Linear-Quadratic Mixed Stackelberg–Nash Stochastic Differential Game with Major–Minor Agents

Jianhui Huang, Kehan Si, Zhen Wu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

6 Citations (Scopus)


In this paper, we study a controlled linear-quadratic-Gaussian large population system combining three types of interactive agents mixed, which are respectively, major leader, minor leaders, and minor followers. In reality, they may represent three typical types of participants involved in market price formation: major supplier, minor suppliers and minor producers. The Stackelberg–Nash–Cournot (SNC) approximate equilibrium is derived from the combination of a major–minor mean-field game (MFG) and a leader–follower Stackelberg game. Although all agents are of forward states in that only initial conditions are specified in their dynamics, our SNC analysis provides an MFG framework that is naturally in a forward–backward state in that both initial and terminal conditions are specified. This result differs from those reported in the literature on standard MFG frameworks, mainly as a result of the adoption of a Stackelberg structure. Through variational analysis, the consistency condition system can be represented by some fully-coupled forward–backward-stochastic-differential-equations with a high-dimensional block structure in an open-loop case. To sufficiently address the related solvability, we also derive the feedback form of the SNC approximate. equilibrium strategy via some coupled Riccati equations. Our study includes various mean-field game models as its special cases.

Original languageEnglish
Pages (from-to)2445 - 2494
Number of pages50
JournalApplied Mathematics and Optimization
Issue number3
Early online date31 Aug 2020
Publication statusPublished - Dec 2021


  • Forward–backward-stochastic-differential-equations (FBSDE)
  • Leader–follower game
  • Major–minor (MM) game
  • Mean-field game (MFG)
  • Open-loop (OL) strategy
  • Stackelberg–Nash–Cournot (SNC) approximate equilibrium

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics


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