Pareto efficiency for the concave order and multivariate comonotonicity

TL;DR

This paper extends the concept of comonotonicity and efficiency in risk-sharing from univariate to multivariate settings using convex duality and optimal transportation techniques.

Contribution

It generalizes the comonotone dominance principle and the equivalence of efficiency and comonotonicity to multivariate risks.

Findings

Characterization of multivariate efficiency via generalized comonotonicity

Use of convex duality and optimal transportation in multivariate risk analysis

Extension of univariate risk-sharing principles to higher dimensions

Abstract

In this paper, we focus on efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson [25], that efficiency is characterized by a comonotonicity condition. The goal of this paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multi-dimensional case. The multivariate setting is more involved (in particular because there is no immediate extension of the notion of comonotonicity) and we address it using techniques from convex duality and optimal transportation.