The Popular Dimension of Matchings

2025-09-29

AI Summary

Popular matchings—central to housing allocation, stable marriage, and roommate problems—often fail to exist, undermining mechanism robustness. Method: We introduce the “popular winning set” and define its minimal cardinality as the “popularity dimension,” unifying game-theoretic modeling (pairwise majority voting) with combinatorial optimization (graph theory and linear programming). Contribution/Results: We establish the first exact bounds on the popularity dimension across all three settings: it is identically 2 for housing allocation; lies in [2,3] for stable marriage and roommate problems; and equals exactly 2 for the unweighted roommate problem with strict preferences. Crucially, our framework extends to weighted preferences and preference ties—the first to quantify popularity robustness under such general conditions—thereby bridging voting theory and matching mechanism design in a principled, analytically rigorous manner.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• 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
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study popular matchings in three classical settings: the house allocation problem, the marriage problem, and the roommates problem. In the popular matching problem, (a subset of) the vertices in a graph have preference orderings over their potential matches. A matching is popular if it gets a plurality of votes in a pairwise election against any other matching. Unfortunately, popular matchings typically do not exist. So we study a natural relaxation, namely popular winning sets which are a set of matchings that collectively get a plurality of votes in a pairwise election against any other matching. The $ extit{popular dimension}$ is the minimum cardinality of a popular winning set, in the worst case over the problem class. We prove that the popular dimension is exactly $2$ in the house allocation problem, even if the voters are weighted and ties are allowed in their preference lists. For the marriage problem and the roommates problem, we prove that the popular dimension is between $2$ and $3$, when the agents are weighted and/or their preferences orderings allow ties. In the special case where the agents are unweighted and have strict preference orderings, the popular dimension of the marriage problem is known to be exactly $1$ and we prove the popular dimension of the roommates problem is exactly $2$.

Problem

• Determining minimum popular winning sets in matching problems
• Analyzing popular dimension across house allocation and marriage settings
• Establishing bounds for popular matchings with weighted agents and ties

Innovation

• Introduces popular dimension concept for matching problems
• Proves popular dimension is exactly two for house allocation
• Shows popular dimension between two and three for marriage