We study a class of stochastic dynamic games that exhibit strategic
complementarities between players; formally, in the games we consider, the
payoff of a player has increasing differences between her own state and the
empirical distribution of the states of other players. Such games can be used
to model a diverse set of applications, including network security models,
recommender systems, and dynamic search in markets. Stochastic games are
generally difficult to analyze, and these difficulties are only exacerbated
when the number of players is large (as might be the case in the preceding
examples).
We consider an approximation methodology called mean field equilibrium to
study these games. In such an equilibrium, each player reacts to only the long
run average state 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 a "smallest" equilibrium
among all those where the equilibrium strategy used by a player is
nondecreasing, and we also show that players converge to each of these
equilibria via natural myopic learning dynamics; as we argue, these dynamics
are more reasonable than the standard best response dynamics. We also provide
sensitivity results, where we quantify how the equilibria of such games move in
response to changes in parameters of the game (e.g., the introduction of
incentives to players).Comment: 56 pages, 5 figure