# Interpolating between matching and hedonic pricing models

## TL;DR

This paper explores a unified theoretical framework connecting matching and hedonic pricing models, leveraging multi-marginal optimal transport theory to analyze stability, support dimension, and uniqueness of solutions.

## Contribution

It introduces a novel connection between matching models and multi-marginal optimal transport, providing bounds on support dimension and conditions for uniqueness and purity of stable matchings.

## Key findings

- Upper bound on support dimension of stable matchings
- Conditions for uniqueness and purity of stable matchings
- Examples satisfying and violating the preference condition

## Abstract

We consider the theoretical properties of a model which encompasses bi-partite matching under transferable utility on the one hand, and hedonic pricing on the other. This framework is intimately connected to tripartite matching problems (known as multi-marginal optimal transport problems in the mathematical literature). We exploit this relationship in two ways; first, we show that a known structural result from multi-marginal optimal transport can be used to establish an upper bound on the dimension of the support of stable matchings. Next, assuming the distribution of agents on one side of the market is continuous, we identify a condition on their preferences that ensures purity and uniqueness of the stable matching; this condition is a variant of a known condition in the mathematical literature, which guarantees analogous properties in the multi-marginal optimal transport problem. We exhibit several examples of surplus functions for which our condition is satisfied, as well as some for which it fails.