This paper investigates a mixed leader-follower differential games problem, where the model involves two players with the same hierarchy in decision making and each player has two controls which act as a leader and a follower, respectively. Specifically, we solve a follower problem with unconstrained controls and obtain the corresponding Nash equilibrium. Then a leader problem with constrained controls is tackled and a pair of optimal constrained controls are presented by a projection mapping. Furthermore, the control weights are allowed to be singular. In this case, we first investigate the uniform convexity of the cost functional whose corresponding states are fully-coupled forward-backward stochastic differential equation. After that, the minimizing sequence of solutions with non-degenerate control weights are constructed to study the weak convergence of the corresponding cost functionals. Finally, two examples are addressed for non-singular and singular cases, respectively.

本文研究了一个混合的领导者与跟随者的差分博弈问题，该模型涉及两个具有相同层次结构的决策者，每个参与者都有两个分别充当领导者和跟随者的控件。具体来说，我们用不受约束的控制来解决跟随者问题，并获得相应的纳什均衡。然后解决了约束控制的领导问题，并通过投影映射给出了一对最优约束控制。此外，控制权重可以是单个的。在这种情况下，我们首先研究成本函数的均匀凸性，其对应状态是完全耦合的前向-后向随机微分方程。之后，构造了具有非退化控制权的最小化解序列，以研究相应成本函数的弱收敛性。最后，分别针对非奇异情况和奇异情况提出了两个示例。