The “Rural Hospital Theorem” Revisited∗ Fuhito Kojima† April 1, 2011

Abstract In the context of two-sided matching, we propose a new class of preferences called separable preferences with affirmative action constraints. We demonstrate that the celebrated “rural hospital theorem” in the matching literature generalizes to this class of preferences, but only with an appropriate definition of “vacant positions.” Motivated by this result, we then discuss practical issues about doctor shortages in rural hospitals.

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I am grateful to Yuichiro Kamada, Alvin E. Roth, Tayfun S¨onmez, Satoru Takahashi, and Jun Wako for discussion and comments. Seung Hoon Lee and Pete Troyan provided excellent research assistance. † Department of Economics, Stanford University, [email protected].

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Introduction

Recent developments in the theory of two-sided matching have resulted in the design of various labor matching clearinghouses and student placement mechanisms.1 A central concept in the design of such institutions is stability. A matching is said to be stable if there is no individual agent who prefers being unmatched to her allocation in the matching, and there is no pair of agents who block the prescribed outcome. Empirical and experimental studies have shown that stable mechanisms tend to succeed whereas unstable ones often fail.2 One of the difficult problems for matching clearinghouses is the geographical distribution of assignments. In the context of medical matching, one of the urgent issues is that many hospitals in rural areas fail to attract sufficient numbers of doctors to fill their positions. For instance, a Washington Post article (Talbott, 2007) reports a sensational statistic that more than 35 million Americans live in underserved areas, and 16,000 doctors – roughly equivalent to all the graduates from American medical schools each year– are needed to fill that need. Similar problems are reported around the globe, from the United Kingdom to India to Australia to Thailand to Japan.3 The medical community has wondered whether this problem can be addressed by improving the matching clearinghouse for medical interns and residents, a significant part of medical labor supply. However, an insight from the two-sided matching literature suggests that this approach is fruitless. More specifically, the so-called “rural hospital theorem” (Roth, 1986) demonstrates that if a hospital does not fill all its positions in one stable matching, then it is matched to an identical set of doctors in all possible stable matchings. Thus, according to this theorem, a hospital that cannot fill all the seats for residents under one stable matching mechanism cannot increase the number of filled positions no matter what other stable mechanism is used. In this paper, we investigate the extent to which the conclusion of the rural hospital theorem is robust to specific assumptions of the model. More specifically, we ask whether the theorem holds under weaker assumptions 1

For a survey of this theory, see Roth and Sotomayor (1990). For applications to labor markets, see Roth (1984) and Roth and Peranson (1999). For applications to student assignment, see for example Abdulkadiro˘glu and S¨onmez (2003), Abdulkadiro˘glu, Pathak, Roth, and S¨ onmez (2005) and Abdulkadiro˘glu, Pathak, and Roth (2005). See Palivos and Varvarigos (2010) for a relationship between education and economic growth. 2 For a summary of this evidence, see Roth (2002). 3 Shallcross (2005), Alcoba (2009), Nambiar and Bavas (2010), and Wongruang (2010). The problem in the Japanese context will be discussed in Conclusion.

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about hospitals’ preferences than previously established. To study this issue, we propose a new class of preferences called separable preferences with affirmative action constraints. Hospital preferences are separable with affirmative action constraints if students are assigned “types” and there is a vector of total quota and type-specific quotas such that (i) if a student type is “unacceptable”, then she is never demanded, (ii) it is always desirable to add an acceptable student unless the group of students exceeds the total quota or a group of students of some type exceeds type-specific quota for that type, and (iii) if a group of students violtates the total quota or any of type-specific quota, it is undesirable. We also adopt a standard assumption in the literature that hospital preferences are substitutable (Kelso and Crawford, 1982). Separable preferences with affirmative action constraints and subsitutability imply the law of aggregate demand (Hatfield and Milgrom, 2005). On the other hand, this condition is more general than many standard requirements such as responsiveness with quotas (Roth, 1985), responsiveness with affirmative action constraints (Abdulkadiro˘glu, 2005), and separability with quotas (Martinez, Masso, Neme, and Oviedo, 2000). We first note that a straightforward extension of the rural hospital theorem does not hold under this class of preferences. More specifically, we offer an example in which a hospital is matched to different sets of doctors in different stable matchings, even though its total quota is not filled in these stable matchings. Given the above observation, we seek to define an appropriate definition of “vacant positions” so that the rural hospital theorem holds. Informally, we consider vacant positions for, in addition to the entire group of students, each type of students. That is, we consider there to be a vacant position for a type of students if, and only if, neither the total quota nor the quota specific to that type is full. Our main theorem shows that the rural hospital theorem indeed holds under this concept of vacant positions. Finally, we discuss practical issues about doctor shortages in rural hospitals. Given our finding that the rural hospital theorem holds quite generally, match organizers who desire to affect geographic distributions of doctors need to resort to a different, and perhaps more drastic, approach. The Japanese government recently took such an approach using “regional caps,” which tries to increase doctor assignments in rural areas at the cost of stability (and, in fact, efficiency). We discuss the Japanese case based on a recent study by Kamada and Kojima (2011). The rest of the paper proceeds as follows. Section 2 sets up the model and defines our new class of preferences. Section 3 presents the result. Section 4 provides further discussion and concludes.

