Robustness of Stable Matchings When Attributes and Salience Determine Preferences

TL;DR

This paper investigates how stable matchings in markets are affected by changes in attribute importance, providing algorithms to measure and compute their robustness to such perturbations and exploring the geometric structure of robustness regions.

Contribution

It formalizes the concept of robustness in stable matchings with respect to salience perturbations and develops polynomial algorithms for verification, maximization, and approximation of robustness.

Findings

Polynomial-time algorithms for stability verification within a radius

Methods to compute maximum robustness radius

Geometric characterization of robustness regions as low-dimensional polytopes

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.