On multi-dimensional risk sharing problems

A well-known result of Landsberger and Meilijson says that efficient risk-sharing rules for univariate risks are characterized by a so-called comonotonicity condition. In this talk, I'll first discuss a multivariate extension of this result (joint work with R.-A. Dana and A. Galichon). Then I will discuss the restrictions (in the form of systems of nonlinear PDEs) efficient risk sharing imposes on individual consumption as a function of aggregate consumption. I'll finally give an identification result on how to recover preferences from the knowledge of the risk sharing (joint work with M. Aloqeili and I. Ekeland).