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arxiv: 2602.16401 · v2 · pith:PKXSHPJ4new · submitted 2026-02-18 · 💱 q-fin.RM · econ.TH· q-fin.MF

Stackelberg Equilibria in Monopoly Insurance Markets with Probability Weighting

Maria Andraos , Mario Ghossoub , Bin Li , Benxuan Shi This is my paper

Pith reviewed 2026-05-22 11:22 UTC · model grok-4.3

classification 💱 q-fin.RM econ.THq-fin.MF
keywords policyholderinsuranceequilibriuminsurerdistortionequilibrialossespricing
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The pith

In monopoly insurance markets with probability weighting, Stackelberg equilibria produce layer-type indemnity functions that provide full coverage where the policyholder is more pessimistic than the insurer about tail losses, are Pareto efficient without welfare gains to the policyholder, and yield

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Insurance markets often involve a single insurer setting prices before customers decide how much coverage to buy. When both the insurer and the customer evaluate risks using distorted probabilities rather than objective ones, the resulting contract takes a specific form. Full insurance is provided only on those parts of the loss distribution where the customer is more pessimistic than the insurer's pricing rule about extreme outcomes. On other layers, no coverage is offered. The paper shows that making the customer more risk averse increases both the total insurance purchased and the insurer's profit. The contracts that arise are efficient in the sense that no other arrangement improves one party's position without harming the other, yet they leave the customer no better off than if no insurance were available. Special cases recover earlier results in the literature.

Core claim

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.

Load-bearing premise

The model presupposes a single policyholder facing a profit-maximizing insurer in a sequential Stackelberg game, with both parties evaluating risks exclusively via distortion risk measures and premium principles whose specific forms determine the relative pessimism comparison that drives the layer structure.

Figures

Figures reproduced from arXiv: 2602.16401 by Benxuan Shi, Bin Li, Maria Andraos, Mario Ghossoub.

Figure 1
Figure 1. Figure 1: b depicts the insurer’s expected profit under the Stackelberg optimal contract as a function of t1. The result shows that the insurer’s expected profit does not vary monotonically with θ or the intersection point t1. Notably, it reaches a maximum around t1 ≈ 0.32. t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T3 (t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3 = 0:3 3 = 0:5 3 = 0:7 (a) Policyholder’s Distortion F… view at source ↗
Figure 2
Figure 2. Figure 2: The case where X follows a truncated exponential distribution. Finally, assume that the random loss X follows a Kumaraswamy distribution with CDF given by FX(x) = 1 − [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The case where X follows a Kumaraswamy distribution. 6. Conclusion In this paper, we study Stackelberg Equilibria (Bowley optima) in a monopolistic centralized sequential-move insurance market. We consider a risk-neutral, profit-maximizing insurer who sets premia using a distortion premium principle, and a single policyholder who seeks to minimize a distortion risk measure. We characterize Stackelberg equi… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper analyzes Stackelberg (Bowley) equilibria in a monopolistic insurance market with a profit-maximizing insurer using a distortion premium principle and a single policyholder minimizing a distortion risk measure. Equilibria are characterized by an optimal indemnity function with a layer-type structure: full coverage on loss layers where the policyholder is more pessimistic than the insurer about tail losses, and zero coverage where the policyholder is less pessimistic. The equilibrium pricing distortion is determined by the policyholder's risk aversion such that prices do not exceed marginal willingness to pay for tail losses. Both coverage and insurer profit increase with the policyholder's risk aversion. Equilibria are Pareto efficient but confer no welfare gain to the policyholder; conversely, any Pareto-optimal contract with no welfare gain to the policyholder arises as an equilibrium. Special cases recover prior results in the literature.

