MULTI-MARGINAL OPTIMAL TRANSPORT AND MULTI-AGENT MATCHING PROBLEMS: UNIQUENESS AND STRUCTURE OF SOLUTIONS

摘要

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 Swiech. Of particular interest, we show that this also yields a partial generalization of the Gangbo-Swiech result to manifolds; alternatively, 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, the class of surpluses studied in, another generalization of Gangbo and Swiech's result.