Complexity of Rational and Irrational Nash Equilibria

We introduce two new natural decision problems, denoted as ? RATIONAL NASH and ? IRRATIONAL NASH, pertinent to the rationality and irrationality, respectively, of Nash equilibria for (finite) strategic games. These problems ask, given a strategic game, whether or not it admits (i) a rational Nash equilibrium where all probabilities are rational numbers, and (ii) an irrational Nash equilibrium where at least one probability is irrational, respectively. We are interested here in the complexities of ? RATIONAL NASH and ? IRRATIONAL NASH. Towards this end, we study two other decision problems, denoted as NASH-EQUIVALENCE and NASH-REDUCTION, pertinent to some mutual properties of the sets of Nash equilibria of two given strategic games with the same number of players. The problem NASH-EQUIVALENCE asks whether or not the two sets of Nash equilibria coincide; we identify a restriction of its complementary problem that witnesses ? RATIONAL NASH. The problem NASH-REDUCTION asks whether or not there is a so called Nash reduction: a suitable map between corresponding strategy sets of players that yields a Nash equilibrium of the former game from a Nash equilibrium of the latter game; we identify a restriction of NASH-REDUCTION that witnesses ? IRRATIONAL NASH. As our main result, we provide two distinct reductions to simultaneously show that (i) NASH-EQUIVALENCE is co- $\mathcal{NP}$ -hard and ? RATIONAL NASH is $\mathcal{NP}$ -hard, and (ii) NASH-REDUCTION and ? IRRATIONAL NASH are both $\mathcal{NP}$ -hard, respectively. The reductions significantly extend techniques previously employed by Conitzer and Sandholm (Proceedings of the 18th Joint Conference on Artificial Intelligence, pp. 765–771, 2003; Games Econ. Behav. 63(2), 621–641, 2008).