Nash Equilibria via Polynomial Equations

We consider the problem of computing a Nash equilibrium in multiple-player games. It is known that there exist games, in which all the equilibria have irrational entries in their probability distributions [19]. This suggests that either we should look for symbolic representations of equilibria or we should focus on computing approximate equilibria. We show that every finite game has an equilibrium such that all the entries in the probability distributions are algebraic numbers and hence can be finitely represented. We also propose an algorithm which computes an approximate equilibrium in the following sense: the strategies output by the algorithm are close with respect to l∞-norm to those of an exact Nash equilibrium and also the players have only a negligible incentive to deviate to another strategy. The running time of the algorithm is exponential in the number of strategies and polynomial in the digits of accuracy. We obtain similar results for approximating market equilibria in the neoclassical exchange model under certain assumptions.