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General Stopping Behaviors of Naive and Non-Committed Sophisticated Agents, with Application to Probability Distortion

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  • Yu-Jui Huang
  • Adrien Nguyen-Huu
  • Xun Yu Zhou

Abstract

We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time-inconsistent. We study stopping decisions of naive agents who reoptimize continuously in time, as well as equilibrium strategies of sophisticated agents who anticipate but lack control over their future selves' behaviors. When the state process is one dimensional and the payoff functional satisfies some regularity conditions, we prove that any equilibrium can be obtained as a fixed point of an operator. This operator represents strategic reasoning that takes the future selves' behaviors into account. We then apply the general results to the case when the agents distort probability and the diffusion process is a geometric Brownian motion. The problem is inherently time-inconsistent as the level of distortion of a same event changes over time. We show how the strategic reasoning may turn a naive agent into a sophisticated one. Moreover, we derive stopping strategies of the two types of agent for various parameter specifications of the problem, illustrating rich behaviors beyond the extreme ones such as "never-stopping" or "never-starting".

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  • Yu-Jui Huang & Adrien Nguyen-Huu & Xun Yu Zhou, 2017. "General Stopping Behaviors of Naive and Non-Committed Sophisticated Agents, with Application to Probability Distortion," Papers 1709.03535, arXiv.org, revised Mar 2019.
  • Handle: RePEc:arx:papers:1709.03535
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    References listed on IDEAS

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    1. Yu-Jui Huang & Adrien Nguyen-Huu, 2018. "Time-consistent stopping under decreasing impatience," Finance and Stochastics, Springer, vol. 22(1), pages 69-95, January.
    2. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
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    Cited by:

    1. Yu-Jui Huang & Zhou Zhou, 2018. "Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time," Papers 1809.09243, arXiv.org, revised Aug 2019.
    2. Zongxia Liang & Fengyi Yuan, 2023. "Weak equilibria for time‐inconsistent control: With applications to investment‐withdrawal decisions," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 891-945, July.
    3. Erhan Bayraktar & Jingjie Zhang & Zhou Zhou, 2021. "Equilibrium concepts for time‐inconsistent stopping problems in continuous time," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 508-530, January.
    4. Markus Dertwinkel‐Kalt & Jonas Frey, 2024. "Optimal Stopping In A Dynamic Salience Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 65(2), pages 885-913, May.
    5. Denis Belomestny & Tobias Hübner & Volker Krätschmer, 2022. "Solving optimal stopping problems under model uncertainty via empirical dual optimisation," Finance and Stochastics, Springer, vol. 26(3), pages 461-503, July.
    6. Yu‐Jui Huang & Zhou Zhou, 2020. "Optimal equilibria for time‐inconsistent stopping problems in continuous time," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 1103-1134, July.
    7. Erhan Bayraktar & Zhenhua Wang & Zhou Zhou, 2023. "Equilibria of time‐inconsistent stopping for one‐dimensional diffusion processes," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 797-841, July.
    8. Yu-Jui Huang & Zhou Zhou, 2021. "A Time-Inconsistent Dynkin Game: from Intra-personal to Inter-personal Equilibria," Papers 2101.00343, arXiv.org, revised Dec 2021.
    9. Yu-Jui Huang & Zhou Zhou, 2022. "A time-inconsistent Dynkin game: from intra-personal to inter-personal equilibria," Finance and Stochastics, Springer, vol. 26(2), pages 301-334, April.
    10. Oumar Mbodji & Traian A. Pirvu, 2023. "Portfolio Time Consistency and Utility Weighted Discount Rates," Papers 2402.05113, arXiv.org.
    11. Camilo Hern'andez & Dylan Possamai, 2020. "Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents," Papers 2002.12572, arXiv.org, revised Jul 2021.
    12. Yu-Jui Huang & Zhou Zhou, 2017. "Optimal Equilibria for Time-Inconsistent Stopping Problems in Continuous Time," Papers 1712.07806, arXiv.org, revised Oct 2018.
    13. Marcel Nutz & Yuchong Zhang, 2019. "Conditional Optimal Stopping: A Time-Inconsistent Optimization," Papers 1901.05802, arXiv.org, revised Oct 2019.
    14. Yu-Jui Huang & Zhou Zhou, 2021. "Strong and Weak Equilibria for Time-Inconsistent Stochastic Control in Continuous Time," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 428-451, May.
    15. Zhou, Zhou & Jin, Zhuo, 2020. "Optimal equilibrium barrier strategies for time-inconsistent dividend problems in discrete time," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 100-108.
    16. Xue Dong He & Zhaoli Jiang & Steven Kou, 2020. "Portfolio Selection under Median and Quantile Maximization," Papers 2008.10257, arXiv.org, revised Mar 2021.
    17. Shuoqing Deng & Xiang Yu & Jiacheng Zhang, 2023. "On time-consistent equilibrium stopping under aggregation of diverse discount rates," Papers 2302.07470, arXiv.org, revised Dec 2023.
    18. Yu-Jui Huang & Zhenhua Wang, 2020. "Optimal Equilibria for Multi-dimensional Time-inconsistent Stopping Problems," Papers 2006.00754, arXiv.org, revised Jan 2021.
    19. Zongxia Liang & Fengyi Yuan, 2021. "Weak equilibria for time-inconsistent control: with applications to investment-withdrawal decisions," Papers 2105.06607, arXiv.org, revised Jun 2023.

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