Three-dimensional stable matching with cyclic preferences Kimmo Eriksson a, ⁎ , Jonas Sjöstrand b , Pontus Strimling a a Department of Mathematics, Mälardalen University, Box 883, 721 23 Västerås, Sweden b Royal Institute of Technology, Sweden Received 30 October 2005; accepted 30 March 2006 Available online 9 May 2006 Abstract We consider stable three-dimensional matchings of three genders (3GSM). Alkan [Alkan, A., 1988. Non- existence of stable threesome matchings. Mathematical Social Sciences 16, 207–209] showed that not all instances of 3GSM allow stable matchings. Boros et al. [Boros, E., Gurvich, V., Jaslar, S., Krasner, D., 2004. Stable matchings in three-sided systems with cyclic preferences. Discrete Mathematics 286, 1–10] showed that if preferences are cyclic, and the number of agents is limited to three of each gender, then a stable matching always exists. Here we extend this result to four agents of each gender. We also show that a number of well-known sufficient conditions for stability do not apply to cyclic 3GSM. Based on computer search, we formulate a conjecture on stability of “strongest link” 3GSM, which would imply stability of cyclic 3GSM. © 2006 Elsevier B.V. All rights reserved. Keywords: Stable matching; 3GSM; Cyclic preferences; Balanced game; Effectivity function JEL classification: C78 1. Introduction The stable marriage problem is: Given a set of men and a set of women, find a matching that is stable in the sense that no man m and woman w who both prefer each other to their current partners in the matching. Gale and Shapley (1962) introduced this problem and gave a constructive proof of the existence of a stable matching for any combination of preferences. The theory of stable matchings has become an important subfield within game theory, as documented Mathematical Social Sciences 52 (2006) 77 – 87 www.elsevier.com/locate/econbase ⁎ Corresponding author. Tel.: +46 21 101533; fax: +46 21 101330. E-mail address: kimmo.eriksson@mdh.se (K. Eriksson). 0165-4896/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2006.03.005