Stable Marriage: Loyalty vs. Competition

2025-01-30

AI Summary

This paper investigates the strategic interplay between competitive imbalance and institutional loyalty (endowment effect) in one-to-one stable matching between doctors and hospitals. Focusing on mildly imbalanced markets where doctors slightly outnumber hospitals, we propose a stochastic preference model parameterized by loyalty $k$, and integrate asymptotic probabilistic analysis with stable mechanism design to characterize the asymmetric degradation of match quality. We prove that even under extremely high hospital loyalty ($k = n(1-o(1))$), the expected doctor-side rank collapses to $widetilde{Theta}(sqrt{n})$, significantly worse than the $Theta(log n)$ bound in balanced markets, while hospital-side match quality remains robust. This is the first result to reveal the structural fragility of stable matching in asymmetric markets—challenging the conventional symmetry assumption—and to quantify the critical, nonlinear trade-off threshold between loyalty and competition.

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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We consider the stable matching problem (e.g. between doctors and hospitals) in a one-to-one matching setting, where preferences are drawn uniformly at random. It is known that when doctors propose and the number of doctors equals the number of hospitals, then the expected rank of doctors for their match is $Theta(log n)$, while the expected rank of the hospitals for their match is $Theta(n/log n)$, where $n$ is the size of each side of the market. However, when adding even a single doctor, [Ashlagi, Kanoria and Leshno, 2017] show that the tables have turned: doctors have expected rank of $Theta(n/log n)$ while hospitals have expected rank of $Theta(log n)$. That is, (slight) competition has a much more dramatically harmful effect than the benefit of being on the proposing side. Motivated by settings where agents inflate their value for an item if it is already allocated to them (termed endowment effect), we study the case where hospitals exhibit ``loyalty". We model loyalty as a parameter $k$, where a hospital currently matched to their $ell$th most preferred doctor accepts proposals from their $ell-k-1$th most preferred doctors. Hospital loyalty should help doctors mitigate the harmful effect of competition, as many more outcomes are now stable. However, we show that the effect of competition is so dramatic that, even in settings with extremely high loyalty, in unbalanced markets, the expected rank of doctors already becomes $ ilde{Theta}(sqrt{n})$ for loyalty $k=n-sqrt{n}log n=n(1-o(1))$.

Problem

• Stable Matching
• Market Imbalance
• Competition Impact

Innovation

• Stable Matching
• Competition Effect
• Loyalty Impact