Approximating the Minmax Value of Three-Player Games within a Constant is as Hard as Detecting Planted Cliques
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چکیده
We consider the problem of approximating the minmax value of a multiplayer 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 socalled planted clique problem in Erdős-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 -Nash equilibria.
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