Efficient resource allocation under multi-unit demand

Games and Economic Behavior 82 (2013) 1–14

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Games and Economic Behavior www.elsevier.com/locate/geb

Efficient resource allocation under multi-unit demand Fuhito Kojima 1 Department of Economics, Stanford University, United States

a r t i c l e

i n f o

Article history: Received 14 March 2012 Available online 25 June 2013 JEL classification: C71 C78 D71 D78 J44 Keywords: Matching Market design Stability Strategy-proofness Efficiency Priority Acyclicity Essential homogeneity

a b s t r a c t We study resource allocation with multi-unit demand, such as the allocation of courses to students. In contrast to the case of single-unit demand, no stable mechanism, not even the (student-proposing) deferred acceptance algorithm, achieves desirable properties: it is not strategy-proof and the resulting allocation is not even weakly efficient under submitted preferences. We characterize the priority structure of courses over students under which stability is consistent with strategy-proofness or efficiency. We show that stability is compatible with strategy-proofness or efficiency if and only if the priority structure is essentially homogeneous. This result suggests that efficient allocation under multi-unit demand is difficult and that the use of stable mechanisms may not deliver desirable outcomes. © 2013 Published by Elsevier Inc.

1. Introduction Suppose that a mechanism designer needs to allocate indivisible goods to a set of agents. Real-life examples include housing allocation in universities and student placement in public schools.2 The goal of the mechanism designer is to assign the goods in an efficient and fair way while eliciting the true preferences of the agents. In many of these problems each good is equipped with a priority ordering over agents. An allocation is stable if any good that an agent prefers to her assignment is allocated (up to its supply) to others who are granted higher priority for it. Stability is a fairness concept in the sense that there is no justified envy. In situations where each agent receives only one good, stable mechanisms have been used in practice. By contrast, stable mechanisms are rarely used in resource allocation problems with multi-unit demand, in which each individual may demand more than one good.3 Drafts in professional sports (in which teams take turns to select several

E-mail address: [email protected]. I am grateful to Ramazan Bora, Eric Budish, John William Hatfield, Taro Kumano, Yusuke Narita, Yuki Takagi, and especially the Associate Editor and a referee for comments. I also thank Jin Chen, Pranathi Gummadi, Seung Hoon Lee, Bobak Pakzad-Hurson, Neil Prasad, Pete Troyan, and especially Fanqi Shi for excellent research assistance. 2 See Abdulkadiro˘glu and Sönmez (1999) and Chen and Sönmez (2002) for application to house allocation, and Balinski and Sönmez (1999) and Abdulkadiro˘glu and Sönmez (2003) for student placement. 3 In fact, we are unaware of the use of a stable mechanism with multi-unit demand in real-life resource allocation. One real example close to it is medical matching in the U.K. (Roth, 1991) but, as mentioned below, even this is not a perfect match because the problem is two-sided. 1

0899-8256/$ – see front matter © 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.geb.2013.06.005

