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