Completely uncoupled dynamics and Nash equilibria

Completely uncoupled dynamics are a repeated play of a game, where every period each player knows only his own action set and the history of his own past actions and payoffs; thus, he does not know anything about the other player's actions and payoffs. The main contributions of the present paper are the following. First, there exist no completely uncoupled dynamics that lead to almost sure convergence of play to pure Nash equilibria in almost all games possessing pure Nash equilibria. Second, the above result does not hold for Nash ε-equilibrium: we exhibit completely uncoupled dynamics that lead to almost sure convergence of play to Nash ε-equilibrium.