Metastability of the Logit Dynamics for Asymptotically Well-Behaved Potential Games

Abstract

Convergence rate and stability of a solution concept are classically measured in terms of “eventually” and “forever,” respectively. In the wake of recent computational criticisms to this approach, we study whether these timeframes can be updated to have states computed “quickly” and stable for “long enough”. Logit dynamics allows irrationality in players’ behavior and may take time exponential in the number of players n to converge to a stable state (i.e., a certain distribution over pure strategy profiles). We prove that every potential game, for which the behavior of the logit dynamics is not chaotic as n increases, admits distributions stable for a super-polynomial number of steps in n no matter the players’ irrationality and the starting profile of the dynamics. The convergence rate to these metastable distributions is polynomial in n when the players are not too rational. Our proofs build upon the new concept of partitioned Markov chains, which might be of independent interest, and a number of involved technical contributions.