Search

This paper proposes a model for incomplete-information games where the knowledge of the players is represented by a Dempster-Shafer belief function. Beyond an extension of the classical definitions, it shows such a game can be transformed into an equivalent hypergraphical complete-information game (without uncertainty), thus generalizing Howson and Rosenthal's theorem to the framework of belief functions and to any number of players. The complexity of this transformation is finally studied and shown to be polynomial in the degree of k -additivity of the mass function. [ABSTRACT FROM AUTHOR]

A game-theoretical concept of an exact (cooperative) game corresponds to the notion of a discrete coherent lower probability, used in the context of imprecise probabilities. The collection of (suitably standardized) exact games forms a pointed polyhedral cone and the paper is devoted to the recognition of extreme rays of that cone, whose generators are called extreme exact games . We give a necessary and sufficient condition for an exact game to be extreme. Our criterion leads to solving a simple linear equation system determined by a certain min-representation of the game. It has been implemented on a computer and a web-based platform for testing the extremity of an exact game is available, which works with a modest number of variables. The paper also deals with different min-representations of a fixed exact game μ , which can be compared with the help of the concept of a tightness structure (of a min-representation) introduced in the paper. The collection of tightness structures (of min-representations of μ ) is shown to be a finite lattice with respect to a refinement relation. We give a method to obtain a min-representation with the finest tightness structure, which construction comes from the coarsest standard min-representation of μ given by the (complete) list of vertices of the core (polytope) of μ . The newly introduced criterion for exact extremity is based on the finest tightness structure. [ABSTRACT FROM AUTHOR]

Experimental results on the Ultimatum Game indicate that receivers may reject non-zero offers, even though that seems irrational. The explanation is that, when players are treated unfairly, they can act against strict rationality. This paper discusses an evolutionary model of the Ultimatum Game describing how populations of players change their behaviour in time. We prove an analytical result that establishes under what conditions receivers tend to reject unfair offers. The response to unfair offers is also shown to be sensitive to different degrees of unfairness. We then introduce a Bayesian game to translate our result from populations to individual players. [ABSTRACT FROM AUTHOR]

Since the seminal work of John Nash, convex combinations of actions are known to guarantee the existence of equilibria in strategic-form games. This paper introduces an alternative notion of randomisation among actions – possibilistic randomisation – and investigates the mathematical consequences of doing so. The framework of possibility theory gives rise to two distinct notions of equilibria both of which are characterised in our main results: a qualitative one based on the Sugeno integral and a quantitative one based on the Choquet integral. Then the two notions of equilibrium are compared against a coordination game with payoff-distinguishable equilibria known as the Weak-link game. [ABSTRACT FROM AUTHOR]

In this paper, we study new classes of value operators for coalitional games with players organized into a boolean algebra. Coalitional games are cooperative models in which players can form coalitions to maximize profit. The basic solution concepts in such game scenarios are value operators, which assign a unique real value to every player, reflecting thus selected principles of economic rationality. Some value concepts were extended beyond the classic coalitional model where every coalition of players can form. In particular, the extension of Shapley value exists for coalitional games in which players are partially ordered, and the feasible coalitions are the corresponding down-sets. Interestingly, this game-theoretic framework was employed in the method called Information Attribution. This method aims to solve the information decomposition problem, which asks for a particular additive decomposition of the mutual information between the input and target random variables. In such information-theoretic games, the players are predictors, and their set has the natural structure of a boolean algebra. Motivated by the original problem, we consider coalitional games where the players form a boolean algebra, and the coalitions are the corresponding down-sets. This more general approach enables us to study various value solution concepts in detail. Namely, we focus on the classes of values that can represent alternatives to the solution of the information decomposition problem, such as random-order values or sharing values. We extend the axiomatic characterization of some classes of values that were known only for the standard coalitional games. [ABSTRACT FROM AUTHOR]