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

For every integrable allocation (X₁, X₂, ..., X_n) of a random endowment Y=Σ_i^=1/nX_i among n agents, there is another allocation (X₁*, X₂*, ..., X_n*) such that for every 1≤i≤n, X_i* is a nondecreasing function of Y (or, (X₁*, X₂*, ..., X_n*) are co-monotone) and X_i* dominates X_i by Second Degree Dominance. If (X₁*, X₂*, ..., X_n*) is a co-monotone allocation of Y=Σ_i^=1/nX_i*, then for every 1≤i≤n, Y is more dispersed than X_i* 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 than 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.