The Distortion of Stable Matching

2026-02-16

AI Summary

This study addresses the problem of designing stable matching algorithms with low distortion to optimize social welfare when only ordinal preferences of agents are known. It first establishes that any purely ordinal algorithm suffers from unbounded distortion. To overcome this limitation, the work proposes two efficient approaches: one requiring just a single cardinal query per agent and another employing a simple randomized strategy—both achieving the optimal distortion of 2. Furthermore, the paper presents an asymptotically query-optimal algorithm that attains a $(1+\varepsilon)$-approximate distortion with $O(\log n / \varepsilon^2)$ queries, and demonstrates significantly reduced query complexity on structured instances. Both theoretical analysis and empirical evaluation confirm the effectiveness and superiority of the proposed methods.

Technology Category

• Search and Optimization: Algorithm Configuration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We initiate the study of distortion in stable matching. Concretely, we aim to design algorithms that have limited access to the agents'cardinal preferences and compute stable matchings of high quality with respect to some aggregate objective, e.g., the social welfare. Our first result is a strong impossibility: the classic Deferred Acceptance (DA) algorithm of Gale and Shapley [1962], as well as any deterministic algorithm that relies solely on ordinal information about the agents'preferences, has unbounded distortion. To circumvent this impossibility, we consider algorithms that either (a) use randomization or (b) perform a small number of value queries to the agents'cardinal preferences. In the former case, we prove that a simple randomized version of the DA algorithm achieves a distortion of $2$, and that this is optimal among all randomized stable matching algorithms. For the latter case, we prove that the same bound of $2$ can be achieved with only $1$ query per agent, and improving upon this bound requires $\Omega(\log n)$ queries per agent. We further show that this query bound is asymptotically optimal for any constant approximation: for any $\varepsilon>0$, there exists an algorithm which uses $O(\log n /\varepsilon^2)$ queries, and achieves a distortion of $1+\varepsilon$. Moreover, under natural structural restrictions on the instances of the problem, we provide improved upper bounds on the number of queries required for a $(1+\varepsilon)$-approximation. We complement our main findings above with theoretical and empirical results on the average-case performance of stable matching algorithms, when the preferences of the agents are drawn i.i.d. from a given distribution.

Problem

• stable matching
• distortion
• cardinal preferences
• social welfare
• ordinal information

Innovation

• distortion
• stable matching
• randomized algorithms
• value queries
• social welfare