Abstract
In the stable roommates (SR) problem we have n agents, where each agent ranks all other agents in strict order of preference. The problem is then to match agents into pairs such that no two agents prefer each other to their matched partners, and this is a stable matching. The stable marriage (SM) problem is a special case of SR, where we have two equal sized sets of agents, men and women, where men rank only women and women rank only men. Every instance of SM admits at least one stable matching, whereas for SR as the number of agents increases the number of instances with stable matchings decreases. So, what will happen if in SM we allow men to rank men and women to rank women, i.e. we relax gender separation? Will stability abruptly disappear? And what happens in a stable roommates scenario if agents do not rank all other agents? Again, is stability uncommon? And finally, what happens if there are an odd number of agents? We present empirical evidence to answer these questions.
C. McCreesh—Supported by the Engineering and Physical Sciences Research Council [grant number EP/K503058/1].
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Notes
- 1.
We assume that every agent ranks himself in last position and can potentially be self-matched.
- 2.
To permute an array of n elements, vary i from n down to 2, randomly select j in the range i to 1 inclusive, then swap the \(i^{th}\) and \(j^{th}\) array elements.
- 3.
We leave any social interpretation of these observations to others.
- 4.
The x axis is cut short due to small sample size for odd n and large p.
- 5.
We thank David Manlove for this explanation.
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Acknowledgements
We would like to thank David Manlove, Augustine Kwanashie, Rob Irving, Ian Gent and Craig Reilly.
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McCreesh, C., Prosser, P., Trimble, J. (2016). Morphing Between Stable Matching Problems. In: Rueher, M. (eds) Principles and Practice of Constraint Programming. CP 2016. Lecture Notes in Computer Science(), vol 9892. Springer, Cham. https://doi.org/10.1007/978-3-319-44953-1_52
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