Editing Stable matching problem (section)

==Rural hospitals theorem==
{{Main|Rural hospitals theorem}}
The rural hospitals theorem concerns a more general variant of the stable matching problem, like that applying in the problem of matching doctors to positions at hospitals, differing in the following ways from the basic {{mvar|n}}-to-{{mvar|n}} form of the stable marriage problem:
* Each participant may only be willing to be matched to a subset of the participants on the other side of the matching.
* The participants on one side of the matching (the hospitals) may have a numerical capacity, specifying the number of doctors they are willing to hire.
* The total number of participants on one side might not equal the total capacity to which they are to be matched on the other side.
* The resulting matching might not match all of the participants.
In this case, the condition of stability is that no unmatched pair prefer each other to their situation in the matching (whether that situation is another partner or being unmatched). With this condition, a stable matching will still exist, and can still be found by the Gale–Shapley algorithm.

For this kind of stable matching problem, the rural hospitals theorem states that:
* 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.