Table of Contents
Game Theory
Volume 2014, Article ID 937070, 11 pages
http://dx.doi.org/10.1155/2014/937070
Research Article

On Perfect Nash Equilibria of Polymatrix Games

1Department of Mathematics and Systems Engineering, MSB-MedTech, 1053 Les Berges du Lac II, Tunisia
2Department of Mathematics and Statistics, KFUPM, Dhahran 31261, Saudi Arabia

Received 29 May 2014; Accepted 10 September 2014; Published 29 September 2014

Academic Editor: Walter Briec

Copyright © 2014 Slim Belhaiza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of -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.