Approximate Nash Equilibria with Near Optimal Social Welfare / 504
Artur Czumaj, Michail Fasoulakis, Marcin Jurdzinski
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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)2, 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 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 ≥ 1/2 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.