Beyond Stability: Improved Efficiency Guarantees for $α$-Stable Matchings

2026-07-20

AI Summary

This work investigates the inherent tension between stability and social welfare optimality in stable matching, introducing α-stability—a refinement wherein agents deviate only if their utility improves by more than a threshold α. The paper provides the first complete characterization of the theoretical trade-off between α-stability and efficiency under asymmetric valuations. It proposes a novel polynomial-time mechanism that inflates the values of an optimal matching before applying the Gale–Shapley algorithm, yielding an α-stable matching with tight efficiency guarantees: when α ≤ μ/(μ+1), the outcome is fully efficient (1-efficient); for larger α, social welfare is at least (1/α)·μ/(μ+1). Moreover, the study establishes that computing a welfare-optimal α-stable matching remains NP-hard even as α approaches 1.

Abstract

Stable matching mechanisms are fundamental to market design but face an inherent tension between stability and social welfare optimality. We study a natural relaxation of stability, termed $α$-stability, which models agents as willing to deviate only when the potential improvement is sufficiently large. Under $α$-stability, no pair of agents can deviate and improve their valuations by more than a factor of $1/α$, with $α\in (0,1]$. We provide a complete characterization of the stability-efficiency tradeoff under asymmetric valuations. This tradeoff depends on the degree of asymmetry $μ\in (0,1]$, which bounds the ratio between agents' valuations for any pair. Our results show that relaxing stability can substantially improve achievable efficiency guarantees. We further present a polynomial-time algorithm that computes an $α$-stable matching attaining the best possible efficiency guarantee. For $α\le μ/(μ+1)$, our algorithm achieves 1-efficiency; for larger $α$, it computes an $α$-stable matching achieving at least $(1/α)\cdot μ/(μ+1)$ of the optimal social welfare. Remarkably, our algorithm inflates the values of an optimal matching and then applies the Gale-Shapley algorithm to the modified instance. Finally, we show that computing an optimal $α$-stable matching is NP-hard, even under slight relaxations of stability, i.e., for $α$ close to 1.

Problem

• stable matching
• social welfare
• efficiency guarantee
• α-stability
• market design

Innovation

• alpha-stability
• stability-efficiency tradeoff
• asymmetric valuations
• polynomial-time algorithm
• Gale-Shapley algorithm