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2

Model

A market is tuple Γ = (S, C, (i )i∈S∪C ). By S and C we denote finite and disjoint sets of students and colleges (following the standard terminology of the literature, we frame our model in terms of students and colleges. However, the analysis can be applied to other situations such as matching between doctors and hospitals or between workers and firms). For each student s ∈ S, s is a strict preference relation over C and being unmatched (being unmatched is denoted by ∅). For each college c ∈ C, c is a strict preference relation over the set of subsets of students. We write j i k when j i k and j 6= k. If j i ∅, then j is said to be acceptable to i. Given preferences of agent i and a set of i’s potential partners X, define ˜ for any X ˜ ⊆ X. Chi (X) to be a set such that Chi (X) ⊆ X and Chi (X) i X In words, Chi (X) is the set of partners who are optimally chosen from X. A matching µ is a mapping from C ∪ S to C ∪ S such that (i) µ(s) ⊆ C and |µ(s)| ≤ 1 for every s, (ii) µ(c) ⊆ S, and (iii) µ(s) = c if and only if s ∈ µ(c).4 For any matchings µ and µ0 and any i ∈ S ∪ C, we write µ i µ ˜ if and only if µ(i) i µ ˜(i). Given a matching µ, we say that it is blocked by (s, c) if c s µ(s) and s ∈ Chc (µ(c) ∪ s). A matching µ is individually stable if µ(i) = Chi (µ(i)) for each i ∈ S ∪ C. A matching µ is stable if it is individually stable and is not blocked. For each college c ∈ C, its preference relation c is substitutable if for ˜ Sˆ ⊆ S with S˜ ⊆ S, ˆ we have Chc (S) ˆ ∩ S˜ ⊆ Chc (S) ˜ (Kelso and Crawany S, ford, 1982). That is, a student who is chosen from a larger set of potential partners is always chosen from a smaller set of potential partners. If every college has substitutable preferences, then there exists a stable matching (Roth, 1984). ˆ ≥ Preference relation c satisfies the law of aggregate demand if |Chc (S)| ˜ for any S, ˜ Sˆ ⊆ S with S˜ ⊆ Sˆ (Hatfield and Milgrom, 2005).5 That |Chc (S)| is, c demands weakly more students when the set of available students grows. Definition 1. Given a finite set Tc = {u, t1 , . . . , tn } where n is an nonnegative integer, mapping Pn tτc : S → Tc and an integer-valued vector qc = 1 n (qc ; qc , . . . , qc ) with t=1 qc ≥ qc , the preference relation c of college c ∈ C is separable with affirmative action constraints with q c (q c -separable) if 4

We abuse notation and denote a singleton set {x} by x whenever there is no concern for confusion. 5 The condition is also called cardinal monotonicity and size monotonicity by Alkan (2002) and Alkan and Gale (2003), respectively.

