Saturday, April 21, 2012

Is the Rural Hospital Problem Really a Problem?

Many medical markets use a version of the deferred acceptance algorithm (Gale Shapley (1962)) to match doctors to residencies or internships (e.g. Roth (1984), Roth and Peranson (1999)). The deferred acceptance algorithm takes in rank order lists of preferences of doctors (over hospitals) and hospitals (over doctors) and outputs a stable match that assigns doctors to hospitals.

It is possible that in a given stable match, some positions (say at undesirable rural hospitals) go unfilled. Because of this, it is sometimes argued that there is a “problem” with rural hospitals being understaffed and thus under-serving rural areas. Maybe there is another stable match that, while worse for some doctors, is better for rural hospitals? It turns out that the answer to that question is no. In a stable match, if a hospital is matched to a set of doctors that does not fill all of its positions, the hospital is matched to the same set of doctors in every stable match (e.g. Roth (1986)). Is there another way to increase the number of doctors that are assigned to rural hospitals? Or, is this even a question worth asking?

Japan recently implemented a program of “regional caps” which restrict the total number of applicants which are allowed to be matched with hospitals in each region. This is an attempt to “match” more doctors to rural areas by limiting the number that can be matched in urban areas. Kamada and Kojima (2011) construct a new mechanism that is stable, strategy-proof and more efficient (from the doctor’s perspective) than the currently implemented one. Basically, they are more flexible in determining which hospitals receive fewer doctors if hospitals in a region exceed the cap (the mechanism in place reduced the number of doctors at each hospital proportionally if the region exceeded the cap).

But is this really better than the original uncapped deferred acceptance algorithm? By entertaining the notation that regional caps are a hard and fast political constraint and calling their solution (constrained) efficient even when the outcome could on net hurt the hospitals (profits, the level of care they provide, etc.) which could lead to an overall socially worse outcome, Kamada and Kojima give the impression that by adopting their solution, the Japanese medical residency market will be “fixed.” In fact, the outcome may be strictly worse than before the regional caps policy was implemented since regional caps are just a type of quota system, and quotas are inefficient. Even if the hospitals are not hurt, doctors that are assigned to less preferable hospitals because of the regional caps bear the costs of the policy. It is hard to tell what the costs are, though. Rank order lists contain only ordinal information, so we don’t know how much doctors and urban hospitals are hurt or how much rural hospitals are helped (welfare comparisons are generally more difficult in matching than standard price theory). 

A contract between a hospital and an applicant usually includes some kind of wage or other benefit. It seems like if a rural hospital is truly underserved it should increase its wage (if the marginal benefit of having the position filled exceeds its cost, then it would be beneficial to the hospital to increase its wage offered with the position in hopes of encouraging more doctors to rate its positions higher). However, year after year, these hospitals seem to prefer to leave the position left unfilled than to increase the wage or any other benefits that would make the position more attractive to potential applicants.

This begs the question, is there even a real problem? If there is, it would have to be because paying more is infeasible or there are externalities not being taken into account. But it may be that the rural hospital problem is just an example of one group extracting rents from another by successfully complaining to a centralized authority. 

Market designers typically take constraints like caps as given and then see how well we can do given the constraints. But I think sometimes it is more important to really question whether these constraints really deserve lexicographic preference over other objectives. What are the real or important constraints is itself an interesting question relevant to design.

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