Approximate Nash Equilibria with Near Optimal Social Welfare

It is known that Nash equilibria and approximate Nash equilibria not necessarily optimize social optima of bimatrix games. In this paper, we show that for every fixed ε > 0, every bimatrix game (with values in [0, 1]) has an ε-approximate Nash equilibrium with the total payoff of the players at least a constant factor, (1 − √ 1− ε), of the optimum. Furthermore, our result can be made algorithmic in the following sense: for every fixed 0 ≤ ε∗ < ε, if we can find an ε∗-approximate Nash equilibrium in polynomial time, then we can find in polynomial time an ε-approximate Nash equilibrium with the total payoff of the players at least a constant factor of the optimum. Our analysis is especially tight in the case when ε ≥ 1 2 . In this case, we show that for any bimatrix game there is an ε-approximate Nash equilibrium with constant size support whose social welfare is at least 2 √ ε − ε ≥ 0.914 times the optimal social welfare. Furthermore, we demonstrate that our bound for the social welfare is tight, that is, for every ε ≥ 12 there is a bimatrix game for which every ε-approximate Nash equilibrium has social welfare at most 2 √ ε− ε times the optimal social welfare.