  In this paper, the known deterministic linear-quadratic Stackelberg game is
revisited, whose open-loop Stackelberg solution actually possesses the nature
of time inconsistency. To handle this time inconsistency, {a two-tier game
framework is introduced, where the upper-tier game works according to
Stackelberg's scenario with a leader and a follower, and two lower-tier
intertemporal games give the follower's and leader's equilibrium response
mappings that mimic the notion of time-consistent open-loop equilibrium control
in existing literature. The resulting open-loop equilibrium solution of the
two-tier game} is shown to be weakly time-consistent in the sense that the
adopted policies will no longer be denied in the future only if past policies
are consistent with the equilibrium policies. On the existence and uniqueness
of such a solution, necessary and sufficient conditions are obtained, which are
characterized via the solutions of several Riccati-like equations.