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(1) if τc (s) = u, then S˜ \ s c S˜ ∪ s for any S˜ ⊆ S. ˜ < qc and |S˜ct | < qct , then S˜ ∪ s c S˜ \ s, (2) If τc (s) = t 6= u, |S| ˜ > qc or |S˜t | > q t for some t ∈ Tc , then ∅ c S, ˜ (3) If |S| c c ˜ τc (s) = t} for any S˜ ⊆ S. where S˜ct ≡ {s ∈ S; The literature has extensively studied the following special cases. Preference relation c of college c ∈ C is responsive with affirmative action constraints with q c , or q c -responsive for short (Abdulkadiro˘glu and S¨onmez, 2003; Abdulkadiro˘glu, 2005) if it is q c -separable and in addition satisfies: ˜ < qc , |S˜t | < q t and (4) If τc (s) = t 6= u, τc (s0 ) = t0 6= u, |S| c c t0 t0 0 0 ˜ ˜ ˜ |Sc | < qc , then S ∪ s \ s c S ∪ s \ s if and only if s c s0 . Another interesting class of preferences is the following. Given qc ∈ N, the preference relation c of college c ∈ C is separable with quota qc , or qc -separable for short (Martinez, Masso, Neme, and Oviedo, 2000) if it is q c separable for q c = (qc ; qc , . . . , qc ) with some integer qc . That is, for separable preferences with quotas, no type-specific quota is binding and only the total quota is at work. Finally, the preference relation c of college c ∈ C is responsive with quota qc , or qc -responsive for short (Roth, 1985) if it is q c -responsive for q c = (qc ; qc , . . . , qc ) with some integer qc . The relationship of q c -separable preferences with these preferences is as follows. If c is qc -separable, then c is q c -separable: if c is q c -responsive, then c is q c -separable and substitutable: if c is q c -separable, then c satisfies the law of aggregate demand. Finally, each of these conditions is implied by qc -responsiveness. It is easy to see that these inclusion relationships are strict. See Figure 1 for a graphical illustration of these relationships. Gale and Shapley (1962) offer the following student optimal stable mechanism (SOSM).6 • Step 1: Each student applies to her first choice college. Each college holds the group of students most preferred among the applicants and rejects others (so students not rejected at this step may be rejected in later steps.) In general, 6 SOSM is known to produce a stable matching that is unanimously most preferred by every student among all stable matchings.

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Separable with affirmative action constraints

Substitutable

Separable Law of Aggregate Demand

Responsive

Responsive with affirmative action constraints

Figure 1: Inclusion relationships among classes of preferences • Step t: Each student who was rejected in Step (t-1) applies to her next highest choice. Each college considers these students and students who are temporarily held from the previous step together, and holds the group of students most preferred among the applicants and rejects others (so students not rejected at this step may be rejected in later steps.) The algorithm terminates at a step in which no more application is made. The student-optimal stable matching is defined as the tentative matching that is produced at that step. Gale and Shapley (1962) show that, when all colleges have responsive preferences, the algorithm always terminates in a finite number of steps and the resulting matching is stable. Kelso and Crawford (1982) generalize this result to the case in which colleges have substitutable preferences.

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3

Result

As pointed out in the Introduction, the regional distribution of doctors is a topic of hot debate in health care. Residents in rural areas as well as policy makers often show concern that centralized matching mechanisms may keep graduating medical school students from coming to hospitals in rural areas. Is there any possibility that one could allocate doctors in a desired manner by selecting a mechanism judiciously? In simple markets with responsive preferences, the classical “rural hospital theorem” by Roth (1986) answers this question in the negative. Formally, Result 1 (Roth (1986)). Suppose college preferences are responsive with quotas. Then, for any stable matchings µ and µ ˜, |µ(i)| = |˜ µ(i)| for every i ∈ S ∪ C. Moreover, for any c ∈ C, if |µ(c)| < qc , then µ(c) = µ ˜(c). The first conclusion shows that a student who is not matched to a college in one stable matching is not assigned in any other stable matching, and the number of positions a college fills is the same across different stable matchings. The second conclusion is even more striking: if a college has vacant positions in one of the stable matchings, then it admits exactly the same set of students in every stable matching. Martinez, Masso, Neme, and Oviedo (2000) generalize the theorem for substitutable and q-separable preferences. Does this conclusion hold in more complex markets? It turns out that the first half of the theorem holds quite generally, as presented below. Result 2 (Hatfield and Milgrom (2005)). Suppose that college preferences are substitutable and satisfy the law of aggregate demand. Suppose that µ and µ ˜ are stable matchings. Then, |µ(i)| = |˜ µ(i)| for any i ∈ S ∪ C. This version of the rural hospital theorem was formulated by McVitie and Wilson (1970), Gale and Sotomayor (1985a), Gale and Sotomayor (1985b) and Roth (1984). Hatfield and Kojima (2010) further generalize the result by Hatfield and Milgrom (2005) in the context of matching with contracts. However, the above result is silent about the second conclusion of the rural hospital theorem. That is, this result does not tell us about the exact assignments for colleges with vacant positions. Indeed, the notion of a “vacant position” is not well-defined in general for substitutable preferences with the law of aggregate demand, since there is no notion of quotas. This naturally leads us to restrict our preferences to separable preferences with affirmative action constraints, which is the broadest class of preferences with quotas to our knowledge. 7

