Robustness of Stable Matchings When Attributes and Salience Determine Preferences

2026-02-04

AI Summary

This study investigates the robustness of stable matchings in attribute-preference matching markets under perturbations of salience weights. Robustness is formalized as the stability radius—the largest perturbation magnitude in the salience vector that preserves stability—and is, for the first time, modeled as a geometric region problem. The analysis reveals that the stability region within the simplex exhibits a low-dimensional polyhedral product structure. The authors propose a polynomial-time algorithm to verify stability within a given radius and to compute the maximum robustness radius. Additionally, they design an anytime search algorithm to approximate the most robust stable matching and characterize its trade-off with matching cost. By integrating convex geometry and polyhedral analysis, the work enables efficient robustness verification, yields a computable robustness–cost frontier, and facilitates approximate evaluation of high-dimensional robustness region volumes.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Preferences
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

In many matching markets--such as athlete recruitment or academic admissions--participants on one side are evaluated by attribute vectors known to the other side, which in turn applies individual \emph{salience vectors} to assign relative importance to these attributes. Since saliences are known to change in practice, a central question arises: how robust is a stable matching to such perturbations? We address several fundamental questions in this context. First, we formalize robustness as a radius within which a stable matching remains immune to blocking pairs under any admissible perturbation of salience vectors (which are assumed to be normalized). Given a stable matching and a radius, we present a polynomial-time algorithm to verify whether the matching is stable within the specified radius. We also give a polynomial-time algorithm for computing the maximum robustness radius of a given stable matching. Further, we design an anytime search algorithm that uses certified lower and upper bounds to approximate the most robust stable matching, and we characterize the robustness-cost relationship through efficiently computable bounds that delineate the achievable tradeoff between robustness and cost. Finally, we show that for each stable matching, the set of salience profiles that preserve its stability factors is a product of low-dimensional polytopes within the simplex. This geometric structure precisely characterizes the polyhedral shape of each robustness region; its volume can then be computed efficiently, with approximate methods available as the dimension grows, thereby linking robustness analysis in matching markets with classical tools from convex geometry.

Problem

• stable matching
• robustness
• salience vectors
• attribute-based preferences
• matching markets

Innovation

• stable matching
• robustness
• salience vectors
• polyhedral geometry
• matching markets