Approximate well-supported Nash equilibria in symmetric bimatrix games

The $\epsilon$-well-supported Nash equilibrium is a strong notion of approximation of a Nash equilibrium, where no player has an incentive greater than $\epsilon$ to deviate from any of the pure strategies that she uses in her mixed strategy. The smallest constant $\epsilon$ currently known for which there is a polynomial-time algorithm that computes an $\epsilon$-well-supported Nash equilibrium in bimatrix games is slightly below $2\mathrm{/}3$. In this paper we study this problem for symmetric bimatrix games and we provide a polynomial-time algorithm that gives a $(1\mathrm{/}2+\delta )$-well-supported Nash equilibrium, for an arbitrarily small positive constant $\delta$.