Leader-Follower Mean Field LQG Games with Multiplicative Noise

TL;DR

This paper develops solutions for leader-follower mean field LQG games with multiplicative noise, providing explicit formulas for optimal costs and strategies using variational analysis and Riccati equations.

Contribution

It introduces a direct approach to solve leader-follower mean field LQG games with multiplicative noise, including new feedback strategies and explicit cost formulas.

Findings

Explicit open-loop and feedback solutions derived

Optimal costs expressed via Riccati equations

Decentralized strategies constructed with matrix maximum principle

Abstract

This paper studies open-loop and feedback solutions to leader-follower mean field linear-quadratic-Gaussian games with multiplicative noise by the direct approach. The leader-follower game involves a leader and many followers, where the state and control weight matrices in their costs are not limited to be positive definite. From variational analysis with mean field approximations, we obtain a set of open-loop controls in terms of solutions to mean field forward-backward stochastic differential equations. By applying the matrix maximum principle, a set of decentralized feedback strategies is constructed. Distinct from traditional works, a cross term has appeared in derivation due to the presence of mean field terms. For open-loop and feedback solutions, the corresponding optimal costs of all players are explicitly given in terms of the solutions to two Riccati equations, respectively.