Metastability of asymptotically well-behaved potential games

Abstract

One of the main criticisms to game theory concerns the assumption of full rationality. Logit dynamics is a decentralized algorithm in which a level of irrationality (a.k.a. “noise”) is introduced in players’ behavior. In this context, the solution concept of interest becomes the logit equilibrium, as opposed to Nash equilibria. Logit equilibria are distributions over strategy profiles that possess several nice properties, including existence and uniqueness. However, there are games in which their computation may take exponential time. We therefore look at an approximate version of logit equilibria, called metastable distributions, introduced by Auletta et al. [4]. These are distributions which remain stable (i.e., players do not go too far from it) for a large number of steps (rather than forever, as for logit equilibria). The hope is that these distributions exist and can be reached quickly by logit dynamics. We identify a class of potential games, that we name asymptotically well-behaved, for which the behavior of the logit dynamics is not chaotic as the number of players increases, so to guarantee meaningful asymptotic results. We prove that any such game admits distributions which are metastable no matter the level of noise present in the system, and the starting profile of the dynamics. These distributions can be quickly reached if the rationality level is not too big when compared to the inverse of the maximum difference in potential. Our proofs build on results which may be of independent interest, including some spectral characterizations of the transition matrix defined by logit dynamics for generic games and the relationship among convergence measures for Markov chains.