Mean-field approximation of forward-looking population dynamics

We study how the equilibrium dynamics of a continuum-population game approximate those of large finite-population games. New agents stochastically arrive to replace exiting ones and make irreversible action choices to maximize the expected discounted lifetime payoffs. The key assumption is that they only observe imperfect signals about the action distribution in the population. We first show that the stochastic process of the action distribution in the finite-population game is approximated by its mean-field dynamics as the population size becomes large, where the approximation precision is uniform across all equilibria. Based on this result, we then establish continuity properties of the equilibria at the large population limit. In particular, each agent becomes almost negligible, in the sense that in equilibrium, each agent's action is almost optimal against the (incorrect) belief that it has no impact on others' actions as presumed in the continuum-population case. Finally, for binary-action supermodular games, we show that when agents are patient, there is a unique equilibrium as observation noise becomes small while the population size becomes large. In this equilibrium, every agent chooses a risk-dominant action, and the population globally converges to the corresponding steady state.