Article

# How Hard Is It to Approximate the Best Nash Equilibrium?

SIAM J. Comput 01/2011; 40:79-91. DOI: 10.1137/090766991

Source: DBLP

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**ABSTRACT:**When confronted with multiple Nash equilibria, decision makers have to refine their choices. Among all known Nash equilibrium refinements, the perfectness concept is probably the most famous one. It is known that weakly dominated strategies of two-player games cannot be part of a perfect equilibrium. In general, this undominance property however does not extend to -player games (E. E. C. van Damme, 1983). In this paper we show that polymatrix games, which form a particular class of -player games, verify the undominance property. Consequently, we prove that every perfect equilibrium of a polymatrix game is undominated and that every undominated equilibrium of a polymatrix game is perfect. This result is used to set a new characterization of perfect Nash equilibria for polymatrix games. We also prove that the set of perfect Nash equilibria of a polymatrix game is a finite union of convex polytopes. In addition, we introduce a linear programming formulation to identify perfect equilibria for polymatrix games. These results are illustrated on two small game applications. Computational experiments on randomly generated polymatrix games with different size and density are provided.Game Theory. 09/2014; 2014. - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider the problem of approximating the minmax value of a mul-tiplayer game in strategic form. We argue that in 3-player games with 0-1 payoffs, approximating the minmax value within an additive constant smaller than ξ/2, where ξ = 3− √ 5 2 ≈ 0.382, is not possible by a polynomial time algorithm. This is based on assuming hardness of a version of the so-called planted clique problem in Erd˝ os-Rényi random graphs, namely that of detecting a planted clique. Our results are stated as reductions from a promise graph problem to the problem of approximating the minmax value, and we use the detection problem for planted cliques to argue for its hardness. We present two reductions: a randomized many-one reduction and a deterministic Turing reduction. The latter, which may be seen as a derandomization of the former, may be used to argue for hardness of approximating the minmax value based on a hardness assumption about deterministic algorithms. Our technique for derandomization is general enough to also apply to related work about equilibria.10/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**This lecture continues last week's theme of somewhat specific data models to inform the analysis of clustering and graph partitioning heuristics. We begin with the classical notion of a mixture model.

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