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

TL;DR

This paper develops a framework for solving linear-quadratic time-inconsistent mean-field Stackelberg differential games by deriving explicit time-consistent equilibrium controls for both leader and follower using coupled Riccati equations and FBSDEs.

Contribution

It introduces a method to obtain explicit, time-consistent Stackelberg equilibrium controls in complex mean-field differential games with nonexponential discounting.

Findings

Explicit equilibrium controls derived for leader and follower.

Solution characterized by coupled Riccati differential equations.

Numerical examples demonstrate solvability of the equations.

Abstract

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.