Polynomial Algorithms for Approximating Nash Equilibria of Bimatrix Games

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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 $\frac{3}{4}$ -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 $\frac{2+\lambda}{4}$ -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.