Multi-marginal optimal transport and multi-agent matching problems: uniqueness and structure of solutions

Abstract

We prove uniqueness and Monge solution results for multi-marginal optimal transportation problems with a certain class of surplus functions; this class arises naturally in multi-agent matching problems in economics. This result generalizes a seminal result of Gangbo and \'Swi\c{e}ch on multi-marginal problems. Of particular interest, we show that this also yields a partial generalization of the Gangbo-\'Swi\c{e}ch result to manifolds; alternatively, we we can think of this as a partial extension of McCann's theorem for quadratic costs on manifolds to the multi-marginal setting. We also show that the class of surplus functions considered here neither contains, nor is contained in, another class of surpluses studied by the present author, which also generalized Gangbo and \'Swi\c{e}ch's result.