Stable Matching under Matroid Rank Valuations

2025-02-27

AI Summary

This paper studies a two-sided matching problem where hospitals have matroid-rank valuations and doctors have unit-demand preferences—either ordinal or cardinal. It establishes the first stability-theoretic framework for such two-stage markets. We propose a sequential assignment mechanism leveraging matroid structure, ensuring stability and strategyproofness for doctors while achieving approximate strategyproofness for hospitals. Under cardinal preferences, the mechanism attains hospital utility optimization—utilitarian or Nash—subject to doctor-welfare maximization. We design a polynomial-time algorithm and rigorously prove its stability and doctor strategyproofness. Notably, we establish that maximizing hospital Nash welfare is NP-hard—a first in matching literature. The mechanism achieves theoretical optimality under both ordinal and cardinal preferences. Our core contribution lies in integrating matroid-rank valuations into stable matching theory, thereby unifying strategic robustness and welfare optimization objectives.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Machine Learning: Learning Preferences or Rankings

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Study two-sided matching with matroid rank valuations.
• Develop algorithms for stable, strategyproof doctor allocations.
• Prove NP-hardness in maximizing hospital Nash welfare.

Innovation

• Matroid rank valuations
• Sequential allocation algorithms
• NP-hard stable allocations