradiction. R V R R ∈ Hence, for all ω Ω, players are Bayesian c-rational at ω. Consequently, s CKR◦ = ∈   s (Ω) = ∗. R  Proof of Corollary 1: (a) Under the restriction of singleton coalitions, (i) R◦ = R by

Proposition 1, and (ii) ∗ = (cf. Luo and Yang [32]). Because the requirement of null- R R event-belief-completeness is void for singleton coalitions, Corollary 1(a) follows directly from Theorem 1(a).

(b) Consider the model ( ) by replacing ∗ with , in the proof of Theorem 1(b). M G R R In this modified model ( ), players are Bayesian rational across the state space Ω, and M G 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 ( ). Then players are Bayesian c-rational at all ω Ω. Therefore, M G ∈ ◦ Ω = CKR. By Theorem 1, for all ω Ω, the strategy profile s∗ = s(ω) is Bayesian ∈ c-rationalizable in . G “Only if”part: Let s∗ be a Bayesian c-rationalizable strategy profile in . Then s∗ ∗. G ∈ R Consider the model ( ) defined in the proof of Theorem 1(b). Then players are Bayesian M G c-rational at all ω Ω. That is, s is an a posteriori Bayesian c-equilibrium in ( ). ∈ M G By the construction of ( ), we can find an a posteriori Bayesian c-equilibrium s in a M G null-event-belief-complete model ( ) such that s∗ = s(ω) for some ω = (si∗, µi )i I Ω. M G | ∈ ∈ 

19 Appendix+: Supplementary Materials

We define an operator for our notion of Bayesian c-rationality in Definition 2. Let

o γ (A, J) all the sets B used in Definition 2B, ≡ ∩ o where A = A(ω, J). Note that γ (A, J) depends not only on coalition J, but also implicitly o on the epistemic state ω. We show, through an example, the “state-dependent”operator γ is not a “sensible”operator in the sense of Ambrus [3]. Consider a two-person game below.

a b a 2, 2 3, 2 . b 2, 3 3, 3

We construct an epistemic model ( ) for this game as follows: M G

Ω = ωaa, ωab, ωba, ωbb with typical state ωxy Ω. • { } ∈

P1(ωaa) = P1(ωab) = ωaa, ωab , P1(ωba) = ωba , P1(ωbb) = ωbb ; • { } { } { } P2(ωaa) = P2(ωba) = ωaa, ωba , P2(ωab) = ωab , P2(ωbb) = ωbb . { } { } { }

For ωxy Ω and i = 1, 2, s (ωxy) = (x, y) and µi (ωxy) ∆∗ ( a, b ) • ∈ 1 |1 ∈ { } such that (1) µ1 a,b (ωaa) = µ1 a,b (ωab) = a+ b, µ1 a,b (ωba) = |{ } |{ } 2 ◦ 2 ◦ |{ } 1 a, and µ1 a,b (ωbb) = 1 b, and (2) µ2 a,b (ωaa) = µ2 a,b (ωba) = { } { } { } 1 ◦ 1 | ◦ | | a + b, µ2 a,b (ωab) = 1 a, and µ2 a,b (ωbb) = 1 b. 2 ◦ 2 ◦ |{ } ◦ |{ } ◦ o Let J = 1, 2 . Then A(ωaa,J) = a, b a, b . By Definition 2, γ (A, J) = b b { } { } × { } o { } × { } for A = A(ωaa,J). Apparently, A is closed under rational behavior, but γ (A, J) fails to be o closed under rational behavior. Thus, the operator γ violates Definition 6(i) in Ambrus [3]. (Also, ( ) is a null-event-belief-complete model.) M G

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