Universal Closest Refinement on Measurable Bipartite Relations

TL;DR

This paper introduces a measure-theoretic framework for the universal closest refinement problem on measurable bipartite relations, establishing a unique structure via convex refinement and equilibrium characterization.

Contribution

It develops a measure-theoretic approach with new tools to identify the universally closest refinement pairs in bipartite relations under divergence criteria.

Findings

Level-optimal maximin criterion characterizes the refinement structure.

Convex refinement minimizes relevant divergence functionals.

Universally closest pairs are characterized by divergence satisfying data-processing inequality.

Abstract

We study the universal closest refinement problem on measurable bipartite relations over standard Borel spaces. Given prescribed side measures, the feasible class consists of finite refinement plans concentrated on the relation and carrying one fixed marginal. The main question is whether this highly nonunique class nevertheless contains a mathematically distinguished class of refinements. We show that the correct one-sided extremal criterion is level-optimal maximin, a levelwise maximin principle formulated through truncation and overflow profiles. We then prove that this structure is exactly the one selected by convex refinement: every minimizer of a strictly convex refinement criterion is level-optimal maximin, while every level-optimal maximin refinement minimizes the full class of relevant proper lower semicontinuous convex divergence functionals. Proportional response then identifies the opposite-side partner and yields a universally closest refinement pair. Our main theorem shows that every such pair is universally closest among all feasible pairs for every divergence satisfying the data-processing inequality under measurable post-processing, and conversely that every universally closest pair has this structure. We also prove a converse paired characterization for strictly convex closest pairs. Finally, we give an equilibrium-theoretic characterization of level-optimal maximin pairs through a naturally associated continuum economy with measure-valued commodities. The measure-theoretic theory requires new tools beyond the finite case, including disintegration, measurable selection, measurable max-flow/min-cut duality, augmentation arguments, and a symmetric density decomposition separating absolutely continuous and singular components.