Optimal allocations with distortion risk measures and mixed risk attitudes

TL;DR

This paper investigates Pareto-optimal risk sharing among heterogeneous agents with distortion risk preferences, reducing complex multi-agent problems to simpler two-agent models and characterizing optimal allocations based on risk attitudes.

Contribution

It introduces a reduction of the multi-agent risk sharing problem to a two-agent framework using infimal convolution of distortion risk measures, providing new conditions for optimality and explicit solutions.

Findings

Optimal sharing is comonotonic for risk-averse agents.

Counter-monotonic sharing occurs among risk-seeking agents.

Explicit characterizations of optimal allocations under specific conditions.

Abstract

We study Pareto-optimal risk sharing in economies with heterogeneous attitudes toward risk, where agents' preferences are modeled by distortion risk measures. Building on comonotonic and counter-monotonic improvement results, we show that agents with similar attitudes optimally share risks comonotonically (risk-averse) or counter-monotonically (risk-seeking). We show how the general $n$-agent problem can be reduced to a two-agent formulation between representative risk-averse and risk-seeking agents, characterized by the infimal convolution of their distortion risk measures. Within this two-agent framework, we establish necessary and sufficient conditions for the existence of optimal allocations, and we identify when the infimal convolution yields an unbounded value. When existence fails, we analyze the problem under nonnegative allocation constraints, and we characterize optima explicitly, under piecewise-linear distortion functions and Bernoulli-type risks. Our findings suggest that the optimal allocation structure is governed by the relative strength of risk aversion versus risk seeking behavior, as intuition would suggest.