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Time-inconsistent mean field and n-agent games under relative performance criteria

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  • Zongxia Liang
  • Keyu Zhang

Abstract

In this paper we study a time-inconsistent portfolio optimization problem for competitive agents with CARA utilities and non-exponential discounting. The utility of each agent depends on her own wealth and consumption as well as the relative wealth and consumption to her competitors. Due to the presence of a non-exponential discount factor, each agent's optimal strategy becomes time-inconsistent. In order to resolve time-inconsistency, each agent makes a decision in a sophisticated way, choosing open-loop equilibrium strategy in response to the strategies of all the other agents. We construct explicit solutions for the $n$-agent games and the corresponding mean field games (MFGs) where the limit of former yields the latter. This solution is unique in a special class of equilibria.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2312.14437
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    References listed on IDEAS

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    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. Guanxing Fu & Chao Zhou, 2021. "Mean Field Portfolio Games," Papers 2106.06185, arXiv.org, revised Apr 2022.
    3. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2012. "Time-Inconsistent Stochastic Linear--Quadratic Control," Post-Print hal-00691816, HAL.
    4. Marín-Solano, Jesús & Navas, Jorge, 2010. "Consumption and portfolio rules for time-inconsistent investors," European Journal of Operational Research, Elsevier, vol. 201(3), pages 860-872, March.
    5. Guanxing Fu, 2022. "Mean Field Portfolio Games with Consumption," Papers 2206.05425, arXiv.org, revised Dec 2022.
    6. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    7. A. Bensoussan & K. Sung & S. Yam, 2013. "Linear–Quadratic Time-Inconsistent Mean Field Games," Dynamic Games and Applications, Springer, vol. 3(4), pages 537-552, December.
    8. Lijun Bo & Shihua Wang & Xiang Yu, 2021. "Mean Field Game of Optimal Relative Investment with Jump Risk," Papers 2108.00799, arXiv.org, revised Feb 2023.
    9. Zhao, Qian & Shen, Yang & Wei, Jiaqin, 2014. "Consumption–investment strategies with non-exponential discounting and logarithmic utility," European Journal of Operational Research, Elsevier, vol. 238(3), pages 824-835.
    10. Rene Carmona & Jean-Pierre Fouque & Li-Hsien Sun, 2013. "Mean Field Games and Systemic Risk," Papers 1308.2172, arXiv.org.
    11. 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.
    12. Ishak Alia & Farid Chighoub & Nabil Khelfallah & Josep Vives, 2021. "Time-Consistent Investment and Consumption Strategies under a General Discount Function," JRFM, MDPI, vol. 14(2), pages 1-27, February.
    13. Tomas Björk & Mariana Khapko & Agatha Murgoci, 2017. "On time-inconsistent stochastic control in continuous time," Finance and Stochastics, Springer, vol. 21(2), pages 331-360, April.
    14. Yushi Hamaguchi, 2019. "Time-inconsistent consumption-investment problems in incomplete markets under general discount functions," Papers 1912.01281, arXiv.org, revised Mar 2021.
    15. Guanxing Fu & Xizhi Su & Chao Zhou, 2020. "Mean Field Exponential Utility Game: A Probabilistic Approach," Papers 2006.07684, arXiv.org, revised Jul 2020.
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    Cited by:

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