This paper studies the backward-forward linear-quadratic-Gaussian (LQG) games with major and minor agents (players). The state of major agent follows a linear backward stochastic differential equation (BSDE) and the states of minor agents are governed by linear forward stochastic differential equations (SDEs). The major agent is dominating as its state enters those of minor agents. On the other hand, all minor agents are individually negligible but their state-average affects the cost functional of major agent. The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies. We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation (BFSDE) system. Next, we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method. Consequently, we obtain the decentralized strategies for major and minor agents which are proved to satisfy the ε-Nash equilibrium property.

中文翻译：

---

具有主要和次要特工的后向线性二次方均场博弈

本文研究了具有主要和次要特工（玩家）的后向线性二次高斯（LQG）游戏。主要主体的状态遵循线性后向随机微分方程（BSDE），次要主体的状态由线性前向随机微分方程（SDE）控制。当主要代理进入其次要代理的状态时，它就处于主导地位。另一方面，所有次要代理人都可以单独忽略，但是它们的状态平均会影响主要代理人的成本功能。制定了在这种后向主要和次要环境中的均值博弈，以分析分散策略。我们首先通过辅助平均场SDE和3×2混合后向-前向随机微分方程（BFSDE）系统导出一致性条件。下一个，我们利用单调性方法来讨论这种BFSDE系统的适定性。因此，我们获得了主要代理人和次要代理人的分散策略，它们被证明满足ε-Nash平衡性质。