Learning Dynamics in Continuous-Time Mean Field Games with Congestion

We study stationary equilibria and learning dynamics in a finite-state continuous-time mean field game with population-dependent action-completion rates generated by a smooth convex congestion potential. In this game, stationary equilibrium populations correspond one-to-one with optimizers of an Eisenberg–Gale-type convex program. We show that this variational structure also yields global convergence of the state–action dynamics under any fixed unichain policy, even though these dynamics are governed by a nonlinear Markov process. We then formulate an equivalent population game over deterministic unichain policies, where a concave ordinal potential gives convergence of best-response dynamics based on stationary long-run payoffs to the policy-equilibrium set. Finally, we study state–policy dynamics that track agents' physical states and committed policies under two revision mechanisms: exogenous opportunities arriving at a Poisson rate, and event-driven opportunities arising at action completions with a specified probability. Under both mechanisms, policy marginals converge to the policy-equilibrium set for every fixed admissible positive revision parameter. As the revision rate or probability tends to zero, the state–action populations associated with the omega-limit points of these dynamics approach the stationary mean field equilibrium populations. These results reveal how equilibrium emerges through repeated policy revisions and give our notion of stationary mean field equilibrium a dynamic foundation.