Mixed Social Optima and Nash equilibrium in Linear-Quadratic-Gaussian Mean-field System

TL;DR

This paper studies a complex large-scale stochastic control system involving a major agent and many minor agents, deriving decentralized strategies and equilibria using advanced mean-field and stochastic analysis techniques.

Contribution

It introduces a novel mean-field forward-backward stochastic differential equation framework for mixed social optimization and Nash equilibrium in LQG systems with both major and minor agents.

Findings

Derived decentralized social strategies via new CC system.

Proved well-posedness of the CC system using discounting.

Established asymptotic social optimality and Nash equilibrium.

Abstract

This paper investigates a class of mixed stochastic linear-quadratic-Gaussian (LQG) 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 are conflictive to that of major agent. Thus, major agent and all minor agents are further competitive to reach some non-zero Nash equilibrium. The control processes enter both diffusion and drift terms of all major and minors' states. This extends the standard setup in which control only enters the drift terms, and such extension brings more modeling difference and technical difficulties, in particular, when dealing with the feedback decentralized strategy via Riccati equation and mean-field consistency condition (CC) representation. Applying the mean-field approximations and person-by-person optimality, we obtain auxiliary control problems for major agent and minor agents, respectively. The decentralized social strategy is derived by a class of new CC system, which is mean-field forward-backward stochastic differential equations. The well-posedness of the 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.