Shared equilibrium strategy profile for a finite number of bi-matrix games

Abstract

We show that a strategy profile is an equilibrium strategy profile for an arbitrary finite set of bi-matrix games if and only if it solves a bi-linear programming problem and the optimal value of the bi-linear programming problem is zero.  An immediate consequence of this result, is that if the payoff matrix for the row player is a convex combination of its payoff matrices in the finite collection and the payoff matrix for the column player is a (possibly different) convex combination of its payoff matrices in the finite collection, then a strategy profile is an equilibrium strategy profile for the pair of matrices if and only if it satisfies the same two conditions.