Open-Loop and Closed-Loop Strategies for Linear Quadratic Mean Field Games: The Direct Approach
Yong Liang, Bing-Chang Wang, Huanshui Zhang
TL;DR
This paper addresses how open-loop and closed-loop strategies differ in Linear Quadratic Mean Field Games under a direct approach. It develops a unified framework to analyze centralized open-loop and closed-loop Nash equilibria for finite-N populations and to design asymptotically decentralized controls in the infinite-population limit, using FBSDEs and Riccati equations. A key finding is that centralized open-loop and closed-loop equilibria can be significantly different for finite N, yet the corresponding asymptotically decentralized Nash equilibria coincide as N grows, with solvability tied to a standard Riccati equation and an asymmetric Riccati equation. Practically, this means decentralized control design in large populations can be robust to the choice between open-loop or closed-loop strategies, and the offline-computed mean-field quantities enable implementation with only local state information.
Abstract
This paper delves into studying the differences and connections between open-loop and closed-loop strategies for the linear quadratic (LQ) mean field games (MFGs) by the direct approach. The investigation begins with the finite-population system for solving the solvability of open-loop and closed-loop systems within a unified framework under the global information pattern. By a comprehensive analysis through variational methods, the necessary and sufficient conditions are obtained for the existence of centralized open-loop and closed-loop Nash equilibria, which are characterized by the solvability of a system of forward-backward stochastic differential equations and a system of Riccati equations, respectively. The connections and disparities between centralized open-loop and closed-loop Nash equilibria are analyzed. Then, the decentralized control is designed by studying the asymptotic solvability for both open-loop and closed-loop systems. Asymptotically decentralized Nash equilibria are obtained by considering the centralized open-loop and closed-loop Nash equilibria in the infinite-population system, which requires a standard and an asymmetric Riccati equations. The results demonstrate that divergences between the centralized open-loop and closed-loop Nash equilibria in the finite-population system, but the corresponding asymptotically decentralized Nash equilibria in the infinite-population system are consistent. Therefore, the choice of open-loop and closed-loop strategies does not play an essential role in the design of decentralized control for LQ MFGs.