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

TL;DR

This paper establishes uniqueness and Monge solution properties for a class of multi-marginal optimal transport problems relevant to economics, extending classical results to manifold settings and clarifying the relationships between different surplus function classes.

Contribution

It generalizes key optimal transport results to multi-agent matching on manifolds, providing new insights into solution structure and uniqueness.

Findings

Proves uniqueness and Monge solutions for specific surplus functions.

Extends classical multi-marginal optimal transport results to manifolds.

Clarifies the relationship between different classes of surplus functions.

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.