Risk sharing under heterogeneous beliefs without convexity

TL;DR

This paper investigates Pareto-optimal risk sharing among agents with heterogeneous, law-invariant risk measures that are not necessarily convex, providing conditions for existence and extending to multidimensional markets.

Contribution

It introduces a simple sufficient condition for the existence of Pareto optima without assuming convexity of risk measures, linking the problem to collapse-to-the-mean results.

Findings

Established existence conditions for Pareto-optimal allocations.

Linked risk sharing to collapse-to-the-mean phenomena.

Extended results to multidimensional security markets.

Abstract

We consider the problem of finding Pareto-optimal allocations of risk among finitely many agents. The associated individual risk measures are law invariant, but with respect to agent-dependent and potentially heterogeneous reference probability measures. Moreover, we assume that the individual risk assessments are consistent with the respective second-order stochastic dominance relations. We do not assume their convexity though. A simple sufficient condition for the existence of Pareto optima is provided. The proof combines local comonotone improvement with a Dieudonn\'e-type argument, which also establishes a link of the optimal allocation problem to the realm of "collapse to the mean" results. Finally, we extend the results to capital requirements with multidimensional security markets.