Across ALL stable matchings, the set of matched participants and the number of filled positions at each hospital is identical. No mechanism design can fix rural hospital shortages.
In any two stable matchings M and M', every hospital fills exactly the same number of positions. Any resident unmatched in one stable matching is unmatched in ALL stable matchings.
1. Run resident-proposing DA → M_R 2. Run hospital-proposing DA → M_H 3. These are the two extreme stable matchings 4. Compare filled positions at each hospital 5. Verify: counts are IDENTICAL
| Hospital | Quota | Type | Ranking |
|---|---|---|---|
| Urban A | 2 | Urban | R1 > R2 > R3 > R4 > R5 |
| Suburban B | 2 | Suburban | R2 > R3 > R1 > R5 > R4 |
| Rural C | 2 | Rural | R3 > R5 > R4 > R1 > R2 |
| Resident | Preferences |
|---|---|
| R1 | A > B > C |
| R2 | A > B > C |
| R3 | B > A > C |
| R4 | A > C > B |
| R5 | B > C > A |
Policy Implication: To help Rural C, you must change the FUNDAMENTALS (pay, location, quality of life), not the algorithm. No stable matching mechanism can increase Rural C's fill rate.
Roth, A.E. (1986). "On the Allocation of Residents to Rural Hospitals: A General Property of Two-Sided Matching Markets." Journal of Economic Theory, 36, 277-288.
Each bar shows the number of filled positions. The Rural Hospitals Theorem guarantees these counts are invariant.