Mixing Time and Stationary Expected Social Welfare of Logit Dynamics

We study logit dynamics [Blume, Games and Economic Behavior, 1993] for strategic games. At every stage of the game a player is selected uniformly at random and she plays according to a noisy best-response dynamics where the noise level is tuned by a parameter β. Such a dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. Our aim is twofold: On the one hand, we are interested in the expected social welfare when the strategy profiles are random according to the stationary distribution of the Markov chain, because we believe it gives a meaningful description of the long-term behavior of the system. 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 the stationary distribution; i.e., the mixing time of the chain. In this paper we study the stationary expected social welfare for the 3-player congestion game that exhibits the worst Price of Anarchy [Christodoulou and Koutsoupias, STOC'05], for 2-player coordination games (the same class of games studied by Blume), and for a simple n-player game. For all these games, we give almost-tight upper and lower bounds on the mixing time of logit dynamics.

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