Noisy Directional Learning and the Logit Equilibrium

We specify a dynamic model in which agents adjust their decisions toward higher payoffs, subject to normal error. This process generates a probability distribution of players’ decisions that evolves over time according to the Fokker–Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This “logit equilibrium” forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.