Long-Run Behavior and Convergence of Dynamic Mean Field Equilibria

We study the behavior of dynamic equilibria in mean field games with large time horizons in a dynamic consumer choice model. We show that if the stationary equilibrium in the associated infinite horizon game is unique, the dynamic equilibria of the finite horizon games converge to the stationary equilibrium of the infinite horizon game as the time horizon tends to infinity. If the stationary equilibrium is not unique, however, the situation becomes more involved. In this case, we show that in addition to convergence to the stationary equilibria, in the long run, the dynamic equilibria circle around randomized stationary equilibria for certain choices of boundary data.