Risk Sharing, Measuring Variability, and Distortion Riskmetrics

We address the problem of sharing risk among agents with preferences modelled by a generalclass of comonotonic additive and law-invariant functionals that need not be either monotoneor convex. Such functionals are called distortion riskmetrics, which include many statisticalmeasures of risk and variability used in portfolio optimization and insurance. The set of Pareto Optimal allocations is characterized under various settings of general or comonotonic risk sharing problems. We solve explicitly Pareto-optimal allocations among agents using the Gini deviation, the mean-median deviation, or the inter-quantile difference as the relevant variability measures. The latter is of particular interest, as optimal allocations are not comonotonic in the presenceof inter-quantile difference agents; instead, the optimal allocation features a mixture of pairwise counter-monotonic structures, showing some patterns of extremal negative dependence.