Approximate Nash Equilibria for Multi-player Games

• Authors: Sébastien Hémon;Michel Rougemont;Miklos Santha
• Venue: SAGT '08 Proceedings of the 1st International Symposium on Algorithmic Game Theory
• Year: 2008

We consider games of complete information with r≥ 2players, and study approximate Nash equilibria in the additive andmultiplicative sense, where the number of pure strategies of theplayers is n. We establish a lower bound of$\sqrt[r-1]{\frac{{\rm ln} n - 2 {\rm ln} {\rm ln} n - {\rm ln}r}{{\rm ln} r}} $ on the size of the support of strategy profileswhich achieve an ε-approximate equilibrium, forεr-1/rin the additive case, andεr- 1 in the multiplicative case. Weexhibit polynomial time algorithms for additive approximation whichrespectively compute an $\frac{r-1}{r}$-approximate equilibriumwith support sizes at most 2, and which extend the algorithms for 2players with better than $\frac{1}{2}$-approximations to computeε-equilibria with εr-1/r. Finally, we investigate the sampling basedtechnique for computing approximate equilibria of Lipton et al.[12] with a new analysis, that instead of Hoeffding's bound usesthe more general McDiarmid's inequality. In the additive case weshow that for 0 εε-approximate Nash equilibrium with support size$\frac{2r {\rm ln} (nr+r)}{\varepsilon^2}$ can be obtained,improving by a factor of rthe support size of [12]. Wederive an analogous result in the multiplicative case where thesupport size depends also quadratically on g-1,for any lower bound gon the payoffs of the players at somegiven Nash equilibrium.