Stable Matching under Matroid Rank Valuations

TL;DR

The paper addresses stable matching in a two-sided market where hospitals have matroid rank valuations over doctors and doctors have ordinal or cardinal unit-demand utilities over hospitals, exploring stability, strategyproofness, and efficiency.For ordinal doctor preferences, it introduces HWSD and Serial Dictatorship mechanisms that achieve stability with doctor-strategyproofness and, in various valuation subclasses, approximate hospital strategyproofness while maximizing hospital welfare; it also proves an impossibility result for hospital-SP and hospital-USW optimization coexistence.For cardinal doctor utilities, it provides a max-doctor-Welfare approach via matroid intersection, then optimizes hospital NSW under the constraint of doctor welfare using a combination of cap-based optimization and local search, accompanied by a hardness result showing NP-hardness of NSW under stability.Overall, the work advances understanding of stable allocations in matroid-constrained two-sided markets, offers practical polynomial-time procedures with provable guarantees, and identifies key open questions on hospital-strategyproofness in the matroid rank valuation setting.

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.

Figures (2)

• Figure 1: The only two stable hospital-welfare maximizing allocations. Red nodes are nodes who receive their best possible outcome, and have no incentive to deviate.
• Figure 2: Illustration of the path transfer executed in the proof of \ref{['lem:fTreport']}. Each hospital has availability slots (gray squares) that can be matched to different doctors (denoted by blue edges). Full blue edges denote the actual matching of doctors to slots, and dashed edges denote possible alternative slots. The doctor $d_1$ is assigned to $h$ under $X_h$, but not under $Y$ (where $h$ receives $T$). We transfer $d_1$ to the slot+hospital it is assigned to under $Y$. That slot is occupied by the doctor $d_2$ who is moved to the spot of $d_3$. Finally, $d_3$ is assigned to their slot with $h_4$ under $Y$, which is unoccupied under $X$. This same path is reversed later in the proof to create the allocation $Y'$ from the allocation $Y$.

Theorems & Definitions (65)

• Definition 2.2: Non-redundancy
• Definition 2.3: Blocking Pair
• Definition 2.4: Stability
• Lemma 2.4
• Example 2.5
• Theorem 3.1
• Lemma 3.3
• proof
• Lemma 3.4
• proof