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.

在本文中，我们研究了一种控制线性-二次高斯大种群系统，该系统结合了三种混合的互动主体，分别是主要领导者，次要领导者和次要跟随者。实际上，它们可能代表参与市场价格形成的三种典型参与者：主要供应商，次要供应商和次要生产者。Stackelberg-Nash-Cournot（SNC）近似平衡是由主要-次要平均场博弈（MFG）和领导者-跟随者Stackelberg博弈的组合得出的。尽管所有代理都处于前向状态，因为它们的动力学只指定了初始条件，但我们的SNC分析提供了一个MFG框架，该框架自然存在于向前和向后状态，同时指定了初始条件和终止条件。该结果与标准MFG框架文献中报道的结果不同，主要是因为采用了Stackelberg结构。通过变分分析，在开环情况下，一致性条件系统可以由具有高维块结构的一些完全耦合的前向后向随机随机微分方程表示。为了充分解决相关的可解性，我们还导出了SNC近似值的反馈形式。通过一些耦合的里卡蒂方程组的平衡策略。我们的研究包括各种平均场博弈模型作为特例。