Survival of the Strictest: Stable and Unstable Equilibria Under Regularized Learning with Partial Information

Abstract

In this paper, we examine the Nash equilibrium convergence properties of no-regret learning in general N -player games. For concreteness, we focus on the archetypal “follow the regularized leader” (FTRL) family of algorithms, and we consider the full spectrum of uncertainty that the players may encounter – from noisy, oracle-based feedback, to bandit, payoff-based information. In this general context, we establish a comprehensive equivalence between the stability of a Nash equilibrium and its support: a Nash equilibrium is stable and attracting with arbitrarily high probability if and only if it is strict (i.e., each equilibrium strategy has a unique best response). This equivalence extends existing continuous-time versions of the “folk theorem” of evolutionary game theory to a bona fide algorithmic learning setting, and it provides a clear refinement criterion for the prediction of the day-to-day behavior of no-regret learning in games.