On the existence of pure epsilon-equilibrium
Bary S. R. Pradelski, Bassel Tarbush
TL;DR
The paper addresses the prevalence of pure $\epsilon$-equilibria in the space of all normal-form games with randomly drawn utilities. It employs the probabilistic method and Chen–Stein Poisson approximation to show that for any $\epsilon>0$, the asymptotic share of games admitting a pure $\epsilon$-equilibrium tends to 1 as the number of agents grows with bounded actions, with an exponential convergence rate. This result extends to the $\epsilon^\star$ version (where all but at most one player nearly optimizes) and is robust to positive dependence among utilities and to well-connected interaction graphs; by contrast, the share of games with a pure Nash equilibrium ($\epsilon=0$) converges to $1-1/e$. When the number of actions grows, the outcome depends on the tail behavior of $F$ through the hazard rate, yielding a nuanced regime where the asymptotic share can lie above or below that benchmark. The findings illuminate how small deviations from perfect rationality can guarantee the existence of stable outcomes in large strategic environments using probabilistic methods and Poisson approximations.
Abstract
We show that for any $ε>0$, as the number of agents gets large, the share of games that admit a pure $ε$-equilibrium converges to 1. Our result holds even for pure $ε$-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 $ε=0$, is asymptotically $1-1/e\approx 0.63$. This suggests that very small deviations from perfect rationality, captured by positive values of $ε$, suffice to ensure the general existence of stable outcomes. We also study the existence of pure $ε$-equilibrium when the number of actions gets large. Our proofs rely on the probabilistic method and on the Chen-Stein method.