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players) are typical examples of environments with multi-unit demand. Another example that has recently attracted the attention of researchers is course allocation in educational institutions such as business schools.4 A pioneering work by Sönmez and Ünver (2010) points out that there are deficiencies in course allocation mechanisms that are currently used at business schools. They propose instead a stable mechanism building on the deferred acceptance mechanism (DA henceforth) of Gale and Shapley (1962). In field and laboratory experiments, their mechanism achieves a significant efficiency gain over existing course-bidding mechanisms (Krishna and Ünver, 2008). However, no stable mechanism, including DA, is (even weakly) efficient or strategy-proof under multi-unit demand.5 While a disappointing result, the fact that stable mechanisms violate both efficiency and strategy-proofness only means that these properties are not guaranteed for each possible priority structure. The current paper thus investigates conditions under which stability is compatible with efficiency and strategy-proofness. More specifically, we provide a characterization of priority structures such that these desiderata hold. Our main result is that there exists an efficient or strategy-proof stable mechanism if and only if the priority structure satisfies “essential homogeneity” in the following sense: all goods have the same priority order over students, except possibly for differences among the very top students who are guaranteed the good if they apply. Thus this result implies that not only is it impossible to guarantee efficiency or strategy-proofness in general, but also that these properties never hold except for the trivial case with almost no variation in priority orders across schools. Moreover, we show that under such a priority structure, a stable mechanism (which is unique) coincides with a multi-unit serial dictatorship with a certain order, in which the first agent receives her most preferred goods up to her demand, the second agent receives his most preferred remaining goods up to his demand, and so forth. Given that serial dictatorships are seriously deficient from the viewpoint of fairness,6 our result suggests that some of the justifications for stable mechanisms, especially DA, under single-unit demand such as student placement may not carry over to situations with multi-unit demand such as course allocation. Resource allocation under multi-unit demand has been the subject of several recent studies.7 Pápai (2001) shows that serial dictatorships–and rules obtained by slightly generalizing the definition–are the only mechanisms that are nonbossy, strategy-proof and efficient. Ehlers and Klaus (2003) and Hatfield (2009) provide analogous characterizations under more special classes of preferences. Since mechanisms axiomatized in these papers are quite restrictive, these results point out difficulties of assigning resources under multi-unit demand. Our paper complements these studies by showing that efficiency or strategy-proofness alone excludes any priority-based mechanism except for serial dictatorships.8 Section 2 presents the model. The main result is presented in Section 3. Section 4 provides discussion, and Section 5 concludes. The proofs are in Appendix A. 2. Model 2.1. Preliminary definitions There are finite and disjoint sets S and C of students and courses (although we frame our model in terms of students and courses, the analysis is applicable to resource allocation problems more generally). Each student s ∈ S has a strict preference relation P s over the set of subsets of C . The weak preference relation associated with P s is denoted by R s , that is, we write C  R s C  (where C  , C  ⊆ C ) if and only if either C  P s C  or C  = C  . Following a standard assumption in the matching literature, we assume that the preference relation of each student is responsive (Roth, 1985), that is, there exists a positive integer q s , called the demand of student s,9 such that: (1) for each C  ⊆ C with |C  |  q s , c ∈ C \ C  and c  ∈ C  , C  ∪ {c } \ {c  } P s C  if and only if c P s c  ,10 (2) for each C  ⊆ C with |C  |  q s and c  ∈ C  , C  P s C  \ {c  } if and only if c  P s ∅, and (3) ∅ P s C  for each C  ⊆ C with |C  | > q s .

4 Yet another example is a labor matching market for doctors in the United Kingdom (Roth, 1991). In the U.K., new doctors sought two positions for their postgraduate training, namely one medical position and one surgical position, and a stable mechanism was used in regions such as Cardiff and Edinburgh. This market is not a perfect match to the model of this paper because it is a two-sided matching market in which not only doctors but also hospital directors are strategic players. However, the analysis of efficiency and incentives for doctors studied in this paper are applicable to this market as well. 5 Sönmez and Ünver (2010) show that students report preferences truthfully under what they call the “price-taking behavior” under DA. We will discuss their analysis in more detail in Section 5. 6 Deterministic serial dictatorships are considered to be highly unfair (for instance, they violate standard fairness properties such as equal treatment of equals and envy-freeness). Although a random serial dictatorship may achieve ex ante fairness, Budish and Cantillon (2012) establish that the ex ante efficiency of a random serial dictatorship can be very low in their data of course allocation at Harvard Business School. 7 There also exists a literature on two-sided many-to-many matching. While the model differs from our object allocation setting in that agents on both sides are active players, many results in that model hold in our setting, with suitable interpretation. See Baïou and Balinski (2000), Echenique and Oviedo (2006), Klaus and Walzl (2009), Hatfield and Kominers (2010), and Hatfield et al. (2012a) for instance. 8 Results of the papers cited here are independent of ours, as there is no logical relationship between the axioms of this paper and those of theirs. 9 We sometimes say that students have multi-unit demand. This statement corresponds to assuming that the student’s demand is two or larger. The demand is often called quota in the literature, although we use the term demand in this paper. 10 We denote singleton set {x} by x when there is no confusion.