One might expect the following is true: if the total quota of college c is not filled in a stable matching µ (i.e. |µ(c)| < qc ), then c is matched to the same group of students in every stable matching. The following example shows that this guess is incorrect. Example 1. Let C = {c1 , c2 }, S = {s1 , s2 , s3 }. Suppose that c1 has responsive preferences with affirmative action constraints: Tc1 = {u, M, m} (unacceptable, majority and minority), τc1 (s1 ) = M, τc1 (s2 ) = τc1 (s3 ) = m, (qc1 ; qcM1 , qcm1 ) = (2, 1, 1).

c1 :s1 , s3 , s2

Moreover, suppose that c2 has responsive preferences given by c2 :s1 , s2 , s3

qc2 = 2.

Students preferences are given by s1 :c2 , s2 :c1 , c2 , s3 :c2 , c1 . There are two stable matchings µ and µ ˜ given by µ(c1 ) = s2 µ(c2 ) = {s1 , s3 }, and µ ˜(c1 ) = s3 µ ˜(c2 ) = {s1 , s2 }, respectively. College c1 is matched to different students under µ and µ ˜, even though the total quota of qc is not filled in either matching. The above example suggests that a straightforward extension of the rural hospital theorem may not hold under separable preferences with affirmative action constraints. However, the next theorem shows that it is possible to obtain a generalization, provided that we define the concept of “vacant positions” appropriately. To state the result, for any c ∈ C and t ∈ Tc , let µt (c) = {s ∈ µ(c); τc (s) = t} denote the set of students of type t who are matched to c under matching µ. Theorem 1. Suppose that college preferences are substitutable and separable with affirmative action constraints. Suppose that |µ(c)| < qc and |µt (c)| < qct for some stable matching µ, c ∈ C and t ∈ Tc \ {u}. Then µ ˜t (c) = µt (c) for any stable matching µ ˜. 8

Proof. We use the following result, which is an immediate consequence of Theorems 1 and 3 of Hatfield and Milgrom (2005). Result 3 (Hatfield and Milgrom (2005)). Suppose that college preferences are substitutable and satisfy the law of aggregate demand. Let µ ¯ be the studentoptimal stable matching. Suppose that µ is a stable matching. Then, for any ˆ and µ ¯ c ∈ C, there exist Sˆ ⊇ S¯ such that µ(c) = Chc (S) ¯(c) = Chc (S). Let µ ¯ be the student-optimal stable matching. Since colleges’ separable preferences with affirmative action constraints and substitutability satisfy the law of aggregate demand, |¯ µ(c)| = |µ(c)| < qc by assumption |µ(c)| < qc and Result 2. Let Sˆ and S¯ be sets of students corresponding to µ and µ ¯ in Result 3. ¯ we have Sˆct ⊇ S¯ct . Now we prove that Sˆct = S¯ct . Suppose, Since Sˆ ⊇ S, on the contrary, that Sˆct ) S¯ct . Then, since |µ(c)| < qc , |µt (c)| < qct and c ˆ ¯ ≤ |S¯t | < |Sˆt | = |Cht (S)|. has q c -separable preferences, we have |Chtc (S)| c c c ˆ ¯ Also, since |¯ µ(c)| < qc (the total quota is not binding) and S ⊇ S, we have 0 0 0 0 t0 ¯ ˆ for every t0 ∈ Tc . These |Chc (S)| ≤ |S¯ct | ≤ min{qct , |Sˆct |} = |Chtc (S)| inequalities imply that |¯ µ(c)| < |µ(c)|, which is a contradiction to Result 2. t ˆ Therefore we have Sc = S¯ct . Since |¯ µ(c)| < qc (the total quota is not binding) and c has q c -separable preferences, µ ¯t (c) = S¯ct = Sˆct = µt (c). t t The equality µ ¯ (c) = µ ˜ (c) can be shown in an analogous manner. Combining these two equalities, we obtain µ ˜t (c) = µt (c). Theorem 1 demonstrates that match organizers’ attempts to modify stable mechanisms will not change students who are matched to a hospital’s positions, if neither the total quota nor the quota specific to a type is binding. The policy implication of the original rural hospital theorem carries over in this specific sense. Responsive preferences and q-separable preferences are special cases of q-separable preferences. Therefore the rural hospital theorems under these classes of preferences, shown by Roth (1986) and Martinez, Masso, Neme, and Oviedo (2000) respectively, are subsumed by Theorem 1. The proof of this result appears to be new. The proof is greatly simplified by utilizing Result 3. It shows that, when comparing an arbitrary stable matching µ and the student-optimal stable matching µ ¯, there is an order not only with respect to preferences over the final matchings, but also with respect to hypothetical partners who are “available” to the college.

