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A Two-layer Stochastic Game Approach to Reinsurance Contracting and Competition

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  • Zongxia Liang
  • Yi Xia
  • Bin Zou

Abstract

We propose a two-layer stochastic game model to study reinsurance contracting and competition in a market with one insurer and two competing reinsurers. The insurer negotiates with both reinsurers simultaneously for proportional reinsurance contracts that are priced using the variance premium principle. The reinsurance contracting between the insurer and each reinsurer is modeled as a Stackelberg game. The two reinsurers compete for business from the insurer and optimize the so-called relative performance, instead of their own surplus, and their competition is settled by a noncooperative Nash game. We obtain a sufficient and necessary condition, related to the competition degrees of the two reinsurers, for the existence of an equilibrium. We show that the equilibrium, if exists, is unique, and the equilibrium strategy of each player is constant, fully characterized in semiclosed form. Furthermore, we obtain interesting sensitivity results for the equilibrium strategies through both analytical and numerical studies.

Suggested Citation

  • Zongxia Liang & Yi Xia & Bin Zou, 2024. "A Two-layer Stochastic Game Approach to Reinsurance Contracting and Competition," Papers 2405.06235, arXiv.org, revised Sep 2024.
    • Handle: RePEc:arx:papers:2405.06235
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    References listed on IDEAS

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    1. Lu, ZhiYi & Meng, LiLi & Wang, Yujin & Shen, Qingjie, 2016. "Optimal reinsurance under VaR and TVaR risk measures in the presence of reinsurer’s risk limit," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 92-100.
    2. Li, Danping & Young, Virginia R., 2022. "Stackelberg differential game for reinsurance: Mean-variance framework and random horizon," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 42-55.
    3. Zhibin Liang & Kam Chuen Yuen, 2016. "Optimal dynamic reinsurance with dependent risks: variance premium principle," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2016(1), pages 18-36, January.
    4. Boonen, Tim J. & Tan, Ken Seng & Zhuang, Sheng Chao, 2021. "Optimal reinsurance with multiple reinsurers: Competitive pricing and coalition stability," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 302-319.
    5. Tan, Ken Seng & Wei, Pengyu & Wei, Wei & Zhuang, Sheng Chao, 2020. "Optimal dynamic reinsurance policies under a generalized Denneberg’s absolute deviation principle," European Journal of Operational Research, Elsevier, vol. 282(1), pages 345-362.
    6. Cao, Jingyi & Li, Dongchen & Young, Virginia R. & Zou, Bin, 2023. "Reinsurance games with two reinsurers: Tree versus chain," European Journal of Operational Research, Elsevier, vol. 310(2), pages 928-941.
    7. Jun Cai & Yichun Chi, 2020. "Responses to discussions on ‘Optimal reinsurance designs based on risk measures: a review’," Statistical Theory and Related Fields, Taylor & Francis Journals, vol. 4(1), pages 26-27, July.
    8. Bjarne Højgaard & Søren Asmussen & Michael Taksar, 2000. "Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation," Finance and Stochastics, Springer, vol. 4(3), pages 299-324.
    9. Chi, Yichun, 2012. "Optimal reinsurance under variance related premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 310-321.
    10. Hanspeter Schmidli, 2001. "Optimal Proportional Reinsurance Policies in a Dynamic Setting," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2001(1), pages 55-68.
    11. Zhou, Ming & Yuen, Kam C., 2012. "Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle," Economic Modelling, Elsevier, vol. 29(2), pages 198-207.
    12. Cao, Jingyi & Li, Dongchen & Young, Virginia R. & Zou, Bin, 2023. "Reinsurance games with $\boldsymbol{{n}}$ variance-premium reinsurers: from tree to chain," ASTIN Bulletin, Cambridge University Press, vol. 53(3), pages 706-728, September.
    13. Li, Danping & Rong, Ximin & Zhao, Hui, 2015. "Time-consistent reinsurance–investment strategy for a mean–variance insurer under stochastic interest rate model and inflation risk," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 28-44.
    14. Jun Cai & Yichun Chi, 2020. "Optimal reinsurance designs based on risk measures: a review," Statistical Theory and Related Fields, Taylor & Francis Journals, vol. 4(1), pages 1-13, July.
    15. Ken Tan & Chengguo Weng & Yi Zhang, 2009. "VAR and CTE Criteria for Optimal Quota-Share and Stop-Loss Reinsurance," North American Actuarial Journal, Taylor & Francis Journals, vol. 13(4), pages 459-482.
    16. Chen, Lv & Shen, Yang, 2019. "Stochastic Stackelberg differential reinsurance games under time-inconsistent mean–variance framework," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 120-137.
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    Cited by:

    1. Cao, Jingyi & Li, Dongchen & Young, Virginia R. & Zou, Bin, 2025. "Co-opetition in reinsurance markets: When Pareto meets Stackelberg and Nash," Insurance: Mathematics and Economics, Elsevier, vol. 125(C).
    2. Zhang, Jinjin & Zhang, Caibin, 2025. "Nash equilibrium in insurance pricing and investment under common shocks," Finance Research Letters, Elsevier, vol. 86(PA).
    3. Tim J. Boonen & Engel John C. Dela Vega & Bin Zou, 2025. "Optimal Dividend, Reinsurance, and Capital Injection Strategies for an Insurer with Two Collaborating Business Lines," Papers 2508.08130, arXiv.org.

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