Stable Matching with Interviews

2025-01-21

AI Summary

Existing bilateral matching research typically assumes fully known preferences, overlooking the critical real-world process wherein agents—such as medical students and residency programs—gradually clarify preferences through iterative interactions (e.g., interviews). This omission leads to inefficiency in practical settings like the medical residency match. This paper is the first to explicitly model interview interactions as a prerequisite for stable matching. We propose both adaptive and non-adaptive algorithms that guarantee interim stability while strictly bounding the number of one-sided interviews at $O(log^2 n)$ or $O(log^3 n)$. Building upon an extended Gale–Shapley framework, our approach integrates random sampling, tiered matching, and probabilistic analysis to uniformly support both sequential proposal and batch-interview market structures. Theoretically, we prove that, with high probability, our algorithms achieve interim stable matchings in tiered random markets—significantly reducing practical matching costs.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Multiagent Systems: Mechanism Design
• Machine Learning: Learning Preferences or Rankings

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In several two-sided markets, including labor and dating, agents typically have limited information about their preferences prior to mutual interactions. This issue can result in matching frictions, as arising in the labor market for medical residencies, where high application rates are followed by a large number of interviews. Yet, the extensive literature on two-sided matching primarily focuses on models where agents know their preferences, leaving the interactions necessary for preference discovery largely overlooked. This paper studies this problem using an algorithmic approach, extending Gale-Shapley's deferred acceptance to this context. Two algorithms are proposed. The first is an adaptive algorithm that expands upon Gale-Shapley's deferred acceptance by incorporating interviews between applicants and positions. Similar to deferred acceptance, one side sequentially proposes to the other. However, the order of proposals is carefully chosen to ensure an interim stable matching is found. Furthermore, with high probability, the number of interviews conducted by each applicant or position is limited to $O(log^2 n)$. In many seasonal markets, interactions occur more simultaneously, consisting of an initial interview phase followed by a clearing stage. We present a non-adaptive algorithm for generating a single stage set of in tiered random markets. The algorithm finds an interim stable matching in such markets while assigning no more than $O(log^3 n)$ interviews to each applicant or position.

Problem

• Preference clarification
• Interactive decision-making
• Matching markets

Innovation

• Improved Algorithms
• Logarithmic Interview Rounds
• Hierarchical Random Markets