Optimal risk sharing with non-monotone monetary functionals

We consider the problem of sharing pooled risks among n economic agents endowed with non-necessarily monotone monetary functionals. In this framework, results of characterization and existence of optimal solutions are easily obtained as extension from the convex risk measures setting. Moreover, the introduction of the best monotone approximation of non-monotone functionals allows us to compare the original problem with the one which involves only ad hoc monotone criterions. The explicit calculation of optimal risk sharing rules is provided for particular cases, when agents are endowed with well known preference relations.