Matchings under Preferences: Strength of Stability and Trade-offs

TL;DR

This paper introduces new concepts of robustness and near stability for matchings under preferences, analyzing their computational complexity and trade-offs with societal optimality criteria.

Contribution

It proposes two quantitative solution concepts for matchings, explores their computational complexity, and provides algorithms for finding socially optimal robust matchings.

Findings

Polynomial-time algorithm for robust matchings

NP-hardness of finding nearly stable matchings

Trade-offs between stability and societal optimality

Abstract

We propose two solution concepts for matchings under preferences: robustness and near stability. The former strengthens while the latter relaxes the classic definition of stability by Gale and Shapley (1962). Informally speaking, robustness requires that a matching must be stable in the classic sense, even if the agents slightly change their preferences. Near stability, on the other hand, imposes that a matching must become stable (again, in the classic sense) provided the agents are willing to adjust their preferences a bit. Both of our concepts are quantitative; together they provide means for a fine-grained analysis of stability for matchings. Moreover, our concepts allow to explore the trade-offs between stability and other criteria of societal optimality, such as the egalitarian cost and the number of unmatched agents. We investigate the computational complexity of finding matchings that implement certain predefined trade-offs. We provide a polynomial-time algorithm that given agents' preferences returns a socially optimal robust matching, and we prove that finding a socially optimal and nearly stable matching is computationally hard.