Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion

For every integrable allocation of a random endowment among n agents, there is another allocation such that for every is a nondecreasing function of Y (or, are co-monotone) and dominates by Second Degree Dominance. If is a co-monotone allocation of then for every Y is more dispersed than in the sense of the Bickel and Lehmann stochastic order. To illustrate the potential use of this concept in economics, consider insurance markets. It follows that unless the uninsured position is Bickel and Lehmann more dispersed then the insured position, the existing contract can be improved so as to raise the expected utility of both parties, regardless of their (concave) utility functions.