Bayesian coalitional rationalizability

In this paper we extend Ambrus's [A. Ambrus, Coalitional rationalizability, Quart. J. Econ. 121 (2006) 903-929] concept of “coalitional rationalizability (c-rationalizability)” to situations where, in seeking mutually beneficial interests, players in groups (i) make use of Bayes' rule in expectation calculations and (ii) contemplate various deviations, i.e., the validity of deviation is checked against any arbitrary sets of strategies, and not just only against restricted subsets of strategies. We offer an alternative notion of Bayesian c-rationalizability suitable for such complex social interactions. We show that Bayesian c-rationalizability possesses nice properties similar to those of conventional rationalizability.     

Interactive beliefs,epistemic independence and strong rationalizability

We use a universal, extensive form interactive beliefs system to provide an epistemic characterization of a weak and a strong notion of rationalizability with independent beliefs. The weak solution concept is equivalent to backward induction in generic perfect information games where no player moves more than once in any play. The strong solution concept is related to explicability (Reny,22) and is outcome-equivalent to backward induction in generic games of perfect information.     

Cooperative games in permutational structure

Summary. By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition being a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. In this paper we consider non-transferable utility games in which for any coalition the set of payoffs depends on a permutation or ordering upon any partition of the coalition into subcoalitions. We call such a game a cooperative game in permutational structure or shortly permutational game. Doing so we extend the scope of the standard cooperative game theory in dealing with economic or political problems. Next we define the concept of core for such games. By introducing balancedness for ordered partitions of coalitions, we prove the nonemptiness of the core of a balanced non-transferable utility permutational game. Moreover we show that the core of a permutational game coincides with the core of an induced game in coalitional structure, but that balancedness of the permutational game need not imply balancedness of the corresponding coalitional game. This leads to a weakening of the conditions for the existence of a nonempty core of a game in coalitional structure, induced by a game in permutational structure. Furthermore, we refine the concept of core for the class of permutational games. We call this refinement the balanced-core of the game and show that the balanced-core of a balanced permutational game is a nonempty subset of the core. The proof of the nonemptiness of the core of a permutational game is based on a new intersection theorem on the unit simplex, which generalizes the well-known intersection theorem of Shapley. Received: October 31, 1995; revised version: February 5, 1997     

On weighted Kalai–Samet solutions for non-transferable utility coalitional form games

We investigate the implications of the axiom of coalitional concavity for non-transferable utility coalitional form games. This axiom says that if the feasible set of some coalition is uncertain whereas the feasible sets of other coalitions are known, then all players in the coalition with the uncertain feasible set should (weakly) benefit from reaching a compromise before the uncertainty is resolved. By imposing this axiom, in addition to other minor axioms, we characterize the weighted Kalai–Samet [Econometrica 53 (1985) 307] solutions: these solutions coincide with the weighted egalitarian solutions on the domain of bargaining problems, and with the weighted Shapley values on the domain of transferable utility coalitional form games.     

Preparation

A product set of pure strategies is a prep set (‘prep’ is short for ‘preparation’) if it contains at least one best reply to any consistent belief that a player may have about the strategic behavior of his opponents. Minimal prep sets are shown to exist in a class of strategic games satisfying minor topological conditions. The concept of minimal prep sets is compared with (pure and mixed) Nash equilibria, rationalizability, minimal curb sets, and persistent retracts.     

Strong Belief and Forward Induction Reasoning

We provide a unified epistemic analysis of some forward-induction solution concepts in games with complete and incomplete information. We suggest that forward induction reasoning may be usefully interpreted as a set of assumptions governing the players' belief revision processes, and define a notion of strong belief to formalize these assumptions. Building on the notion of strong belief, we provide an epistemic characterization of extensive-form rationalizability and the intuitive criterion, as well as sufficient epistemic conditions for the backward induction outcome in generic games with perfect information. Journal of Economic Literature Classification Numbers: C72, D82.     

