Mixed Social Optima and Nash Equilibrium in Linear-Quadratic-Gaussian Mean-Field System

This article investigates a class of mixed stochastic linear-quadratic-Gaussian social optimization and Nash game in the context of a large-scale system. Two types of interactive agents are involved: a major agent and a large number of weakly coupled minor agents. All minor agents are cooperative to minimize the social cost as the sum of their individual costs, whereas such social cost is conflictive to that of the major agent. Thus, the major agent and all minor agents are further competitive to reach some nonzero-sum Nash equilibrium. Applying the mean-field approximations and person-by-person optimality, we obtain auxiliary control problems for the major agent and minor agents, respectively. The decentralized social strategy is derived by a class of new consistency condition (CC) system, which consists of mean-field forward-backward stochastic differential equations. The well-posedness of CC system is obtained by the discounting method. The related asymptotic social optimality for minor agents and Nash equilibrium for major-minor agents are also verified.