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The set of all responsive preferences for a student is denoted by P .11 In what follows, we often use notation that specifies preferences over individual courses and the demand only. For example, P s : c , c  , ∅; q s = 2 means that student s has responsive preferences with demand 2 such that s likes {c } strictly better than {c  } and {c  } strictly better than ∅. Although there may exist multiple preferences over sets of courses that are consistent with these restrictions, it turns out that the conclusions stated in this paper can be made for every responsive preference relation that satisfies the conditions imposed on individual students and the demand. Thus, in such a case, we use the above notation to denote arbitrary responsive preferences satisfying the given conditions. Each course c ∈ C is equipped with a priority (or priority order) c , which is a strict, complete, and transitive binary relation over S.12 For each c ∈ C , qc is the supply of c. We write  = (c )c ∈C and q C = (qc )c ∈C , and refer to the pair (, q C ) as a priority structure. A problem is specified by a tuple ( S , C , P , , q C ). A matching μ is a function that specifies assignment of courses to students. Formally, matching μ = (μs )s∈ S assigns a seat at courses μs ⊆ C to each student s, with seats in each course c assigned to at most qc students. We write μc = {s ∈ S | c ∈ μs } for the set of students who are assigned seats at course c. Matching μ is (Pareto) efficient if there exists no matching μ such that μs R s μs for each s ∈ S and μs P s μs for at least one s ∈ S. Matching μ is blocked by (s, c ) ∈ S × C if s ∈ / μc and either (1) c P s ∅ and |μs | < q s or (2) c P s c  for some c  ∈ μs , and   either (1) |μc | < qc or (2) s c s for some s ∈ μc . A matching μ is individually rational if μs R s C  for each s ∈ S and C  ⊆ μs . A matching μ is stable if it is individually rational and is not blocked. We refer to a tuple ( S , C , , q C ) as an environment and consider a situation where only student preferences are private information while the environment ( S , C , , q C ) is exogenously given. A mechanism is a function ϕ from P | S | to the set of matchings. It is efficient if ϕ ( P ) is an efficient matching for each P ∈ P | S | . It is stable if ϕ ( P ) is a stable matching for each P ∈ P | S | . It is strategy-proof if ϕs ( P ) R s ϕs ( P s , P −s ) for each P ∈ P | S | , s ∈ S and P s ∈ P . Observe that course priorities are fixed and publicly known, so the definition of strategy-proofness requires incentive compatibility for students only. The assumption that course priories are fixed is usually satisfied in practical resource allocation problems such as course allocation, as courses are not active agents but merely goods to be consumed (see Sönmez and Ünver, 2010). Also in the main text we assume that the demand q s of each student is private information, but in Section 4.5 we show that the conclusion of this paper is robust to different informational structures. Given P , the (student-proposing) deferred acceptance (DA) algorithm is defined as follows (see Gale and Shapley, 1962, for the original definition for the single-unit demand case; see Roth and Sotomayor, 1990, for extensions to the multi-unit demand case).

• Step 1: Each student s applies to her q s most preferred courses (if any). Each course rejects the lowest-ranking students in excess of its supply among those who applied to it, keeping the remaining students tentatively (so students not rejected at this step may be rejected in later steps). In general, for each t  2,

• Step t: Each student s who was not tentatively matched to q s courses in Step (t − 1) applies to her next highest acceptable courses up to demand if any.13 Each course considers these students and students who are tentatively held from the previous step together, and rejects the lowest-ranking students in excess of its supply, keeping the remaining students tentatively (so students not rejected at this step may be rejected in later steps).

