Stochastic control of mean field models with mixed players

Abstract

We consider mean field stochastic systems consisting of a major player and a large number of minor players. We study decentralized strategies for both game and social optimization problems. For the game problem, the objective is to obtain asymptotic Nash equilibria. For the social optimization problem, the objective is to nearly minimize a weighted sum of the individual costs. A very peculiar feature of the two problems is that the presence of the major player causes a lack of sufficient statistics for decentralized decision-making. This difficulty is overcome by a state space augmentation technique, which leads to stochastic, rather than deterministic, mean field approximations. For the social optimization problem, we apply a social cost perturbation argument to the augmented model to design decentralized strategies. © 2011 IFAC.