The rural hospitals theorem (RHT) is a fundamental theorem in the theory of stable matching.
It considers the problem of matching doctors to hospitals for residency, where each doctor is matched to a single hospital but each hospital has several positions for doctors. The total number of positions is larger than the total number of doctors, so some hospitals inevitably remain with unfilled positions. Usually, rural hospitals are less wanted than urban hospitals, so they often remain with many empty positions. This raised the question of whether the mechanism used to match doctors to hospitals can be changed in order to help these rural hospitals.[1]
The rural hospitals theorem answers this question negatively assuming all preferences are strict (i.e., no doctor is indifferent between two hospitals and no hospital is indifferent between two doctors). The theorem has two parts:
The set of assigned doctors, and the number of filled positions in each hospital, are the same in all stable matchings.
Any hospital that has some empty positions in some stable matching, receives exactly the same set of doctors in all stable matchings.
In other words: changing the matching mechanism (as long as it produces stable matchings) will not help the rural hospitals in any way: they will not receive more doctors, nor better doctors.
The theorem is robust in two-sided matching, since it applies to one-to-one and many-to one matchings, and can be extended to many-to-many matching.[2]
^Roth, Alvin E. (1986-03-01). "On the Allocation of Residents to Rural Hospitals: A General Property of Two-Sided Matching Markets". Econometrica. 54 (2): 425–427. doi:10.2307/1913160. ISSN 0012-9682. JSTOR 1913160.
^Klijn, Flip; Yazıcı, Ayşe (2014-10-01). "A many-to-many 'rural hospital theorem'" (PDF). Journal of Mathematical Economics. 54: 63–73. doi:10.1016/j.jmateco.2014.09.003. ISSN 0304-4068.
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