Stable Matching under Matroid Rank Valuations

TL;DR

This paper explores stable matching algorithms in a two-sided market where hospitals have matroid rank valuations and doctors have either ordinal or cardinal preferences, providing new algorithms and complexity results.

Contribution

It introduces new algorithms for stable matchings under matroid rank valuations and analyzes their properties, including strategyproofness and welfare maximization.

Findings

Algorithms guarantee stability and strategyproofness for doctors.

Maximizes doctor welfare and hospital welfare under certain conditions.

Computing approximate hospital Nash welfare is NP-hard.

Abstract

We study a two-sided matching model where one side of the market (hospitals) has combinatorial preferences over the other side (doctors). Specifically, we consider the setting where hospitals have matroid rank valuations over the doctors, and doctors have either ordinal or cardinal unit-demand valuations over the hospitals. While this setting has been extensively studied in the context of one-sided markets, it remains unexplored in the context of two-sided markets. When doctors have ordinal preferences over hospitals, we present simple sequential allocation algorithms that guarantee stability, strategyproofness for doctors, and approximate strategyproofness for hospitals. When doctors have cardinal utilities over hospitals, we present an algorithm that finds a stable allocation maximizing doctor welfare; subject to that, we show how one can maximize either the hospital utilitarian or hospital Nash welfare. Moreover, we show that it is NP-hard to compute stable allocations that approximately maximize hospital Nash welfare.