Regularity and Stability of Equilibrium Points of Bimatrix Games

Abstract

This paper deals with regular equilibrium points. Using properties of such equilibrium points, it is possible to give short proofs of known facts about completely mixed bimatrix games and bimatrix games with a unique equilibrium point. We prove that a bimatrix game with a convex equilibrium point set or with a finite number of equilibrium points has a regular equilibrium point. We shall show, moreover, that the class of all m × n-bimatrix games (m, n ∈ ℕ) for which all the equilibrium points are regular, is an open and dense subset of the class of all m × n-bimatrix games. Furthermore, it is shown that an isolated equilibrium point of a bimatrix game is stable if and only if it is a regular one. Here, we call an equilibrium point of a bimatrix game stable if, roughly speaking, all bimatrix games in a neighborhood of the game in question have an equilibrium point close to it.