Optimal allocations with distortion risk measures and mixed risk attitudes

2025-10-20

AI Summary

This paper investigates Pareto-optimal risk sharing under heterogeneous risk preferences, employing distortion risk measures. Addressing agents’ divergent attitudes—both risk-averse and risk-seeking—we reduce the multi-agent problem to a representative two-agent framework. Existence of optimal allocations is characterized via infimal convolution, and necessary and sufficient conditions for optimality are established; notably, unboundedness of the infimal convolution implies nonexistence of an optimal solution. Methodologically, we innovatively integrate comonotonic/countermonotonic improvements, piecewise-linear distortion functions, and Bernoulli-type utility modeling. For the first time, we derive explicit closed-form optimal allocations under general constraints. Our analysis rigorously demonstrates that the relative strength of agents’ risk attitudes fundamentally determines the structure of the optimal risk allocation, thereby providing both theoretical foundations and computationally tractable mechanisms for risk distribution in heterogeneous economies.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Reasoning under Uncertainty: Decision/Utility Theory
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

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.

Problem

• Optimizing risk sharing among agents with mixed risk attitudes
• Reducing multi-agent allocation to representative two-agent framework
• Characterizing optimal allocations under distortion risk measures

Innovation

• Reduces n-agent problem to two-agent formulation
• Uses infimal convolution of distortion risk measures
• Characterizes optima under nonnegative allocation constraints