A linear quadratic (LQ) stochastic optimization problem with delay involving weakly-coupled large population is investigated in this paper. Different to classic mean field (MF) game, here agents cooperate with each other to minimize the so-called social objective. With the aid of delayed person-by-person optimality principle, one arrives at an auxiliary LQ delayed control problem by decentralized information. A decentralized strategy is obtained by feat of an MF type anticipated forward-backward stochastic differential delay equation (AFBSDDE) consistency condition. The discounting method with delay feature is employed to solve the consistency condition system. Finally, by some estimates of AFBSDDEs we derive the asymptotic social optimality.

本文研究了涉及弱耦合大群体的线性二次（LQ）随机优化问题。与经典的平均场（MF）游戏不同，这里智能体相互合作以最小化所谓的社会目标。借助延迟个体最优性原理，通过分散信息得到辅助LQ延迟控制问题。分散策略是通过 MF 型预期前向-后向随机微分延迟方程 (AFBSDDE) 一致性条件获得的。采用具有延迟特性的折扣方法来求解一致性条件系统。最后，通过对 AFBSDDE 的一些估计，我们得出渐近社会最优性。