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.

本文研究了一类涉及大量弱耦合异质代理的随机线性二次高斯（LQG）动态优化问题。“异质”是指代理具有不同类型的参数，因此它们在统计上并不相同。具体而言，此处考虑离散类型的异类代理，它们比同质类型的代理更实用，同时比连续体类型的代理更易于处理异构代理。与经过精心研究的均场博弈不同，这些代理商通过合作将团队正式化，以最大程度地降低某些社会成本。而且，与标准的社会最优文献不同，这里的状态是通过某种后向随机微分方程（BSDE）演变而来的，在该方程中指定了最终条件而不是初始条件。因此，相关的社会成本是由一些递归功能表示的量，初始状态被考虑。应用逐个个人最优性的向后版本，我们基于去中心化信息为每个代理构建辅助控制问题。分散的社会策略是由一类新的一致性条件（CC）系统派生的，这些系统是均值字段类型的前向后向随机微分方程（FBSDE）。这种一致性条件系统的适定性是通过Riccati解耦方法获得的。相关的渐近社会最优性也得到了验证。