Robustness of Stable Matchings When Attributes and Salience Determine Preferences

TL;DR

This work studies robustness of stable matchings when one side evaluates candidates via salience-weighted attributes, and salience vectors may drift within an $oldsymbol{\ell_p}$ radius $r$ with a budget $k$ of coordinates allowed to change. It introduces a continuous robustness framework, provides polynomial-time verification and maximum-radius computation for fixed attribute dimension $m$, and develops an anytime search to identify the most robust stable matching using rotation posets and convex relaxations. It also links robustness to welfare-aware costs, offering polynomial-time proxy bounds and a constructive optimization approach via max-weight closure, while revealing a product-structured geometry of the robustness region across the salience space. Overall, the results integrate convex geometry, polyhedral descriptions, and classic stable-matching tools to quantify and optimize stability under structured preference perturbations with practical implications for market design.

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.

Figures (2)

• Figure 1: Example of $\mathcal{P}_\mu(b)$ for $m=3$: the shaded polygon inside the simplex $\Lambda_2$.
• Figure 2: Rotation schematic (exposed at $\mu$). Gray edges show the current pairs $(a_i,b_i)$; arrows show the reassignment in $\operatorname{elim}(\mu,\rho)$.

Theorems & Definitions (43)

• Definition 3.1: Robustness Verification (RV)
• Lemma 3.1
• proof
• Theorem 3.2: Polynomial-time verification via support enumeration
• proof
• Remark 3.1: Strict vs. non-strict inequalities
• Definition 4.1: Pairwise thresholds and robustness radius
• Lemma 4.1
• proof
• Theorem 4.2: Polynomial-time computation via pairwise thresholds