Mean field equilibrium in dynamic games with complementarities

We study stochastic dynamic games with a large number of players, where players are coupled via their payoff functions. We consider mean field equilibrium for such games: in such an equilibrium, each player reacts to only the long run average state of other players. In this paper we focus on a special class of stochastic games, where a player experiences strategic complementarities from other players; formally the payoff of a player has increasing differences between her own state and the aggregate empirical distribution of the states of other players. We find necessary conditions for the existence of a mean field equilibrium in such games. Furthermore, as a simple consequence of this existence theorem, we obtain several natural monotonicity properties. We show that there exist a “largest” and “smallest” equilibrium among all those where the equilibrium strategy used by a player is nondecreasing. We also show that natural best response dynamics converge to each of these equilibria.