Rural hospitals theorem, Part 1

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 stable matchings. Then

1. the set of assigned doctors is the same: for every $d\in D$, $d$ is matched under $\mu$ if and only if $d$ is matched under $\nu$;
2. the number of filled positions at each hospital is the same:

This is Part 1 of the rural hospitals theorem (Roth 1984): changing the stable mechanism cannot change who is employed or how many positions each hospital fills.

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.