New Complexity Results about Nash Equilibria

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.