On Perfect Nash Equilibria of Polymatrix Games

Slim Belhaiza

doi:10.1155/2014/937070

Abstract

When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, the perfectness concept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, this undominance property however does not extend to n-player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of n-player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided.

Highlights

Interest for game theoretic applications has been growing in engineering, management and political sciences
Decision makers can often be represented by autonomous agents such as hardware or software, which are unable to distinguish between a set of Nash equilibria unless a refinement procedure is used
This paper presents a new characterization of perfect Nash equilibria for polymatrix games

Summary

Introduction

Interest for game theoretic applications has been growing in engineering, management and political sciences. A polymatrix game is a confrontation of n players (n ≥ 2) in a normal and noncooperative context. Polymatrix games form a particular class of n-player games. A polymatrix game G[(Aij)i=j ] with n players is such that player i’s payoff relative to player j’s decisions is independent from the remaining players’ choices. N} as the set of all players, each player i ∈ N controls a finite set of pure strategies Si = {si1, .

Results

Discussion

Conclusion