Linear-Quadratic Mean-Field-Game for Stochastic Delayed Systems

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24 Citations (Scopus)


This paper studies the linear-quadratic mean-field game (MFG) for a class of stochastic delayed systems. We consider a large-population system, where the dynamics of each agent is modeled by a stochastic differential delayed equation. The consistency condition is derived through an auxiliary system, which is an anticipated forward-backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such an AFBSDDE system can be obtained using a continuation method. Thus, the MFG strategies can be defined on an arbitrary time horizon, not necessary on a small time horizon by a commonly used contraction mapping method. Moreover, the decentralized strategies are verified to satisfy the ϵ-Nash equilibrium property. For illustration, three special cases of delayed systems are further explored, for which the closed-loop and open-loop MFG strategies are derived, respectively.

Original languageEnglish
Pages (from-to)2722-2729
Number of pages8
JournalIEEE Transactions on Automatic Control
Issue number8
Publication statusPublished - 1 Aug 2018


  • Anticipated forward-backward stochastic differential equation with delay (AFBSDDE)
  • continuation method
  • input delay
  • mean-field game (MFG)
  • stochastic differential equation with delay (SDDE)
  • ϵ-Nash equilibrium

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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