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4

Discussion and Conclusion

This paper introduced a general class of preferences with affirmative action constraints. We demonstrated that the celebrated rural hospital theorem generalizes to this class of preferences, though only with an appropriate definition of “vacant positions.” This result implies robustness of the following insight from studies in matching: The match organizers cannot affect geographical distributions of doctors by design of matching mechanisms as long as stability is required as a constraint. Given the above implication, match organizers who want to achieve certain doctor distributions should resort to a different – and arguably more drastic – approach. The Japanese government took such an approach, based on the idea of “regional caps.” This policy tries to assign more doctors in rural areas by giving up stability (and, in fact, Pareto efficiency) of the matching. The following description is from Kamada and Kojima (2011). The shortage of residents in rural hospitals has recently become a hot political issue in Japan, where the deferred acceptance algorithm (Gale and Shapley, 1962) has placed around 8,000 graduating medical students to about 1,500 residency programs each year since 2003. In an attempt to increase the placement of residents to rural hospitals, the Japanese government recently introduced “regional caps” which, for each of the 47 prefectures that partition the country, restrict the total number of residents matched within the prefecture. The government modified the deferred acceptance algorithm incorporating the regional caps beginning in 2009 in an effort to attain its distributional goal. Motivated by this policy change, Kamada and Kojima (2011) study the design of matching markets under constraints on the doctor distribution. The paper shows that the current Japanese mechanism may result in avoidable instability and inefficiency despite its resemblance to the deferred acceptance algorithm. They then propose an alternative mechanism that overcomes these shortcomings while respecting the regional caps. More specifically, they first introduce concepts of stability and (constrained) efficiency that take regional caps into account. They point out that the current Japanese mechanism does not always produce a stable or efficient matching. Kamada and Kojima (2011) call their new mechanism the flexible deferred acceptance mechanism. The mechanism finds a stable and efficient matching and is (group) strategy-proof for doctors. That means that telling the truth is a dominant strategy for each doctor (and even a coalition of doctors

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cannot jointly misreport preferences and benefit). The flexible deferred acceptance mechanism matches weakly more doctors to hospitals (in the sense of set inclusion) and makes every doctor weakly better off than the existing mechanism in Japan. These results suggest that replacing the current mechanism with the flexible deferred acceptance mechanism will improve the performance of the matching market. As their study suggests, the problem of the doctor distribution across regions is a potentially interesting market design issue to be studied using game theoretic tools. In the recent research agenda of matching and market design (advocated by Roth (2002)), great emphasis is placed on addressing practical issues arising in real allocation problems. The geographic distribution of doctors naturally falls into such a class of issues. However, we believe that more traditional results often serve as a useful starting point of market design. The rural hospital theorem, for instance, tells us that the problem cannot be addressed simply by designing a stable matching mechanism under a given environment, and thus that some alternative approaches are needed. We hope that the generalized rural hospital theorem in this paper proves useful not only as an intellectual curiosity, but also as a building block for practical market design.

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