IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2605.25824.html

Mean-field game of mean-variance portfolio management with peer-based relative risk aversion

Author

Listed:
  • Weilun Cheng
  • Zongxia Liang
  • Sheng Wang
  • Xiang Yu

Abstract

This paper investigates a mean-field game (MFG) problem for mean-variance (MV) portfolio management, highlighting a new type of relative performance encoded by the peer-based risk aversion. Specifically, the risk aversion is formulated as a piecewise form that depends on whether the individual's wealth is above or below the population average. Due to the inherent time-inconsistency in the MV criterion, together with the piecewise risk aversion, we encounter a class of time-inconsistent MFG, new to the literature. Our goal is to seek a mean-field equilibrium, characterized by a forward-backward stochastic differential equation (FBSDE) system and a mean-field consistency condition. The new challenge stems from the discontinuous coefficients induced by the piecewise risk aversion. In response, we first propose a smooth regularization technique and obtain the existence of the equilibrium in the intra-personal game for the representative agent by establishing the solution to the discontinuous multi-dimensional FBSDE. Next, by invoking fixed-point arguments and convergence analysis as smoothing regularization vanishes, we conclude the existence of the mean-field equilibrium in the time-inconsistent MFG.

Suggested Citation

  • Weilun Cheng & Zongxia Liang & Sheng Wang & Xiang Yu, 2026. "Mean-field game of mean-variance portfolio management with peer-based relative risk aversion," Papers 2605.25824, arXiv.org.
    • Handle: RePEc:arx:papers:2605.25824
    as

