  We address the problem of sharing risk among agents with preferences modelled
by a general class of comonotonic additive and law-based functionals that need
not be either monotone or convex. Such functionals are called distortion
riskmetrics, which include many statistical measures 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
presence of inter-quantile difference agents; instead, the optimal allocation
features a mixture of pairwise counter-monotonic structures, showing some
patterns of extremal negative dependence.
