Rural hospitals theorem

Consider a many-to-one matching market with a finite set $D$ of doctors and a finite set $H$ of hospitals, where hospital $h$ has $q_h$ positions and all preferences are strict. For a matching $\mu$, write $\mu(h)\subseteq D$ for the set of doctors assigned to $h$. Let $\mu$ and $\nu$ be any two stable matchings. Then

1. a doctor is matched under $\mu$ if and only if she is matched under $\nu$;
2. every hospital fills the same number of positions: $|\mu(h)|=|\nu(h)|$ for all $h\in H$;
3. every hospital with an empty position under $\mu$ receives the same doctors under $\nu$:

This is the rural hospitals theorem (Roth 1984, 1986): as long as the mechanism produces stable matchings, changing it cannot give under-subscribed ("rural") hospitals more doctors or different doctors.

Formalization Note Uses the model RuralHospitals.Market (definition RuralHospitals_Market): finite types D (doctors) and H (hospitals); strict preferences given by injective rank functions on Option H / Option D (smaller rank = preferred, none = unmatched / empty position); responsive hospital preferences; capacities capacity : H → ℕ. An assignment is a map μ : D → Option H, and assigned μ h is the finite set of doctors placed at h.