Paper: arXiv 2602.16401

Authors: Maria Andraos, Mario Ghossoub, Bin Li, Benxuan Shi

Abstract

We study Stackelberg Equilibria (Bowley optima) in a monopolistic centralized sequential-move insurance market, with a profit-maximizing insurer who sets premia using a distortion premium principle, and a single policyholder who seeks to minimize a distortion risk measure. We show that equilibria are characterized as follows: In equilibrium, the optimal indemnity function exhibits a layer-type structure, providing full insurance over any loss layer on which the policyholder is more pessimistic than the insurer’s pricing functional about tail losses; and no insurance coverage over loss layers on which the policyholder is less pessimistic than the insurer’s pricing functional about tail losses. In equilibrium, the optimal pricing distortion function is determined by the policyholder’s degree of risk aversion, whereby prices never exceed the policyholder’s marginal willingness to insure tail losses. Moreover, we show that both the insurance coverage and the insurer’s expected profit increase with the policyholder’s degree of risk aversion. Additionally, and echoing recent work in the literature, we show that equilibrium contracts are Pareto efficient, but they do not induce a welfare gain to the policyholder. Conversely, any Pareto-optimal contract that leaves no welfare gain to the policyholder can be obtained as an equilibrium contract. Finally, we consider a few examples of interest that recover some existing results in the literature as special cases of our analysis.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical game-theoretic and risk-theoretic framework (Stackelberg equilibria with distortion risk measures) extending previous literature to non-concave probability weighting functions. However, it is purely mathematical economics/actuarial theory without any empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Characterize Stackelberg Equilibria in Monopoly Insurance Markets with Probability Weighting] --> B[Inputs: Policyholder Risk Measure & Insurer Distortion Pricing Model]
    B --> C[Methodology: Sequential Game Optimization & Tail Pessimism Comparison]
    C --> D[Derive Optimal Indemnity Layer Structure & Equilibrium Pricing Distortion]
    D --> E1[Finding 1: Layer-type coverage based on relative tail pessimism]
    D --> E2[Finding 2: Pareto efficient contracts yielding zero welfare gain to policyholder]
    D --> E3[Finding 3: Higher policyholder risk aversion increases coverage & insurer profit]

Cited References (60)

