Abstract
We consider the problem of approximating the minmax value of a multi-player game in strategic form. We argue that in three-player games with 0-1 payoffs, approximating the minmax value within an additive constant smaller than ξ/2, where \(\xi = \frac{3-\sqrt5}{2} \approx 0.382\), 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 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 randomised many-one reduction and a deterministic Turing reduction. The latter, which may be seen as a derandomisation 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 derandomisation is general enough to also apply to related work about ε-Nash equilibria.
Keywords
- Nash Equilibrium
- Pure Strategy
- Polynomial Time Algorithm
- Polynomial Time Approximation Scheme
- Fully Polynomial Time Approximation Scheme
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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Eickmeyer, K., Hansen, K.A., Verbin, E. (2012). Approximating the Minmax Value of Three-Player Games within a Constant is as Hard as Detecting Planted Cliques. In: Serna, M. (eds) Algorithmic Game Theory. SAGT 2012. Lecture Notes in Computer Science, vol 7615. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33996-7_9
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DOI: https://doi.org/10.1007/978-3-642-33996-7_9
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