The so-called risk sharing problem has two variants: (1) How can an aggregation of individual utilities within a system of agents be maximised by means of distributing a certain good, or (2) how can an aggregation of individual risks be minimised by means of redistributing a certain loss the system incurs? The problem has a long history in economics and financial mathematics, and  solving it in relevant infinite dimensional spaces is very challenging. Substantial results have been obtained in the context of Pareto efficiency of such sharing schemes when law-invariant (risk) preferences are involved. These rank Savage acts -- random variables -- in a way which only depends on their distribution under a fixed reference probability measure. It was shown that Pareto efficient allocations (i.e. sharing schemes) exist.

For problem (1) of maximising aggregated utility, the talk presents a unifying framework in which solutions to the risk sharing problem can be found. As applications, we will discuss the existence of weakly Pareto efficient allocations, Pareto efficient allocations, and core allocations.

In a second part, we solve the risk sharing problem (2) for a wide class of capital requirements determining individual risks. These will not be law-invariant functionals, but build on a law-invariant ingredient.

The talk is based on joint work with Gregor Svindland.