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Entropy Regularization for Mean Field Games with Learning

Author

Listed:
  • Xin Guo

    (Department of Industrial Engineering and Operations Research, University of California, Berkeley, California 94720; Tsinghua-UC Berkeley Shenzhen Institute, Shenzhen 518055, China)

  • Renyuan Xu

    (Epstein Department of Industrial and Systems Engineering, University of Southern California, Los Angeles, California 90089; Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom)

  • Thaleia Zariphopoulou

    (Departments of Mathematics and IROM, The University of Texas at Austin, Austin, Texas 78712)

Abstract

Entropy regularization has been extensively adopted to improve the efficiency, the stability, and the convergence of algorithms in reinforcement learning. This paper analyzes both quantitatively and qualitatively the impact of entropy regularization for mean field games (MFGs) with learning in a finite time horizon. Our study provides a theoretical justification that entropy regularization yields time-dependent policies and, furthermore, helps stabilizing and accelerating convergence to the game equilibrium. In addition, this study leads to a policy-gradient algorithm with exploration in MFG. With this algorithm, agents are able to learn the optimal exploration scheduling, with stable and fast convergence to the game equilibrium.

Suggested Citation

  • Xin Guo & Renyuan Xu & Thaleia Zariphopoulou, 2022. "Entropy Regularization for Mean Field Games with Learning," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 3239-3260, November.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:4:p:3239-3260
    DOI: 10.1287/moor.2021.1238
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    References listed on IDEAS

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    1. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
    2. Ben Hambly & Renyuan Xu & Huining Yang, 2020. "Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon," Papers 2011.10300, arXiv.org, revised Jun 2021.
    3. Haoran Wang & Xun Yu Zhou, 2019. "Continuous-Time Mean-Variance Portfolio Selection: A Reinforcement Learning Framework," Papers 1904.11392, arXiv.org, revised May 2019.
    4. Josef Hofbauer & William H. Sandholm, 2002. "On the Global Convergence of Stochastic Fictitious Play," Econometrica, Econometric Society, vol. 70(6), pages 2265-2294, November.
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    Cited by:

    1. Sun, Zhongshi & Jia, Guangyan, 2026. "Robust policy iteration for the continuous-time stochastic H∞ control problem with unknown dynamics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PA), pages 430-448.
    2. Dianetti, Jodi & Dumitrescu, Roxana & Ferrari, Giorgio & Xu, Renyuan, 2025. "Entropy Regularization in Mean-Field Games of Optimal Stopping," Center for Mathematical Economics Working Papers 755, Center for Mathematical Economics, Bielefeld University.
    3. Boyu Wang & Xuefeng Gao & Lingfei Li, 2026. "Reinforcement learning for continuous-time optimal execution: actor–critic algorithm and error analysis," Finance and Stochastics, Springer, vol. 30(2), pages 597-655, April.
    4. Mononen, Lasse, 2025. "On Preference for Simplicity and Probability Weighting," Center for Mathematical Economics Working Papers 748, Center for Mathematical Economics, Bielefeld University.
    5. Yu Li & Yuhan Wu & Shuhua Zhang, 2025. "The Exploratory Multi-Asset Mean-Variance Portfolio Selection using Reinforcement Learning," Papers 2505.07537, arXiv.org.
    6. Xin Guo & Anran Hu & Renyuan Xu & Junzi Zhang, 2023. "A General Framework for Learning Mean-Field Games," Mathematics of Operations Research, INFORMS, vol. 48(2), pages 656-686, May.
    7. Xiang Cheng & Zhuo Jin & Hailiang Yang & George Yin, 2026. "A Hybrid Deep Reinforcement Learning Method for Insurance Portfolio Management," Journal of Optimization Theory and Applications, Springer, vol. 208(1), pages 1-42, January.
    8. Min Dai & Yuchao Dong & Yanwei Jia & Xun Yu Zhou, 2026. "Merton's Problem with Recursive Perturbed Utility," Papers 2602.13544, arXiv.org.

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