  The aims of this study are twofold. First, we consider an optimal risk
allocation problem with non-convex preferences. By establishing an infimal
representation for distortion risk measures, we give some necessary and
sufficient conditions for the existence of optimal and asymptotic optimal
allocations. We will show that, similar to a market with convex preferences, in
a non-convex framework with distortion risk measures the boundedness of the
optimal risk allocation problem depends only on the preferences. Second, we
consider the same optimal allocation problem by adding a further assumption
that allocations are co-monotone. We characterize the co-monotone optimal risk
allocations within which we prove the "marginal risk allocations" take only the
values zero or one. Remarkably, we can separate the role of the market
preferences and the total risk in our representation.
