On the stability of the logit dynamics in population games

We study the asymptotic stability of the logit evolutionary dynamics in population games, possibly with multiple heterogenous populations. For general population games, we prove that, on the one hand, strict Nash equilibria are asymptotically stable under the logit dynamics for low enough noise levels, on the other hand, a globally exponentially stable logit equilibrium exists for sufficiently large noise levels. This suggests the emergence of bifurcations in population games admitting multiple strict Nash equilibria, as observed in numerous examples. We then provide sufficient conditions on the population game structure for the existence of globally asymptotically stable logit equilibria for every noise level. The considered class of monotone separable games finds applications, e.g., in routing games on series compositions of networks with parallel routes when there are multiple populations of users that differ in the reward functions.