Deterministic Dynamic Stackelberg Games: Time-Consistent Open-Loop Solution

TL;DR

This paper introduces a two-tier game framework to address time inconsistency in deterministic linear-quadratic Stackelberg games, ensuring weak time consistency of open-loop solutions through Riccati-like equations.

Contribution

It proposes a novel two-tier game approach to achieve time-consistent open-loop solutions in Stackelberg games, with conditions characterized by Riccati-like equations.

Findings

The two-tier framework ensures weak time consistency of solutions.

Necessary and sufficient conditions for existence and uniqueness are derived.

Solutions are characterized via Riccati-like equations.

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, 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.