  Noncooperative game theory provides a normative framework for analyzing
strategic interactions. However, for the toolbox to be operational, the
solutions it defines will have to be computed. In this paper, we provide a
single reduction that 1) demonstrates NP-hardness of determining whether Nash
equilibria with certain natural properties exist, and 2) demonstrates the
#P-hardness of counting Nash equilibria (or connected sets of Nash equilibria).
We also show that 3) determining whether a pure-strategy Bayes-Nash equilibrium
exists is NP-hard, and that 4) determining whether a pure-strategy Nash
equilibrium exists in a stochastic (Markov) game is PSPACE-hard even if the
game is invisible (this remains NP-hard if the game is finite). All of our
hardness results hold even if there are only two players and the game is
symmetric.
  Keywords: Nash equilibrium; game theory; computational complexity;
noncooperative game theory; normal form game; stochastic game; Markov game;
Bayes-Nash equilibrium; multiagent systems.
