Stackelberg Equilibria in Monopoly Insurance Markets with Probability Weighting

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.

• The paper characterizes Stackelberg equilibria for arbitrary policyholder distortion functions, showing that optimal coverage follows layers where the insurer’s pricing distortion is below the policyholder’s weighting function.
• Inverse-S-shaped probability weighting produces a deductible with full coverage for extreme losses, while weak or strong risk aversion yields full insurance and VaR weighting produces capped coverage.
• The paper proves that equilibrium contracts are individually rational and Pareto optimal while the monopolist extracts all consumer surplus, although profits vary non-monotonically with distortion parameters and loss distributions.

Setting and model

The paper studies a monopoly insurance market as a two-stage Stackelberg (Bowley) game. A risk-neutral, profit-maximizing insurer moves first by choosing a pricing distortion function $g$, which induces a distortion premium principle $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$. A single policyholder then responds by selecting an indemnity function $I$ that minimizes a distortion risk measure $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$ of their end-of-period exposure. Indemnities are restricted to 1-Lipschitz functions satisfying the no-sabotage condition of Carlier and Dana, which rules out ex post moral hazard from loss misreporting; indemnity and retention are thereby comonotone.

The principal contribution relative to prior work—most closely Cheung, Yam and Zhang's risk-adjusted Bowley reinsurance model—is the removal of curvature restrictions on the policyholder's distortion function $T$. Earlier analyses assumed either strict concavity (strong risk aversion) or a VaR-type distortion. Here $T$ may be any distortion function, accommodating inverse-S-shaped (Tversky–Kahneman), S-shaped (Prelec), and Hurwicz-style weightings, all empirically better supported than concave distortions.

The two-step characterization

Both stages are reformulated on quantile space using comonotonic additivity: for admissible retention quantiles $q \in \mathcal{Q}_L$,

$$\rho^{Pol}(q,g) = \int_0^1 q'(t)\,[T(1-t) - g(1-t)]\,dt + \int_0^1 (F_X^{-1})'(t)\, g(1-t)\,dt.$$

Policyholder's problem. For fixed $g$, minimization is pointwise in the marginal retention. The optimal indemnity has a layer structure: full coverage ($\kappa = 1$) on loss layers where $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$0 (policyholder more pessimistic than the pricing functional); no coverage ($\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$1) where the inequality reverses; and arbitrary marginal indemnification within feasibility where the two coincide. This extends Assa's marginal indemnity approach to a setting where the premium principle itself is endogenous and strategic.

Insurer's problem. Writing insurer profit via an auxiliary function $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$2, the profit maximizes pointwise, yielding a conjugate-based characterization: the optimal pricing distortion satisfies $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$3 whenever $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$4 (i.e., $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$5), and otherwise lies in an interval determined by monotonicity constraints. Equilibrium profit takes the closed form

$\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$6

which depends only on the regions where the policyholder is weakly risk averse. The equilibrium pricing distortion is thus pinned down by the policyholder's own risk perception: prices never exceed the policyholder's marginal willingness to insure tail losses.

Comparative statics and equilibrium contract forms

Two comparative results follow directly. If $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$7 pointwise, then both equilibrium coverage and equilibrium insurer profit are higher under $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$8. Since strong risk aversion implies weak risk aversion (via a concave transformation of $\Pi_g(I(X)) = \int I(X)\, dg\circ\mathbb{P}$9), the same monotonicity holds under strong risk aversion. Notably, the paper concedes that no analogous monotonicity exists when the loss distribution changes rather than preferences—a limitation of the comparative analysis.

Three canonical contract forms emerge as special cases:

Policyholder type	Equilibrium contract	Pricing distortion
Weakly/strongly risk averse ($I$0)	Full insurance	$I$1
VaR at level $I$2	Full coverage below $I$3, capped above	Step-function distortion
Inverse-S-shaped	Straight deductible $I$4; upper tail fully ceded	Coincides with $I$5 below $I$6

The first two recover Cheung–Yam–Zhang as special cases. The third is new relative to that literature: for inverse-S-shaped distortions—where the policyholder overweights both very small and very large tail probabilities—the equilibrium is a deductible contract under which extreme losses are fully transferred to the insurer. This contrasts sharply with the VaR case, where the upper tail is uninsured, and illustrates how the shape of $I$7, not merely its level of risk aversion, determines contract form.

Numerical experiments with the Tversky–Kahneman family $I$8 (for $I$9) over uniform, truncated exponential, and Kumaraswamy loss distributions show that equilibrium insurer profit is not monotone in the intersection point $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$0: it attains an interior maximum (around $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$1 across specifications). With right-skewed losses (larger $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$2, or smaller $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$3 in Kumaraswamy), the profit-maximizing $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$4 shifts toward smaller values, indicating that greater policyholder concern for extremes raises the insurer's achievable profit when large losses become less likely—but when the underlying risk increases in first-order stochastic dominance, equilibrium profit declines for all $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$5. These non-monotonicities underscore that the clean preference-based comparative statics do not extend to distributional changes.

Welfare properties: a pair of welfare theorems

The paper establishes a version of the two welfare theorems for this market. Pareto-optimal contracts are exactly those solving $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$6. Every Stackelberg equilibrium contract is individually rational and Pareto optimal, but leaves the policyholder exactly indifferent between participation and non-participation: $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$7. Conversely, any Pareto-optimal contract satisfying this indifference condition can be induced by some Stackelberg equilibrium mechanism (constructed by setting the pricing distortion equal to $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$8).

These results confirm, in a general-distortion setting, the surplus-extraction phenomenon documented by Boonen–Ghossoub and Ghossoub–Zhu: the monopolist extracts the entire consumer surplus while the allocation remains efficient. The efficiency result is therefore consistent with—and here does not depend on—the restrictive assumptions of earlier papers, since it holds for arbitrary distortion functions.

Limitations and open questions

Several restrictions bear on interpretation. First, the characterization requires $\rho^{Pol}(Z) = \int Z\, dT\circ\mathbb{P}$9 strictly increasing, ensuring differentiable-a.e. quantiles; atoms in the loss distribution fall outside the analysis. Second, equilibria are generally non-unique: wherever $T$0 coincides with $T$1 at the identity, the marginal indemnity $T$2 is arbitrary within feasibility, so the equilibrium set is a correspondence rather than a singleton. Third, the no-sabotage constraint excludes indemnity schedules that could mitigate moral hazard through other channels, and the market is centralized with perfect information—asymmetric-information extensions (adverse selection, belief heterogeneity) are outside scope. Fourth, the numerical finding of interior profit maxima in $T$3 is established only for specific parametric families; a general characterization of when equilibrium profit is maximized over ISSD families remains open. Finally, whether the layer-structure characterization survives under alternative premium principles or multiple policyholders interacting with heterogeneous distortion functions is not addressed here.

Conclusion

This paper delivers a complete, assumption-light characterization of Stackelberg equilibria in monopoly insurance markets with distortion-based preferences and distortion pricing. Equilibrium contracts have a layer structure governed by the comparison between $T$4 and $T$5 on tail probabilities; the equilibrium pricing distortion aligns with the policyholder's weak risk aversion; coverage and insurer profit increase monotonically with risk aversion; and equilibrium contracts constitute precisely the individually rational Pareto optima. The treatment of inverse-S-shaped distortions—yielding deductible contracts with fully ceded tails—is the substantive departure from prior work, and the welfare-theorem pair confirms the robustness of monopoly surplus extraction beyond concave-distortion settings.