Pillage and property

This paper introduces a class of coalitional games, called pillage games, as a model of Hobbesian anarchy. Any coalition can pillage, costlessly and with certainty, any less powerful coalition. Power is endogenous, so a pillage game does not have a characteristic function, but pillage provides a domination concept that defines a stable set, which represents an endogenous balance of power. Every stable set contains only finitely many allocations, and can be represented as a farsighted core. Additional results are obtained for particular games, including the game in which the power of each coalition is determined by its total wealth.     

Recent results on belief,knowledge and the epistemic foundations of game theory

We provide a self-contained, selective overview of the literature on the role of knowledge and beliefs in game theory. We focus on recent results on the epistemic foundations of solution concepts, including correlated equilibrium, rationalizability in dynamic games, forward and backward induction.     

Dynamic coalitional equilibrium

We study coalition formation processes of Konishi and Ray (2003) [27]. It is shown that an absorbing and deterministic process of coalition formation that also forms an equilibrium - satisfies a coalitional one-deviation property - does exist if one allows the process to be history dependent. All such dynamic equilibrium processes of coalition formation are characterized. Absorbing outcomes of dynamic equilibrium processes are also identified. It is shown that they always constitute a subset of the largest consistent set of Chwe (1994) [11]. A procedure that identifies a dynamic equilibrium process of coalition formation in finite time is constructed.     

Coalition formation as a dynamic process

We study coalition formation as an ongoing, dynamic process, with payoffs generated as coalitions form, disintegrate, or regroup. A process of coalition formation (PCF) is an equilibrium if a coalitional move to some other state can be “justified” by the expectation of higher future value, compared to inaction. This future value, in turn, is endogenous: it depends on coalitional movements at each node. We study existence of equilibrium PCFs. We connect deterministic equilibrium PCFs with unique absorbing state to the core, and equilibrium PCFs with multiple absorbing states to the largest consistent set. In addition, we study cyclical as well as stochastic equilibrium PCFs.     

Semi-core equivalence

Summary. The main requirements for equivalence of the core of an economy and the Walrasian equilibrium allocations are largeness and the freedom to form almost arbitrary coalitions in the Edgeworthian barter processes. We investigate whether constraints on coalition formation and coalitional barter alter these insights. Our notion of the semi-core imposes a restriction on the collection of formable coalitions that does not affect the fundamental equivalence property. Using our concept of the contract-core we show that additional constraints on coalitional barter can only be alleviated within an environment with sufficiently many formable coalitions: besides a contract-core equivalence theorem we show certain non-equivalence results.Received: December 11, 1995; revised version: September 17, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Information transmission in coalitional voting games

A core allocation of a complete information economy can be characterized as one that would not be unanimously rejected in favor of another feasible alternative by any coalition. We use this test of coalitional voting in an incomplete information environment to formalize a notion of resilience. Since information transmission is implicit in the Bayesian equilibria of such voting games, this approach makes it possible to derive core concepts in which the transmission of information among members of a coalition is endogenous. Our results lend support to the credible core of Dutta and Vohra [Incomplete information, credibility and the core, Math. Soc. Sci. 50 (2005) 148-165] and the core proposed by Myerson [Virtual utility and the core for games with incomplete information, Mimeo, University of Chicago, 2005] as two that can be justified in terms of coalitional voting.     

Individual preference rankings compatible with prices, income distributions and total resources

The compatibility of a given ranking with a dataset consisting of prices, income distributions and total resources is shown to be equivalent to the existence of a solution to a set of linear equalities and inequalities. Their structure makes their solution easier to compute than the solutions of Afriat’s inequalities that characterize the rationalizability of a finite set of individual consumption data. Exploiting this structure, we also give new proofs of the rationalizability of finite data sets where total resources are close to being collinear and the contractibility and pathconnectedness of the set that consists of rationalizable finite datasets.     

