Zero-Sum Stackelberg Stochastic Linear-Quadratic Differential Games

TL;DR

This paper explicitly derives the Stackelberg equilibrium for zero-sum stochastic LQ differential games, revealing its equivalence to the Nash equilibrium and providing a linear feedback solution.

Contribution

It introduces a method to explicitly solve the Stackelberg equilibrium via Riccati equations and shows its equivalence to Nash equilibrium in stochastic LQ games.

Findings

Stackelberg and Nash equilibria are identical in this setting.

The equilibrium can be represented as a linear state feedback.

Two Riccati equations characterize the equilibrium solutions.

Abstract

The paper is concerned with a zero-sum Stackelberg stochastic linear-quadratic (LQ, for short) differential game over finite horizons. Under a fairly weak condition, the Stackelberg equilibrium is explicitly obtained by first solving a forward stochastic LQ optimal control problem (SLQ problem, for short) and then a backward SLQ problem. Two Riccati equations are derived in constructing the Stackelberg equilibrium. An interesting finding is that the difference of these two Riccati equations coincides with the Riccati equation associated with the zero-sum Nash stochastic LQ differential game, which implies that the Stackelberg equilibrium and the Nash equilibrium are actually identical. Consequently, the Stackelberg equilibrium admits a linear state feedback representation, and the Nash game can be solved in a leader-follower manner.