Comonotonicity and Pareto Optimality, with Application to Collaborative Insurance

Two by-now folkloric results in the theory of risk sharing are that (i) any feasible allocation is convex-order-dominated by a comonotonic allocation; and (ii) an allocation is Pareto optimal for the convex order if and only if it is comonotonic. Here, comonotonicity corresponds to the no-sabotage condition, which aligns the interests of all parties involved. Several proofs of these two results have been provided in the literature, mostly based on the comonotonic improvement algorithm of Landsberger and Meilijson (1994) and a limit argument based on the Martingale Convergence Theorem. However, no proof of (i) is explicit enough to allow for an easy algorithmic implementation in practice; and no proof of (ii) provides a closed-form characterization of Pareto optima. In this paper, we provide novel proofs of these foundational results. Our proof of (i) is based on the theory of majorization and an extension of a result of Lorentz and Shimogaki (1968), which allows us to provide an explicit algorithmic construction that can be easily implemented. In addition, our proof of (ii) leads to a crisp closed-form characterization of Pareto-optimal allocations in terms of alpha-quantiles (mixed quantiles). An application to collaborative insurance, or decentralized risk sharing, illustrates the relevance of these results.