  1. Optimal Excess-of-Loss Reinsurance Contract with Ambiguity Aversion in the Principal-Agent Model (2020)
  2. A Continuous-Time Theory of Reinsurance Chains (2020)
  3. Stackelberg Reinsurance Chain under Model Ambiguity (2024)
  4. Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay (2022)
  5. A Hybrid Stochastic Differential Reinsurance and Investment Game with Bounded Memory (2022)
  6. A Stackelberg Reinsurance–Investment Game with Asymmetric Information and Delay (2021)
  7. Robust Reinsurance Contracts with Risk Constraint (2020)
  8. Stackelberg Differential Game for Reinsurance: Mean-Variance Framework and Random Horizon (2022) — link
and 52 more
  1. On a New Paradigm of Optimal Reinsurance: A Stochastic Stackelberg Differential Game Between an Insurer and a Reinsurer (2018)
  2. Stochastic Stackelberg Differential Reinsurance Games under Time-Inconsistent Mean–Variance Framework (2019) — link
  3. Stackelberg Differential Game for Insurance under Model Ambiguity: General Divergence (2023)
  4. Stackelberg Differential Game for Insurance under Model Ambiguity (2022) — link
  5. T he D ual T heory of C hoice under R isk (1987)
  6. G eneralized E xpected U tility T heory - T he R ank- D ependent M odel (1993)
  7. R isk A version in the T heory of E xpected U tility with R ank D ependent P obabilities (1987)
  8. Pareto E fficient I nsurance C ontracts when the I nsurer’s C ost F unction is D iscontinuous (2003)
  9. R earrangement I nequalities in N on-convex I nsurance M odels (2005)
  10. Testing Hurwicz Expected Utility (2023)
  11. The P robability W eighting F unction (1998)
  12. Bilateral M onopoly (1928)
  13. S tochastic F inance: A n I ntroduction in D iscrete T ime – $4^ th $ ed. (2016)
  14. Subgame Perfect Nash Equilibria in Large Reinsurance Markets — → 2506.07291
  15. C omparative R isk A version in RDEU with A pplications to O ptimal U nderwriting of S ecurities I ssuance (2021)
  16. Bowley-Optimal Convex-Loaded Premium Principles (2025)
  17. Optimal Insurance under Wang’s Premium Principle (1999)
  18. R isk A version in the S mall and in the L arge (1964)
  19. Risk-Adjusted Bowley Reinsurance under Distorted Probabilities (2019)
  20. On Optimal Reinsurance Policy with Distortion Risk Measures and Premiums (2015)
  21. Optimal Reinsurance under VaR and CTE Risk Measures (2008)
  22. Optimal Reinsurance from the Perspectives of Both an Insurer and a Reinsurer (2016)
  23. Premium Calculation by Transforming the Layer Premium Density (1996)
  24. Axiomatic Characterization of Insurance Prices (1997)
  25. Pareto-Optimal Reinsurance Arrangements under General Model Settings (2017)
  26. Optimal Retention for a Stop-Loss Reinsurance under the VaR and CTE Risk Measures (2007)
  27. Optimal Design of Reinsurance Contracts under Adverse Selection with a Continuum of Types — → 2504.17468
  28. Bowley Insurance with Expected Utility Maximization of the Policyholders (2024)
  29. Optimal Reinsurance with General Premium Principles (2013)
  30. Optimal Reinsurance Designs Based on Risk Measures: A Review (2020)
  31. Optimal Reinsurance with Multiple Reinsurers: Distortion Risk Measures, Distortion Premium Principles, and Heterogeneous Beliefs (2021)
  32. Reinsurance Games with Two Reinsurers: Tree versus Chain (2023)
  33. Multidimensional Screening and Menu Design in Health Insurance Markets (2022)
  34. Optimal Insurance with Belief Heterogeneity and Incentive Compatibility (2020)
  35. The R einsurer’s M onopoly and the B owley S olution (1985)
  36. Risk- A djusted B owley R einsurance under D istorted P robabilities (2019)
  37. A Bowley Solution with Limited Ceded Risk for a Monopolistic Reinsurer (2020)
  38. On C onvex P rinciples of P remium C alculation (1985)
  39. On a N ew P aradigm of O ptimal R einsurance: A S tochastic S tackelberg D ifferential G ame between an I nsurer and a R einsurer (2018)
  40. Short C ommunication: O ptimal I nsurance to M aximize E xponential U tility W hen P remium I s C omputed by a C onvex F unctional (2024)
  41. Bowley R einsurance with A symmetric I nformation on the I nsurer’s R isk P references (2021)
  42. Bowley R einsurance with A symmetric I nformation: A F irst- B est S olution (2022)
  43. Stackelberg D ifferential G ame for I nsurance under M odel A mbiguity (2022)
  44. S tackelberg E quilibria with M ultiple P olicyholders (2024)
  45. Equilibria and E fficiency in a R einsurance M arket (2023)
  46. Risk E xchange I : A U nification of S ome E xisting R esults (1992)
  47. Bowley Solution of a Mean–Variance Game in Insurance (2021)
  48. Uncertainty and the W elfare E conomics of M edical C are (1963)
  49. Optimal I nsurance P olicy I ndemnity S chedules (1983)
  50. Bowley vs. Pareto Optima in Reinsurance Contracting (2023)
  51. OPTIMAL I NSURANCE D ESIGN UNDER R ANK- D EPENDENT E XPECTED U TILITY (2015)
  52. Advances in P rospect T heory: C umulative R epresentation of U ncertainty (1992)

2 of 60 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.