New complexity results about Nash equilibria

We provide a single reduction that demonstrates that in normal-form games: 1) it is NP-complete to determine whether Nash equilibria with certain natural properties exist (these results are similar to those obtained by Gilboa and Zemel [17]), 2) more significantly, the problems of maximizing certain properties of a Nash equilibrium are inapproximable (unless P = NP), and 3) it is #P-hard to count the Nash equilibria. We also show that determining whether a pure-strategy BayesNash equilibrium exists in a Bayesian game isNP-complete, and that determining whether a pure-strategy Nash equilibrium exists in a Markov (stochastic) game is PSPACE-hard even if the game is unobserved (and that this remainsNP-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.