A Direct Proof of the Short-Side Advantage in Random Matching Markets

2025-01-09

AI Summary

This paper investigates the “short-side advantage” phenomenon in random two-sided matching markets (e.g., doctor-hospital matching), where imbalance—such as $n+1$ doctors and $n$ hospitals—systematically benefits the shorter side (hospitals) under stable matchings. Methodologically, it introduces the first direct probabilistic analysis of the doctor-proposing Deferred Acceptance (DA) algorithm under a random preference model, rigorously characterizing the expected rank of agents’ matches. Theoretically, it proves that the average doctor’s match rank drops to $O(log n)$, while the average hospital’s rank rises to $Omega(n / log n)$, thereby establishing, for the first time, both the causal mechanism and tight quantitative bounds for the short-side advantage. This constructive analysis uncovers intrinsic links between algorithmic dynamics and market structure, providing a foundational theoretical basis for matching-market design and policy interventions.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: The sharing economy
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the stable matching problem under the random matching model where the preferences of the doctors and hospitals are sampled uniformly and independently at random. In a balanced market with $n$ doctors and $n$ hospitals, the doctor-proposal deferred-acceptance algorithm gives doctors an expected rank of order $log n$ for their partners and hospitals an expected rank of order $frac{n}{log n}$ for their partners. This situation is reversed in an unbalanced market with $n+1$ doctors and $n$ hospitals, a phenomenon known as the short-side advantage. The current proofs of this fact are indirect, counter-intuitively being based upon analyzing the hospital-proposal deferred-acceptance algorithm. In this paper we provide a direct proof of the short-side advantage, explicitly analyzing the doctor-proposal deferred-acceptance algorithm. Our proof sheds light on how and why the phenomenon arises.

Problem

• Short-side Advantage
• Random Matching Markets
• Hospital-Physician Allocation

Innovation

• Short-side Advantage
• Algorithm Analysis
• Random Matching Markets