Uniqueness of stationary equilibrium payoffs in coalitional bargaining

We study a model of sequential bargaining in which, in each period before an agreement is reached, the proposer?s identity is randomly determined, the proposer suggests a division of a pie of size one, each other agent either approves or rejects the proposal, and the proposal is implemented if the set of approving agents is a winning coalition for the proposer. The theory of the fixed point index is used to show that stationary equilibrium expected payoffs of this coalitional bargaining game are unique. This generalizes Eraslan [34] insofar as: (a) there are no restrictions on the structure of sets of winning coalitions; (b) different proposers may have different sets of winning coalitions; (c) there may be a positive probability that no proposer is selected.     

Chernoff's dual axiom,revealed preference and weak rational choice functions

A new and a weaker concept of rationalizability is introduced and its implications on the properties of individual choice are investigated. Especially, (i) it is shown that a property of choice functions, called Chernoff's dual axiom in this paper, plays a crucial role for the rationality of choice, and (ii) weakening the concept of rationalizability facilitates characterization of revealed preference relations.     

Strong rationalizability for two-player noncooperative games

Summary. The paper introduces a version of rationalizability that ignores strategies that are supported by negligible sets of beliefs, where a negligible set is one whose Lebesgue measure is zero. The theory is developed solely for the special case of point rationalizability; conditions are then derived under which point rationalizability entails no loss of generality. When these conditions obtain, the predictions yielded by this approach are often (although not always) a significant reduction over what is predicted by rationalizability. Received: September 10 1996; revised version: July 18, 1997     

Strategic Rationality Orderings and the Best Rationalization Principle

In a finite game fix a space of extended probabilities over strategies and a profile of best response correspondences. A profile ofrationality orderingsis then given by an ordered partition of the set of strategies of each player, representing different degrees of rationality, where at-leastk+1-rational strategies are best responses against extended probabilities reflecting at leastkdegrees of rationality. This solution can be constructed inductively, providing a Bayesian foundation for controversial deletion procedures such as extensive form rationalizability and iterated weak dominance. Focusing on extensive games, this approach formalizes the best rationalization principle.Journal of Economic LiteratureClassification Number: C72.     

Partition function bargaining with public demands

In the partition function bargaining problem the value of a coalition depends on the coalition structure in which it is embedded. This paper applies the demand-making bargaining game of coalition formation to the three-player partition function bargaining problem. The values of some embedded coalitions appear to be strategically irrelevant. The strategically relevant values constitute a coalition function bargaining problem. A classification in terms of the associated coalition function bargaining problem is provided. For one of the three classes that are distinguished the set of equilibrium outcomes closely relates to the core of the associated coalitional problem. For another class of problems the equilibrium outcome corresponds to the Von Neumann-Morgenstern outcome. Received: February 14, 2000; revised version: December 13, 2002 RID="*" ID="*" I would like to thank Harold Houba and Gerard van der Laan for valuable discussions, and an anonymous referee for helpful comments. This research was conducted while I was at the Vrije Universiteit Amsterdam.     

Core-stable rings in auctions with independent private values

We propose a semi-cooperative game theoretic approach to check whether a given coalition is stable in a Bayesian game with independent private values. The ex ante expected utilities of coalitions, at an incentive compatible (noncooperative) coalitional equilibrium, describe a (cooperative) partition form game. A coalition is core-stable if the core of a suitable characteristic function, derived from the partition form game, is not empty. As an application, we study collusion in auctions in which the bidders? final utility possibly depends on the winner?s identity. We show that such direct externalities offer a possible explanation for cartels? structures (not) observed in practice.     

Rationalizability in large games

This paper characterizes both point-rationalizability and rationalizability in large games when societal responses are formulated as distributions or averages of individual actions. The sets of point-rationalizable and rationalizable societal responses are defined and shown to be convex, compact and equivalent to those outcomes that survive iterative elimination of never best responses, under point-beliefs and probabilistic beliefs, respectively. Given the introspection and mentalizing that rationalizability notions presuppose, one motivation behind the work is to examine their viability in situations where the terms rationality and full information can be given a more parsimonious, and thereby a more analytically viable, expression.