Sampling from the Gibbs Distribution in Congestion Games

Abstract

Logit dynamics is a form of randomized game dynamics where players have a bias towards strategic deviations that give a higher improvement in cost [3, 6]. It is used extensively in practice, but not well-understood from a theoretical perspective. In congestion (or potential) games [8], the dynamics converges to the so-called Gibbs distribution over the set of all strategy profiles, when interpreted as a Markov chain (see, e.g., [2]). In general, logit dynamics might converge slowly to the Gibbs distribution (i.e., the corresponding Markov chain is slowly mixing), but beyond that, not much is known about its algorithmic aspects, nor that of the Gibbs distribution. In this work, we are interested in the following two questions for congestion games: (1) Is there an efficient algorithm for sampling from the Gibbs distribution? (2) If yes, does there also exist natural randomized dynamics that converges quickly to the Gibbs distribution? These questions can be seen as dynamic analogues of two of the most important questions regarding the computation of Nash equilibria: i) Can a Nash equilibrium be computed efficiently? ii) If yes, does there also exist natural player dynamics (e.g., best response dynamics) quickly reaching a Nash equilibrium?