Deterministic dynamic Stackelberg games: Time-consistent open-loop solution

Abstract

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: the upper-tier game works according to Stackelberg’s scenario between a leader and a follower, and the outcomes of two lower-tier intertemporal games characterize, respectively, the follower’s and the leader’s response controls that mimic the notion of time-consistent open-loop equilibrium control in existing literature. The obtained open-loop equilibrium of this 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.