Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stackelberg Equilibria in Monopoly Insurance Markets with Probability Weighting

Maria Andraos, Mario Ghossoub, Bin Li, Benxuan Shi

TL;DR

This paper analyzes Stackelberg equilibria (Bowley optima) in a monopolistic insurance market where the insurer uses a distortion premium principle and the policyholder evaluates risk via a distortion risk measure with distortion $T$. Using a two-step, quantile-based approach, the authors show that the policyholder's optimal indemnity under a fixed pricing distortion has a layer-type structure determined by the comparison of the pricing distortion with the policyholder's tail-pessimism, and that the insurer's optimal pricing distortion $g^*$ is driven by the policyholder's risk aversion, ensuring prices do not exceed the marginal willingness to insure tail losses. They prove that equilibria are Pareto efficient but extract all surplus from the policyholder, and they characterize several important special cases: full insurance under higher risk aversion, VaR-based coverage limits, and deductible contracts under inverse-S-shaped distortions. The findings connect to and unify results across TVaR, VaR, and deductible schemes, while highlighting how equilibrium outcomes depend on both the loss distribution and the shape of probability weighting. Overall, the work provides a rigorous framework for understanding how probability weighting and distortion risk preferences shape optimal insurance contracts in a monopolistic, sequential-move setting, with implications for contract design and welfare analysis.

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.

Stackelberg Equilibria in Monopoly Insurance Markets with Probability Weighting

TL;DR

This paper analyzes Stackelberg equilibria (Bowley optima) in a monopolistic insurance market where the insurer uses a distortion premium principle and the policyholder evaluates risk via a distortion risk measure with distortion . Using a two-step, quantile-based approach, the authors show that the policyholder's optimal indemnity under a fixed pricing distortion has a layer-type structure determined by the comparison of the pricing distortion with the policyholder's tail-pessimism, and that the insurer's optimal pricing distortion is driven by the policyholder's risk aversion, ensuring prices do not exceed the marginal willingness to insure tail losses. They prove that equilibria are Pareto efficient but extract all surplus from the policyholder, and they characterize several important special cases: full insurance under higher risk aversion, VaR-based coverage limits, and deductible contracts under inverse-S-shaped distortions. The findings connect to and unify results across TVaR, VaR, and deductible schemes, while highlighting how equilibrium outcomes depend on both the loss distribution and the shape of probability weighting. Overall, the work provides a rigorous framework for understanding how probability weighting and distortion risk preferences shape optimal insurance contracts in a monopolistic, sequential-move setting, with implications for contract design and welfare analysis.

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.
Paper Structure (23 sections, 14 theorems, 116 equations, 3 figures)

This paper contains 23 sections, 14 theorems, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Preferences
  4. Risk Aversion
  5. Market Mechanisms and Contracts
  6. A Sequential-Move Game
  7. Characterization of Stackelberg Equilibria
  8. The Policyholder's Problem
  9. The Insurer's Problem
  10. Pareto Efficiency of Stackelberg Equilibria
  11. Examples
  12. Optimality of Full Insurance
  13. Optimality of Coverage Limit Contracts
  14. Optimality of Deductible Contracts
  15. Conclusion
  16. ...and 8 more sections

Key Result

Proposition 2.5

Consider two policyholders whose preferences $\succcurlyeq$ and $\succcurlyeq^*$ over $B(\mathcal{F})$ admit representations by distortion risk measures $\rho^{Pol}$ and ${\rho^{*}}^{Pol}$ respectively. Let $T$ and $T^*$ denote the respective distortion functions of each policyholder. Then the follo

Figures (3)

  • Figure 1: The case where $X$ follows a uniform distribution.
  • Figure 2: The case where $X$ follows a truncated exponential distribution.
  • Figure 3: The case where $X$ follows a Kumaraswamy distribution.

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4: Comparative risk aversion
  • Proposition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.9
  • Remark 2.10
  • Definition 2.11
  • ...and 27 more