  In this paper, we focus on efficient risk-sharing rules for the concave
dominance order. For a univariate risk, it follows from a comonotone dominance
principle, due to Landsberger and Meilijson [25], that efficiency is
characterized by a comonotonicity condition. The goal of this paper is to
generalize the comonotone dominance principle as well as the equivalence
between efficiency and comonotonicity to the multi-dimensional case. The
multivariate setting is more involved (in particular because there is no
immediate extension of the notion of comonotonicity) and we address it using
techniques from convex duality and optimal transportation.
