Approximate Nash Equilibria for Multi-player Games

We consider games of complete information with r ≥ 2 players, and study approximate Nash equilibria in the additive and multiplicative sense, where the number of pure strategies of the players is n. We establish a lower bound of r−1 √ lnn−2 ln lnn−ln r ln r on the size of the support of strategy profiles which achieve an ε-approximate equilibrium, for ε < r−1 r in the additive case, and ε < r− 1 in the multiplicative case. We exhibit polynomial time algorithms for additive approximation which respectively compute an r−1 r -approximate equilibrium with support sizes at most 2, and which extend the algorithms for 2 players with better than 1 2 -approximations to compute εequilibria with ε < r−1 r . Finally, we investigate the sampling based technique for computing approximate equilibria of Lipton et al.[12] with a new analysis, that instead of Hoeffding’s bound uses the more general McDiarmid’s inequality. In the additive case we show that for 0 < ε < 1, an ε-approximate Nash equilibrium with support size 2r ln(nr+r) ε2 can be obtained, improving by a factor of r the support size of [12]. We derive an analogous result in the multiplicative case where the support size depends also quadratically on g−1, for a lower bound g on the payoffs of the players at some given Nash equilibrium.