Reversibility and Mixing Time for Logit Dynamics with Concurrent Updates

Logit dynamics [Blume, Games and Economic Behavior, 1993] is a randomized best response dynamics where at every time step a player is selected uniformly at random and she chooses a new strategy according to the “logit choice function”, i.e. a probability distribution biased towards strategies promising higher payoffs, where the bias level corresponds to the degree of rationality of the agents. While the logit choice function is a very natural behavioral model for approximately rational agents, the specific revision process that selects one single player per time step seems less justified. In this paper we thus focus on the dynamics where at every time step every player simultaneously updates her strategy according to the logit choice function. We call such a dynamics the “all-logit”, as opposed to the classical “one-logit” dynamics. The all-logit dynamics for a game induces an ergodic Markov chain over the set of strategy profiles which is significantly different from the Markov chain induced in the one-logit case. In this paper we first highlight similarities and differences between the two dynamics with some simple examples of two-player games; we then give a characterization of the class of games such that the Markov chains induced by the all-logit dynamics are reversible and we show it is a subclass of potential games; finally, we analyze the mixing time of the all-logit dynamics for a well-known coordination game.