Stable matching as transport

TL;DR

This work builds a bridge between stable matching with aligned preferences and parametric optimal transport, introducing cost functions $c_\alpha(x,y)=\frac{1-\exp(\alpha\,u(x,y))}{\alpha}$ to interpolate between fairness and stability as $\alpha$ varies. It proves that $\alpha>0$ yields approximate stability and $\alpha<0$ yields egalitarian outcomes, with utilitarian welfare recovered at $\alpha\to0$, and establishes existence and structure results via concave transport concepts. The authors derive universal welfare and fairness bounds for stable matchings and show that large, misaligned markets can be well approximated by aligned models. They apply the framework to school choice, ride-sharing, and spatial matching, and extend the theory to ordinal conditions for alignment and to multi-sided markets, illustrating the practical relevance of stability-induced inequality and the tractability of the approach. Overall, the paper provides a unified, transport-based lens on stability, efficiency, and equity in broad matching contexts, with implications for market design and policy.

Abstract

This paper links matching markets with aligned preferences to optimal transport theory. We show that stability, efficiency, and fairness emerge as solutions to a parametric family of optimal transport problems. The parameter reflects society's preferences for inequality. This link offers insights into structural properties of matchings and trade-offs between objectives; showing how stability can lead to welfare inequalities, even among similar agents. Our model captures supply-demand imbalances in contexts like spatial markets, school choice, and ride-sharing. We also show that large markets with idiosyncratic preferences can be well approximated by aligned preferences, expanding the applicability of our results.

Figures (6)

• Figure 1: Two solutions to the matching problem.
• Figure 2: The density is above the axis for $X$, and below for $Y$.
• Figure 3: Forbidden patterns in stable matchings.
• Figure 4: The matchings which satisfy no-crossing for three values of $\theta$.
• Figure 5: Dependence of $\theta$ on $\alpha$ in the optimal transportation problem with cost $c_\alpha$ and distribution from Figure \ref{['fig:runningexample']}. The optimal choice of $\theta$ for each $\alpha$ is given by the curve where the two grey regions meet.

Theorems & Definitions (58)

• Example 1
• Example 2
• Example 3
• Lemma 1
• proof
• Lemma 2
• Theorem 1
• Corollary 1
• Example 4: Non-uniqueness of stable matchings
• Theorem 2