Linear quadratic mean field social optimization: Asymptotic solvability

Abstract

This paper considers a linear-quadratic (LQ) mean field control problem involving a major player and a large number of minor players, where the dynamics and costs depend on random parameters. The objective is to optimize a social cost as a weighted sum of the individual costs under decentralized information. We apply the person-by-person optimality principle in team decision theory to the finite population model to construct two limiting variational problems whose solutions, subject to the requirement of consistent mean field approximations, yield a system of forwardbackward stochastic differential equations (FBSDEs). We show the existence and uniqueness of a solution to the FBSDEs and obtain decentralized strategies nearly achieving social optimality in the original large but finite population model. ).

Linear quadratic mean field social optimization: Asymptotic solvability and decentralized control

“…More importantly, here we take a new perspective by adopting a notion called asymptotic solvability. Some early analysis has been presented in the conference paper [28]. The asymptotic solvability approach was initially developed in LQ mean field games [30,31]; roughly that approach attempts to answer such a question for the games: Does there exist an intrinsic low dimensional object that governs the large system's solutions to generate a good asymptotic behavior when the population sizes tend to infinity.…”