On the Continuity of the Feasible Set Mapping in Optimal Transport

TL;DR

This paper provides a shorter proof of the continuity of the feasible set mapping in optimal transport problems and explores its application to assignment games with unknown distributions.

Contribution

It introduces a novel, concise proof of the continuity of the feasible set correspondence in optimal transport and applies it to transferable utility matching problems.

Findings

Established a shorter proof of the continuity result

Demonstrated stability of optimal transport plans

Applied the result to assignment games with unknown types

Abstract

Consider the set of probability measures with given marginal distributions on the product of two complete, separable metric spaces, seen as a correspondence when the marginal distributions vary. In problems of optimal transport, continuity of this correspondence from marginal to joint distributions is often desired, in light of Berge's Maximum Theorem, to establish continuity of the value function in the marginal distributions, as well as stability of the set of optimal transport plans. Bergin (1999) established the continuity of this correspondence, and in this note, we present a novel and considerably shorter proof of this important result. We then examine an application to an assignment game (transferable utility matching problem) with unknown type distributions.