    Download full text from publisher

    File URL: https://arxiv.org/pdf/2605.25824
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Gilles-Edouard Espinosa & Nizar Touzi, 2015. "Optimal Investment Under Relative Performance Concerns," Mathematical Finance, Wiley Blackwell, vol. 25(2), pages 221-257, April.
    2. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2012. "Time-Inconsistent Stochastic Linear--Quadratic Control," Post-Print hal-00691816, HAL.
    3. Guohui Guan & Jiaqi Hu & Zongxia Liang, 2025. "N-player and mean field games among fund managers considering excess logarithmic returns," Papers 2503.02722, arXiv.org.
    4. Guanxing Fu & Chao Zhou, 2023. "Mean field portfolio games," Finance and Stochastics, Springer, vol. 27(1), pages 189-231, January.
    5. Daniel Lacker & Thaleia Zariphopoulou, 2019. "Mean field and n‐agent games for optimal investment under relative performance criteria," Mathematical Finance, Wiley Blackwell, vol. 29(4), pages 1003-1038, October.
    6. Botond Kőszegi & Matthew Rabin, 2006. "A Model of Reference-Dependent Preferences," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 121(4), pages 1133-1165.
    7. R. H. Strotz, 1955. "Myopia and Inconsistency in Dynamic Utility Maximization," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 23(3), pages 165-180.
    8. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2017. "Time-Inconsistent Stochastic Linear--Quadratic Control: Characterization and Uniqueness of Equilibrium," Post-Print hal-01139343, HAL.
    9. Yu‐Jui Huang & Li‐Hsien Sun, 2026. "Partial Information in a Mean‐Variance Portfolio Selection Game," Mathematical Finance, Wiley Blackwell, vol. 36(2), pages 309-329, April.
    10. Guanxing Fu, 2023. "Mean field portfolio games with consumption," Mathematics and Financial Economics, Springer, volume 17, number 4, June.
    11. Panagiotis E. Souganidis & Thaleia Zariphopoulou, 2024. "Mean field games with unbounded controlled common noise in portfolio management with relative performance criteria," Mathematics and Financial Economics, Springer, volume 18, number 10, June.
    12. Hanqing Jin & Xun Yu Zhou, 2008. "Behavioral Portfolio Selection In Continuous Time," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 385-426, July.
    13. Ludovic Tangpi & Xuchen Zhou, 2024. "Optimal investment in a large population of competitive and heterogeneous agents," Finance and Stochastics, Springer, vol. 28(2), pages 497-551, April.
    14. Erhan Bayraktar & Zhenhua Wang, 2025. "On Time‐Inconsistency in Mean‐Field Games," Mathematical Finance, Wiley Blackwell, vol. 35(3), pages 613-635, July.
    15. Guohui Guan & Xiang Hu, 2022. "Time-Consistent Investment and Reinsurance Strategies for Mean–Variance Insurers in N-Agent and Mean-Field Games," North American Actuarial Journal, Taylor & Francis Journals, vol. 26(4), pages 537-569, November.
    16. Guohui Guan & Zongxia Liang, 2026. "Robust n-Agent Heterogeneous Investment-Consumption Game Under $$\alpha $$ α -Maxmin Mean-Variance-Utility Criterion," Journal of Optimization Theory and Applications, Springer, vol. 208(1), pages 1-38, January.
    17. Tomas Björk & Agatha Murgoci & Xun Yu Zhou, 2014. "Mean–Variance Portfolio Optimization With State-Dependent Risk Aversion," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 1-24, January.
    18. Botond Koszegi & Matthew Rabin, 2007. "Reference-Dependent Risk Attitudes," American Economic Review, American Economic Association, vol. 97(4), pages 1047-1073, September.
    19. Suleyman Basak & Georgy Chabakauri, 2010. "Dynamic Mean-Variance Asset Allocation," The Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 2970-3016, August.
    20. Zongxia Liang & Keyu Zhang, 2023. "Time-inconsistent mean field and n-agent games under relative performance criteria," Papers 2312.14437, arXiv.org, revised Apr 2024.
    21. Guohui Guan & Jiaqi Hu & Zongxia Liang, 2025. "N-player and mean field games among fund managers considering excess logarithmic returns," Annals of Operations Research, Springer, vol. 349(3), pages 1663-1691, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yue Cao & Zongxia Liang & Sheng Wang & Xiang Yu, 2025. "Equilibrium Portfolio Selection under Utility-Variance Analysis of Log Returns in Incomplete Markets," Papers 2511.05861, arXiv.org, revised Dec 2025.
    2. Marcel Nutz & Yuchong Zhang, 2019. "Conditional Optimal Stopping: A Time-Inconsistent Optimization," Papers 1901.05802, arXiv.org, revised Oct 2019.
    3. Zongxia Liang & Jianming Xia & Fengyi Yuan, 2023. "Dynamic portfolio selection for nonlinear law-dependent preferences," Papers 2311.06745, arXiv.org, revised Nov 2023.
    4. Guanxing Fu & Ulrich Horst, 2025. "Mean Field Portfolio Games with Epstein-Zin Preferences," Papers 2505.07231, arXiv.org, revised Jun 2026.
    5. Lijun Bo & Yijie Huang & Xiang Yu, 2025. "Mean Field Game of Optimal Tracking Portfolio," Papers 2505.01858, arXiv.org, revised Apr 2026.
    6. Zongxia Liang & Sheng Wang & Jianming Xia, 2024. "An Integral Equation in Portfolio Selection with Time-Inconsistent Preferences," Papers 2412.02446, arXiv.org, revised Jan 2025.
    7. Xue Dong He & Xun Yu Zhou, 2021. "Who Are I: Time Inconsistency and Intrapersonal Conflict and Reconciliation," Papers 2105.01829, arXiv.org.
    8. Zongxia Liang & Sheng Wang & Jianming Xia & Fengyi Yuan, 2024. "Dynamic portfolio selection under generalized disappointment aversion," Papers 2401.08323, arXiv.org, revised Mar 2024.
    9. Guojiang Shao & Zuo Quan Xu & Qi Zhang, 2025. "Competitive optimal portfolio selection under mean-variance criterion," Papers 2511.05270, arXiv.org.
    10. Guanxing Fu & Ulrich Horst, 2025. "Mean Field Portfolio Games with Epstein-Zin Preferences," Rationality and Competition Discussion Paper Series 540, CRC TRR 190 Rationality and Competition.
    11. Bingyan Han & Chi Seng Pun & Hoi Ying Wong, 2023. "Robust Time-inconsistent Linear-Quadratic Stochastic Controls: A Stochastic Differential Game Approach," Papers 2306.16982, arXiv.org, revised Apr 2025.
    12. Yan, Tingjin & Wong, Hoi Ying, 2020. "Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 105-119.
    13. De Gennaro Aquino, Luca & Sornette, Didier & Strub, Moris S., 2023. "Portfolio selection with exploration of new investment assets," European Journal of Operational Research, Elsevier, vol. 310(2), pages 773-792.
    14. Guohui Guan & Zongxia Liang & Yi Xia, 2025. "Many-insurer robust games of reinsurance and investment under model uncertainty in incomplete markets," Mathematics and Financial Economics, Springer, volume 19, number 4, June.
    15. Ling Wang & Mei Choi Chiu & Hoi Ying Wong, 2021. "Time-consistent mean-variance reinsurance-investment problem with long-range dependent mortality rate," Papers 2112.06602, arXiv.org.
    16. Dylan Possamai & Mateo Rodriguez Polo, 2026. "Here, there and everywhere: state-dependent time-inconsistent stochastic control," Papers 2603.22022, arXiv.org.
    17. Dylan Possamai & Chiara Rossato, 2025. "Variance strikes back: sub-game--perfect Nash equilibria in time-inconsistent $N$-player games, and their mean-field sequel," Papers 2512.08745, arXiv.org.
    18. Liyuan Wang & Zhiping Chen, 2019. "Stochastic Game Theoretic Formulation for a Multi-Period DC Pension Plan with State-Dependent Risk Aversion," Mathematics, MDPI, vol. 7(1), pages 1-16, January.
    19. Alia, Ishak & Chighoub, Farid & Sohail, Ayesha, 2016. "A characterization of equilibrium strategies in continuous-time mean–variance problems for insurers," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 212-223.
    20. Nicole Bauerle & Tamara Goll, 2025. "Relative portfolio optimization via a value at risk based constraint," Papers 2503.20340, arXiv.org, revised Jun 2025.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2605.25824. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: https://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.