Counter-monotonic Risk Sharing with Heterogeneous Distortion Risk Measures

We study risk sharing among agents with preferences modeled by heterogeneous distortion risk measures, who are not necessarily risk averse. Pareto optimality for agents using risk measures is often studied through the lens of inf-convolutions, because allocations that attain the inf-convolution are Pareto optimal, and the converse holds true under translation invariance. Our main focus is on groups of agents who exhibit varying levels of risk seeking. Under mild assumptions, we derive explicit solutions for the unconstrained inf-convolution and the counter-monotonic inf-convolution, which can be represented by a generalization of distortion risk measures.

💡 Research Summary

This paper investigates optimal risk sharing among agents whose preferences are modeled by heterogeneous distortion risk measures, allowing for both risk‑averse and risk‑seeking behavior. The authors focus on the role of counter‑monotonic (CM) allocations—structures opposite to the classic comonotonic (CO) allocations that dominate in risk‑averse settings.

The setting is an atomless probability space ((\Omega,\mathcal{F},\mathbb{P})) with a total loss (X) belonging to a suitable space (typically (L^{\infty}), (L^{+}) or (L^{-})). Each agent (i) is equipped with a distortion risk metric (\rho_{h_i}) defined by a function (h_i) belonging to the class (HBV) of bounded‑variation functions; when (h_i) is increasing and normalized it becomes a genuine distortion risk measure. The collection ({h_i}_{i=1}^n) may be heterogeneous, and the functions need not be concave, thereby capturing risk‑seeking attitudes.

Risk sharing is formalized through the inf‑convolution \