Mixing Time and Stationary Expected Social Welfare of Logit Dynamics

We study “logit dynamics” [Blume, Games and Economic Behavior, 1993] for strategic games. This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a “noisy” best-response where the noise level is tuned by a parameter $\beta$. Such a dynamics defines a family of ergodic Markov chains, indexed by $\beta$, over the set of strategy profiles. We believe that the stationary distribution of these Markov chains gives a meaningful description of the long-term behavior for systems whose agents are not completely rational. Our aim is twofold: On the one hand, we are interested in evaluating the performance of the game at equilibrium, i.e. the expected social welfare when the strategy profiles are random according to the stationary distribution. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to its stationary distribution; i.e., the “mixing time” of the chain. In this paper we study the stationary expected social welfare for the 3-player CK game, for 2-player coordination games, and for two simple $n$-player games. For all these games, we also give almost tight upper and lower bounds on the mixing time of logit dynamics. Our results show two different behaviors: in some games the mixing time depends exponentially on $\beta$, while for other games it can be upper bounded by a function independent of $\beta$.

💡 Research Summary

The paper investigates logit dynamics, a stochastic revision process introduced by Blume (1993) for strategic games, where at each discrete time step a player is selected uniformly at random and updates her strategy according to a “noisy” best‑response. The noise level is controlled by a parameter β (inverse temperature): as β → ∞ the update becomes a deterministic best‑response, while β → 0 yields a completely random move. This rule defines, for every β, an ergodic Markov chain on the finite set of pure‑strategy profiles. Because the chain is reversible with respect to a Gibbs distribution, the stationary distribution πβ has the closed form
πβ(s) ∝ exp(β·W(s)),
where W(s) denotes the social welfare (sum of players’ utilities) of profile s. Consequently, the stationary expected social welfare Eπβ