The algorithm terminates at the first step at which no student applies to a course. Each student tentatively accepted by a course at the terminal step is allocated a seat in that course, resulting in a matching which we denote by DA( P ). The deferred acceptance mechanism (also known as the student-optimal stable mechanism), denoted DA, produces matching DA( P ) for every P ∈ P | S | . It is well known that DA is a stable mechanism (Gale and Shapley, 1962). 2.2. Problems under multi-unit demand In addition to stability, another reason that DA is considered very desirable is that it is strategy-proof if each student s has a single-unit demand, that is, q s = 1 (Dubins and Freedman, 1981; Roth, 1982). However, DA is not strategy-proof under multi-unit demand as seen in the following example. Example 1. Let S = {1, 2}, C = {a, b}, 1 a 2, 2 b 1, and qa = qb = 1. Consider student preferences P = ( P 1 , P 2 ) such that: 11 The assumption that all students have responsive preferences can be relaxed. In Section 4.4, we show that the results hold in the case in which preferences of every student are substitutable. 12 As we are primarily interested in resource allocation problems such as course assignment, we assume that every student is acceptable to every course. 13 An alternative definition of the algorithm would be to have each student apply to only one additional course at each step. All our results are unchanged under this alternative formulation because, for each input, the result of such an algorithm coincides with the one produced by the algorithm employed in this paper.

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P 1 : b, a, ∅;

q 1 = 2,

P 2 : a, b, ∅;

q 2 = 1.

At the first step of DA under preference profile P , student 1 applies to both a and b while student 2 applies to a only. Course b keeps the only applicant, student 1, while course a keeps 1 and rejects 2. Rejected from her first choice, course a, student 2 then applies to her second choice, course b. Faced with the current match, student 1, and the new applicant, student 2, course b keeps 2 while rejecting 1. Student 1 has applied to all her acceptable courses, so the algorithm terminates. Thus

DA1 ( P ) = {a},

DA2 ( P ) = {b}.

Then consider reported preferences of student 1, P 1 : b, ∅; q1 = 1. We write P  = ( P 1 , P −1 ). At the first step of DA under P  , student 1 applies to b only and student 2 applies to a only, and both of them are kept. Thus the algorithm terminates immediately, so

 

DA1 P  = {b},

 

DA2 P  = {a}.

Since DA1 ( P  ) = {b} P 1 {a} = DA1 ( P ), DA is not strategy-proof. Any stable mechanism can be profitably manipulated by a student whenever DA can be profitably manipulated by a student (see Pathak and Sönmez, 2013).14 Thus this example demonstrates that, under multi-unit demand, no stable mechanism is strategy-proof. Regarding efficiency, DA weakly Pareto dominates any other stable mechanism (Gale and Shapley, 1962). However, even under single-unit demand, DA is not necessarily efficient. Ergin (2002) provides a characterization of priority structures under which DA is efficient. Definition 1. (See Ergin, 2002.) A priority structure (, q C ) is acyclic if there exist no a, b ∈ C and i , j , k ∈ S such that

• i a j a k and k b i, and • there exist S a , S b ⊂ S \ {i , j , k} such that S a ∩ S b = ∅, | S a | = qa − 1, | S b | = qb − 1, s a j for each s ∈ S a , and s b i for each s ∈ S b . Acyclicity requires that there be only little variation in priority orders across different courses. To illustrate the basic idea of this condition, consider the special case in which each course has only one seat (that is, qc = 1 for each c ∈ C ). Then, the second condition of the definition is vacuously met by S a = S b = ∅. Thus in this case, acyclicity requires that there be no courses a and b and students i, j and k such that course a orders i before j and j before k while course b orders k before i. As shown by Theorem 2 of Ergin, 2002, this is equivalent to the property that the priority ranking of each student varies at most by one across all courses.15 The definition of acyclicity is more complicated in the general case, but the basic idea is unchanged and the concept limits the admissible variation in priorities. Acyclicity plays an essential role for efficiency under single-unit demand. Specifically, Ergin (2002) shows that DA is efficient if and only if the priority structure is acyclic. Although acyclicity is a strong requirement, some variation in priorities is allowed across different courses. Thus there are some, though limited, cases in which DA is efficient. Under multi-unit demand, however, inefficiency problems are more serious. First, the student-optimal stable matching is not necessarily efficient even if the priority structure is acyclic. To see this point, note that the priority structure in Example 1 is acyclic, but DA1 ( P  ) P 1 DA1 ( P ) and DA2 ( P  ) P 2 DA2 ( P ), thus DA( P ) is not efficient. Second, under single-unit demand, DA is weakly efficient for each priority structure in the sense that there is no matching strictly preferred by every student, but this conclusion also fails under multi-unit demand. To see this point, notice that in Example 1 both students 1 and 2 strictly prefer DA( P  ) to DA( P ) according to preference profile P . Given these negative conclusions under multi-unit demand, a natural question to investigate is whether a mechanism achieves desirable properties in a specific environment. More specifically, we investigate conditions on the priority structure (, q C ) under which a stable mechanism is strategy-proof or efficient. The following concept will prove to play a central role in our analysis. Definition 2. A priority structure (, q C ) is essentially homogeneous if there exist no a, b ∈ C and i , j ∈ S such that

