Social Optima of Backward Linear-Quadratic-Gaussian Mean-Field Teams

Xinwei Feng, Jianhui Huang, Shujun Wang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

4 Citations (Scopus)


This paper studies a class of stochastic linear-quadratic-Gaussian (LQG) dynamic optimization problems involving a large number of weakly-coupled heterogeneous agents. By “heterogeneous,” we mean agents are endowed with different types of parameters thus they are not statistically identical. Specifically, discrete-type heterogeneous agents are considered here which are more practical than homogeneous-type agents, and at the same time, more tractable than continuum-type heterogeneous agents. Unlike well-studied mean-field-game, these agents formalize a team with cooperation to minimize some social cost functional. Moreover, unlike standard social optima literature, the state here evolves by some backward stochastic differential equation (BSDE) in which the terminal instead initial condition is specified. Accordingly, the related social cost is represented by some recursive functional for which the initial state is considered. Applying a backward version of person-by-person optimality, we construct an auxiliary control problem for each agent based on decentralized information. The decentralized social strategy is derived by a class of new consistency condition (CC) systems, which are mean-field-type forward-backward stochastic differential equations (FBSDEs). The well-posedness of such consistency condition system is obtained via Riccati decoupling method. The related asymptotic social optimality is also verified.

Original languageEnglish
Pages (from-to)651-694
Number of pages44
JournalApplied Mathematics and Optimization
Issue number1
Early online date4 May 2021
Publication statusPublished - Dec 2021


  • Asymptotic social optima
  • Backward person-by-person optimality
  • Discrete-type heterogeneous system
  • Initially mixed-coupled FBSDE
  • LQG recursive control
  • Mean-field team

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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