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arXiv · 2504.13496

Open-Loop and Closed-Loop Strategies for Linear Quadratic Mean Field Games: The Direct Approach

Yong Liang·Bing-Chang Wang·Huanshui Zhang
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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.

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Yong Liang, Bing-Chang Wang, Huanshui Zhang. 2025-04-18. Open-Loop and Closed-Loop Strategies for Linear Quadratic Mean Field Games: The Direct Approach. https://arxiv.org/abs/2504.13496

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