Polynomial algorithms for approximating Nash equilibria of bimatrix games

We focus on the problem of computing an ϵ-Nash equilibrium of a bimatrix game, when ϵ is an absolute constant. We present a simple algorithm for computing a -Nash equilibrium for any bimatrix game in strongly polynomial time and we next show how to extend this algorithm so as to obtain a (potentially stronger) parameterized approximation. Namely, we present an algorithm that computes a -Nash equilibrium, where λ is the minimum, among all Nash equilibria, expected payoff of either player. The suggested algorithm runs in time polynomial in the number of strategies available to the players.