Linear-quadratic Stochastic Stackelberg Differential Games with Affine Constraints

TL;DR

This paper develops a comprehensive framework for solving linear-quadratic stochastic Stackelberg differential games with affine constraints, deriving optimal strategies via Riccati equations and duality theory, and establishing conditions for solution uniqueness.

Contribution

It introduces a novel approach combining Riccati equations and Lagrangian duality to characterize optimal strategies and solution uniqueness in constrained stochastic Stackelberg games.

Findings

Derived feedback strategies for leader and follower.

Established duality and KKT conditions for the problem.

Provided examples illustrating the theoretical results.

Abstract

This paper investigates the non-zero-sum linear-quadratic stochastic Stackelberg differential games with affine constraints, which depend on both the follower's response and the leader's strategy. With the help of the stochastic Riccati equations and the Lagrangian duality theory, the feedback expressions of optimal strategies of the follower and the leader are obtained and the dual problem of the leader's problem is established. Under the Slater condition, the equivalence is proved between the solutions to the dual problem and the leader's problem, and the KKT condition is also provided for solving the dual problem. Then, the feedback Stackelberg equilibrium is provided for the linear-quadratic stochastic Stackelberg differential games with affine constraints, and a new positive definite condition is proposed for ensuring the uniqueness of solutions to the dual problem. Finally, two non-degenerate examples with indefinite coefficients are provided to illustrate and to support our main results.