Date of Original Version
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Abstract or Description
We provide a single reduction that demonstrates that in normal-form games: (1) it is N P-complete to determine whether Nash equilibria with certain natural properties exist (these results are similar to those obtained by Gilboa and Zemel [Gilboa, I., Zemel, E., 1989. Nash and correlated equilibria: Some complexity considerations. Games Econ. Behav. 1, 80–93]), (2) more signiﬁcantly, the problems of maximizing certain properties of a Nash equilibrium are inapproximable (unless P = N P), and (3) it is #P-hard to count the Nash equilibria. We also show that determining whether a pure-strategy Bayes–Nash equilibrium exists in a Bayesian game is N P-complete, and that determining whether a pure-strategy Nash equilibrium exists in a Markov (stochastic) game is P S PAC E-hard even if the game is unobserved (and that this remains N P-hard if the game has ﬁnite length). All of our hardness results hold even if there are only two players and the game is symmetric.
Games and Economic Behavior, 63, 2, 621-641.