Linear-Quadratic Time-Inconsistent Mean-Field Type Stackelberg Differential Games: Time-Consistent Open-Loop Solutions

In this article, we consider the linear-quadratic time-inconsistent mean-field type leader-follower Stackelberg differential game with an adapted open-loop information structure. The objective functionals of leader and follower include conditional expectations state control (mean field) variables, cost parameters could be general nonexponential discounting depending on initial time. As stated in existing literature, these two settings induce time inconsistency optimal solutions. Given arbitrary leader, first obtain follower's (time consistent) equilibrium its feedback representation terms nonsymmetric coupled Riccati equations (RDEs) backward stochastic equation (SDE). This provides rational behavior follower, characterized by forward-backward SDE (FBSDE). We then leader's explicit RDEs under FBSDE constraint induced follower. With solvability RDEs, controls constitute time-consistent equilibrium. Finally, numerical examples are provided to check RDEs.