On the existence of pure epsilon-equilibrium

We show that for any $\epsilon>0$, as the number of agents gets large, the share of games that admit a pure $\epsilon$-equilibrium converges to 1. Our result holds even for pure $\epsilon$-equilibrium in which all agents, except for at most one, play a best response. In contrast, it is known that the share of games that admit a pure Nash equilibrium, that is, for $\epsilon=0$, is asymptotically $1-1/e\approx 0.63$. This suggests that very small deviations from perfect rationality, captured by positive values of $\epsilon$, suffice to ensure the general existence of stable outcomes. We also study the existence of pure $\epsilon$-equilibrium when the number of actions gets large. Our proofs rely on the probabilistic method and on the Chen-Stein method.