A new solution concept for the roommate problem: Q-stable matchings

The aim of this paper is to propose a new solution concept for the roommate problem with strict preferences. We introduce maximum irreversible matchings and consider almost stable matchings (Abraham et al. [3]) and maximum stable matchings (Tan [32], [34]). These solution concepts are all core consistent. We find that almost stable matchings are incompatible with the other two concepts. Hence, to solve the roommate problem we propose matchings that lie at the intersection of the maximum irreversible matchings and maximum stable matchings, which we call Qstable matchings. We construct an efficient algorithm for computing one element of this set for any roommate problem. We also show that the outcome of our algorithm always belongs to an absorbing set (Inarra et al. [18]). ∗This research is supported by the Spanish Ministry of Science and Innovation (ECO201231346 and ECO2013-44879-R), by the Basque Government (IT-568-13), by the Government of Andalusia (Project for Excellence in Research SEJ1436) and by COST action ICI205 on Computational Social Choice. Péter Biró also acknowledges support from the Hungarian Academy of Sciences under its Momentum Programme (LD-004/2010), and the Hungarian Scientific Research Fund, OTKA, grant no. K108673. †Institute of Economics, Research Centre for Economic and Regional Studies, Hungarian Academy of Sciences, H-1112, Budaörsi út 45, Budapest, Hungary. Email: [email protected]. Part of his work on this paper was conducted while he was visiting professor at the Department of Economics at Stanford University in 2014. ‡BRiDGE, University of the Basque Country (UPV/EHU) and Instituto de Economı́a Pública. Email: [email protected] §GLOBE, Departamento de Teoŕıa e Historia Económica, Universidad de Granada. Email: [email protected]