Mean Field Equilibrium in Dynamic Games with Strategic Complementarities

Abstract

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 (for example, the introduction of incentives to players).

Opinion Dynamics and Stubbornness Via Multi-Population Mean-Field Games

“…We refer to Appendix A for a general treatment of the problem and the derivation of the relative PDEs. Because of the complexity of the optimization problem, as noticed in [16], it is unlikely that the agents will follow such an approach.…”

“…Nevertheless, we will see in the next section that this can be done by assuming that agents are myopic, in the sense of [16]. By myopic we mean that, when choosing their strategy at time t, agents consider the future populations' distributions as constant and equal to the present ones.…”

“…Within the class of linear control strategies, we determine analytically those that are optimal if the agents adopt the myopic perspective discussed in [16].…”

“…, and c i (t) in (16) are consequences of the fact that (17) presents an affine structure as in (4) in Assumption 1. The Gaussianity of the populations' distributions and the time evolution of means and variances as in equation (11), follow from Lemma 2.2, thanks to the fact that controls are affine.…”

“…However, in [16], it is pointed out that determining the solutions of a mean-field game is often impractical from a computational point of view; moreover, it does not naturally correspond to any reasonable dynamic process that agents are likely to follow in practice. For this reason, in [16] it is assumed that the agents implement myopic learning dynamics.…”