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Approximate Nash Equilibria for Multi-player Games

  • Sébastien Hémon
  • Michel de Rougemont
  • Miklos Santha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4997)

Abstract

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 \(\sqrt[r-1]{\frac{{\rm ln} n - 2 {\rm ln} {\rm ln} n - {\rm ln} r}{{\rm ln} r}} \) on the size of the support of strategy profiles which achieve an ε-approximate equilibrium, for \(\varepsilon < \frac{r-1}{r}\) in the additive case, and ε< r − 1 in the multiplicative case. We exhibit polynomial time algorithms for additive approximation which respectively compute an \(\frac{r-1}{r}\)-approximate equilibrium with support sizes at most 2, and which extend the algorithms for 2 players with better than \(\frac{1}{2}\)-approximations to compute ε-equilibria with \(\varepsilon < \frac{r-1}{r}.\) 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 \(\frac{2r {\rm ln} (nr+r)}{\varepsilon^2}\) 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 any lower bound g on the payoffs of the players at some given Nash equilibrium.

Keywords

Nash Equilibrium Mixed Strategy Pure Strategy Polynomial Time Algorithm Additive Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sébastien Hémon
    • 1
    • 2
  • Michel de Rougemont
    • 1
    • 3
  • Miklos Santha
    • 1
  1. 1.CNRS-LRI, Univ. Paris-SudOrsayFrance
  2. 2.LRDE-EPITA F-94276 Le Kremlin-BicetreFrance
  3. 3.Univ. Paris IIParisFrance

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