Abstracting Nash equilibria of supermodular games

Supermodular games are a well known class of noncooperative games which find significant applications in a variety of models, especially in operations research and economic applications. Supermodular games always have Nash equilibria which are characterized as fixed points of multivalued functions on complete lattices. Abstract interpretation is here applied to set up an approximation framework for Nash equilibria of supermodular games. This is achieved by extending the theory of abstract interpretation in order to cope with approximations of multivalued functions and by providing some methods for abstracting supermodular games, thus obtaining approximate Nash equilibria which are shown to be correct within the abstract interpretation framework

Abstract Interpretation of Supermodular Games

Supermodular games find significant applications in a variety of models, especially in operations research and economic applications of noncooperative game theory, and feature pure strategy Nash equilibria characterized as fixed points of multivalued functions on complete lattices. Pure strategy Nash equilibria of supermodular games are here approximated by resorting to the theory of abstract interpretation, a well established and known framework used for designing static analyses of programming languages. This is obtained by extending the theory of abstract interpretation in order to handle approximations of multivalued functions and by providing some methods for abstracting supermodular games, in order to obtain approximate Nash equilibria which are shown to be correct within the abstract interpretation framework

1-concave basis for TU games

The first stage of research, twenty years ago, on the subclass of 1-convex TU games dealt with its characterization through some regular core structure. Appealing abstract and practical examples of 1-convex games were missing until now. Both drawbacks are solved. On the one hand, a generating set for the cone of 1-concave cost games is introduced with clear affinities to the unanimity games taking into account the complementary transformation on coalitions. The dividends within this new game representation are used to characterize the 1-concavity constraint as well as to investigate the core property of the Shapley value for cost games. We present a simple formula to compute the nucleolus and the τ-value within the class of 1-convex/1-concave games and show that in a 1-convex/1-concave game there is an explicit relation between the nucleolus and the Shapley value. On the other hand, an appealing practical example of 1-concave cost game has cropped up not long ago in Sales’s Ph.D study of Catalan university library consortium for subscription to journals issued by Kluwer publishing house, the so-called library cost game which turn out to be decomposable into the abstract 1-concave cost games of the generating set mentioned above

Positional Games

Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201