[Paper Review] How good are Popular Matchings?

A M Krishnapriya, Meghana Nasre+2 more|arXiv (Cornell University)|Apr 28, 2018

Game Theory and Voting SystemsEconomics, Econometrics and Finance4 citations

TL;DR

This paper proposes a novel algorithm to compute popular matchings in the Hospital-Resident problem with Lower Quotas (HRLQ), circumventing prior restrictions on preference lists. It demonstrates empirically that popular matchings outperform stable and envy-free matchings in key practical metrics—achieving up to 10% larger matchings, fewer blocking pairs, and higher resident satisfaction—making popularity a strong alternative to stability in both HRLQ and standard HR settings.

ABSTRACT

In this paper, we consider the Hospital Residents problem (HR) and the Hospital Residents problem with Lower Quotas (HRLQ). In this model with two sided preferences, stability is a well accepted notion of optimality. However, in the presence of lower quotas, a stable and feasible matching need not exist. For the HRLQ problem, our goal therefore is to output a good feasible matching assuming that a feasible matching exists. Computing matchings with minimum number of blocking pairs (MinBP) and minimum number of blocking residents (MinBR) are known to be NP-Complete. The only approximation algorithms for these problems work under severe restrictions on the preference lists. We present an algorithm which circumvents this restriction and computes a popular matching in the HRLQ instance. We show that on data-sets generated using various generators, our algorithm performs very well in terms of blocking pairs and blocking residents. Yokoi (ISAAC 2017) recently studied envy-free matchings for the HRLQ problem. We propose a simple modification to Yokoi's algorithm to output a maximal envy-free matching. We observe that popular matchings outperform envy-free matchings on several parameters of practical importance, like size, number of blocking pairs, number of blocking residents. In the absence of lower quotas, that is, in the Hospital Residents (HR) problem, stable matchings are guaranteed to exist. Even in this case, we show that popularity is a practical alternative to stability. For instance, on synthetic data-sets generated using a particular model, as well as on real world data-sets, a popular matching is on an average 8-10% larger in size, matches more number of residents to their top-choice, and more residents prefer the popular matching as compared to a stable matching. Our comprehensive study reveals the practical appeal of popular matchings for the HR and HRLQ problems.

Motivation & Objective

• To address the absence of stable and feasible matchings in HRLQ instances with lower quotas.
• To propose a practical alternative to stability using popularity as an optimality criterion in HRLQ and HR settings.
• To empirically evaluate popular matchings against stable and envy-free matchings on synthetic and real-world data-sets.
• To develop a scalable algorithm for computing popular matchings in HRLQ without requiring complete preference lists.
• To compare popularity with existing approaches like Min-BP and Min-BR, which are NP-hard to optimize.

Proposed method

• Proposes a new algorithm to compute a maximum cardinality popular matching in HRLQ instances, avoiding the CL-restriction of prior approximation algorithms.
• Introduces a simple modification to Yokoi’s envy-free matching algorithm to compute a maximal envy-free matching for comparison.
• Employs a comprehensive experimental evaluation on synthetic data-sets generated via the Shuffle model and real-world data from IIT-M and HR with couples.
• Measures performance across multiple metrics: matching size, number of blocking pairs, number of blocking residents, and resident preference for the matching.
• Uses statistical comparison between popular matchings (Mp), stable matchings (Ms), and maximum cardinality matchings (Mm) across various data-sets.
• Applies data generators including the Shuffle model and real elective allocation data to ensure representativeness and validity of results.

Experimental results

Research questions

• RQ1 How does the size of a popular matching compare to that of a stable matching in HR and HRLQ instances?
• RQ2 To what extent do popular matchings reduce the number of blocking pairs and blocking residents compared to stable and envy-free matchings?
• RQ3 How does the popularity of a matching compare to stability in terms of resident satisfaction and top-choice assignment?
• RQ4 Can a popular matching be computed efficiently in HRLQ without requiring complete preference lists?
• RQ5 How do popular matchings perform on real-world data-sets from elective allocation systems?

Key findings

• On synthetic data-sets generated via the Shuffle model, popular matchings are on average 8-10% larger than stable matchings.
• In real-world data from IIT-M elective allocation, popular matchings achieve up to 23.81% higher matching size gain over stable matchings.
• Popular matchings reduce the number of blocking pairs by up to 16.76% compared to stable matchings on synthetic data.
• A majority of residents prefer the popular matching over the stable matching, with up to 18.02% preferring it in the Shuffle model.
• The number of residents matched to their top-choice hospital increases by up to 18% in popular matchings compared to stable matchings on the Shuffle model.
• In real-world IIT-M data, popular matchings outperform stable matchings in all key metrics, including a 24.06% increase in top-choice assignments in the Aug–Nov 2017 data-set.