Stackelberg Equilibria in Monopoly Insurance Markets with Probability Weighting

TL;DR

This paper analyzes Stackelberg equilibria in a monopolistic insurance market with probability weighting, revealing how optimal indemnity and pricing functions depend on risk perceptions and risk aversion.

Contribution

It characterizes the structure of equilibrium contracts and pricing in a monopolistic insurance setting with distortion risk measures, extending existing results.

Findings

Optimal indemnity functions have a layer structure based on pessimism levels.

Equilibrium prices are bounded by policyholder’s marginal willingness to insure tail losses.

Insurance coverage and expected profit increase with policyholder's risk aversion.

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.