Significance. If the layer-structure characterization and monotonicity results hold rigorously, the work contributes a precise game-theoretic treatment of insurance design under probability weighting and distortion risk measures. The explicit connection between relative pessimism and layer coverage, together with the Pareto-efficiency equivalence, strengthens the behavioral insurance literature and provides a foundation for analyzing monopoly markets with non-expected-utility agents. The recovery of existing results as special cases adds to the paper's integrative value.

major comments (2)
  1. [§3] §3 (main theorem on layer structure): the definition of 'more pessimistic about tail losses' must be stated explicitly as a pointwise or integral ordering between the policyholder's distortion g and the insurer's pricing distortion h (e.g., g(p) > h(p) or g'(p) > h'(p) for survival probabilities p in the relevant range). Without a precise, non-circular criterion that handles possible crossings of concave distortions, the unambiguous layer decomposition claimed in the abstract does not follow for arbitrary distortions.
  2. [Theorem on monotonicity] Theorem on monotonicity of coverage and profit: the proof that both quantities increase with the policyholder's risk aversion relies on the specific functional forms of the distortions; the manuscript should supply the explicit comparative-statics argument or envelope condition that establishes this without additional parametric restrictions.
minor comments (2)
  1. [§2] Notation for the survival functions and distortion functions should be introduced once in §2 and used consistently; occasional redefinition of g and h in later sections reduces readability.
  2. [equilibrium pricing result] The statement that 'prices never exceed the policyholder's marginal willingness to insure tail losses' would benefit from an explicit equation reference linking the equilibrium premium to the derivative of the policyholder's distortion risk measure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help improve the clarity and rigor of the presentation. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (main theorem on layer structure): the definition of 'more pessimistic about tail losses' must be stated explicitly as a pointwise or integral ordering between the policyholder's distortion g and the insurer's pricing distortion h (e.g., g(p) > h(p) or g'(p) > h'(p) for survival probabilities p in the relevant range). Without a precise, non-circular criterion that handles possible crossings of concave distortions, the unambiguous layer decomposition claimed in the abstract does not follow for arbitrary distortions.

    Authors: We agree that an explicit definition strengthens the result. In the proof of the main theorem in §3, the layer decomposition is obtained by comparing the policyholder's distortion g and the insurer's distortion h pointwise on survival probabilities. To address crossings explicitly, we will revise the manuscript to define the sets of layers where the policyholder is more pessimistic as the intervals on which g(p) > h(p), with the indemnity function switching at any crossing points of g and h. This pointwise criterion is non-circular and follows directly from the first-order optimality conditions without additional assumptions. We will update the statement in the abstract and the theorem to reference this definition. revision: yes

  2. Referee: [Theorem on monotonicity] Theorem on monotonicity of coverage and profit: the proof that both quantities increase with the policyholder's risk aversion relies on the specific functional forms of the distortions; the manuscript should supply the explicit comparative-statics argument or envelope condition that establishes this without additional parametric restrictions.

    Authors: The monotonicity is shown in the manuscript by differentiating the equilibrium indemnity and profit expressions with respect to the risk-aversion parameter, using the maintained concavity and monotonicity properties of the distortions. To make the argument fully explicit and general, we will add a dedicated comparative-statics subsection that applies the envelope theorem to the insurer's value function. This establishes the monotonicity of coverage and profit under the paper's maintained assumptions on g and h, without requiring further parametric restrictions on their functional forms. revision: yes

Circularity Check

0 steps flagged

No circularity: equilibrium layer structure derived from sequential optimization

full rationale

The paper formulates a Stackelberg game in which the insurer chooses a distortion-based premium principle to maximize expected profit and the policyholder chooses an indemnity schedule to minimize a distortion risk measure. The layer-type indemnity structure is obtained by comparing the two distortion functions pointwise on survival probabilities to determine where the policyholder's marginal valuation exceeds the insurer's pricing functional. This comparison is an input to the model (the specific forms of the distortions), not a fitted parameter or self-referential definition. Equilibrium efficiency and monotonicity results likewise follow directly from the first-order conditions of the two optimization problems. No self-citation is load-bearing for the central claims, and no known empirical pattern is merely renamed. The derivation is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions from risk measure theory and game theory rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Insurer sets premia via a distortion premium principle and policyholder minimizes a distortion risk measure.
    This modeling choice incorporates probability weighting and is invoked to define the pricing and risk evaluation in the Stackelberg game.

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Reference graph

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