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Unformatted text preview: THE ROOMMATES PROBLEM REVISITED THAYER MORRILL UNIVERSITY OF MARYLAND OCTOBER 2007 Abstract. One of the oldest but least understood matching prob- lems is Gale and Shapley’s (1962) “roommates problem”: is there a stable way to assign 2 N students into N roommate pairs? Unlike the classic marriage problem or college admissions problem, there need not exist a stable solution to the roommates problem. How- ever, the traditional notion of stability ignores the key physical constraint that roommates require a room, and it is therefore too restrictive. Recognition of the scarcity of rooms motivates replac- ing stability with Pareto optimality as the relevant solution con- cept. This paper proves that a Pareto optimal assignment always exists in the roommates problem, and it provides an efficient algo- rithm for finding a Pareto improvement starting from any status quo . In this way, the paper reframes a classic matching problem, which previously had no general solution, to become both solvable and economically more meaningful. Email address: [email protected] I would like to thank Larry Ausubel, Peter Cramton, Melinda Sandler Morrill, and Daniel Vincent for their assistance throughout this project. I would also like to thank Daniel Aromi and Jonah Gelbach for their helpful comments. 1. Introduction Economics is often defined as the study of how to efficiently allocate scarce resources. As such, assignment problems are at the heart of economics. Two-sided matching theory asks how to best match two unalike objects such as students and schools, residents and hospitals, or kidneys and people in need of a transplant. A different but related question is how to best pair two like objects. Examples of these one- sided matches include roommates at a university, lab partners in a science class, and partners in a police force. Two-sided matching theory has been well studied by economists who have created an elegant and applicable theory. One-sided matching theory has been comparatively neglected 1 . This neglect is likely due to the very paper that introduced it. In their classic 1962 paper College Admissions and the Stability of Marriage Gale and Shapley introduce both the marriage problem and the room- mates problem. While Gale and Shapley prove a stable match always exist in a two-sided market, they demonstrate that a stable pairing need not exist in a one-sided market. Since a stable match need not exist, economists have been stymied in their attempts to find and an- alyze solutions to this important assignment problem. Unfortunately, this has led many economists to turn their attention elsewhere, and as a result, the economics literature on this classic problem is sparse....
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- Spring '08
- Game Theory, pareto, Pareto Improvement