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On the Existence of Three-Dimensional Stable Matchings with Cyclic Preferences

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Abstract

We study the three-dimensional stable matching problem with cyclic preferences. This model involves three types of agents, with an equal number of agents of each type. The types form a cyclic order such that each agent has a complete preference list over the agents of the next type. We consider the open problem of the existence of three-dimensional matchings in which no triple of agents prefer each other to their partners. Such matchings are said to be weakly stable. We show that contrary to published conjectures, weakly stable three-dimensional matchings need not exist. Furthermore, we show that it is NP-complete to determine whether a weakly stable three-dimensional matching exists. We achieve this by reducing from the variant of the problem where preference lists are allowed to be incomplete. We generalize our results to the k-dimensional stable matching problem with cyclic preferences for k ≥ 3.

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Acknowledgments

We are grateful to the anonymous referees for their many insightful comments, which led to significant improvements in the presentation.

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  1. Department of Computer Science, University of Texas at Austin, Austin, TX, USA

    Chi-Kit Lam & C. Gregory Plaxton

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  1. Chi-Kit Lam
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  2. C. Gregory Plaxton
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Correspondence to C. Gregory Plaxton.

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This article belongs to the Topical Collection: Special Issue on Algorithmic Game Theory (SAGT 2019) Guest Editors: Dimitris Fotakis and Vangelis Markakis

A preliminary version of this work appears in [14].

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Lam, CK., Plaxton, C.G. On the Existence of Three-Dimensional Stable Matchings with Cyclic Preferences. Theory Comput Syst 66, 679–695 (2022). https://doi.org/10.1007/s00224-021-10055-8

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  • DOI: https://doi.org/10.1007/s00224-021-10055-8

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