We focus on the problem of computing an
-Nash equilibrium of a bimatrix game, when
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
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.