• i a j and j b i, and • there exist S a , S b ⊂ S \ {i , j } such that | S a | = qa − 1, | S b | = qb − 1, s a j for each s ∈ S a , and s b i for each s ∈ S b . 14

This property is stated as Lemma 1 in Pathak and Sönmez (2013). This condition proved to be also necessary and sufficient for a number of non-manipulability properties of stable mechanisms (Haeringer and Klijn, 2009; Kesten, 2012; Kojima, 2011). Ehlers and Erdil (2010) generalize acyclicity to coarse priorities, while Kumano (2009) generalizes the condition to acceptant and substitutable priorities as defined by Kojima and Manea (2010a). 15

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As the name suggests, essentially homogeneous priority structures allow for almost no variation in priorities between different courses. To illustrate the basic idea, consider the special case in which each course has only one seat, that is, qc = 1 for each c ∈ C . Then the second condition of the definition regarding sets S a and S b is vacuously satisfied by S a = S b = ∅. Thus in this case, essential homogeneity requires that there be no courses a and b and students i and j such that i has a higher priority for a than j but j has a higher priority for b than i. This is equivalent to the requirement that the priority orderings are identical across all courses. Essential homogeneity in the general case, with course supplies potentially larger than one, is more complicated as can be seen in Definition 2. Still, the only variation in the priority ordering allowed involves the top qc students for course c. Such a student is admitted to course c whenever she applies to it, so how highly she is ordered within those top students does not affect the allocation. In particular, it can be easily seen that essential homogeneity implies acyclicity but not vice versa. Recent literature has defined conditions analogous to essential homogeneity: (strong) X-acyclicity by Haeringer and Klijn (2009) and virtual homogeneity due to Hatfield et al. (2012b). It is easy to see that essential homogeneity is stronger than (strong) X-acyclicity but weaker than virtual homogeneity. 3. The main result With the concepts introduced in the last section, we present our main theorem. This result offers a characterization of priority structures under which the stability is compatible with properties such as efficiency and strategy-proofness. Theorem 1. For environment ( S , C , , q C ), the following conditions are equivalent. (1) There exists a stable and efficient mechanism. (2) There exists a stable and strategy-proof mechanism. (3) The priority structure (, q C ) is essentially homogeneous. Proof. Appendix A.1.

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It is well known that DA weakly Pareto dominates any other stable mechanism (Gale and Shapley, 1962). Also any stable mechanism can be profitably manipulated by a student whenever DA can be profitably manipulated by a student (see Pathak and Sönmez, 2013). Thus this theorem is equivalent to the statement that DA is efficient if and only if it is strategy-proof, and each of these properties holds if and only if the priority structure is essentially homogeneous. While the proof of Theorem 1 is somewhat involved and thus relegated to Appendix A, part of its intuition can be obtained by reviewing DA in Example 1 (our proof focuses on DA, which is without loss of generality as pointed out above). First, we illustrate how strategy-proofness fails in that example; note that the priority structure in that example violates essential homogeneity. If student 1 reports her preferences truthfully, she applies to both courses a and b as she has demand 2. Her application to a causes student 2 to be rejected from his first choice, course a. This rejection in turn induces 2 to apply to b, which rejects student 1. By contrast, if 1 reports that b is her only acceptable choice, then no one is rejected in the first step. This causes the algorithm to terminate immediately, producing a matching in which student 1 is matched to her more preferred choice, course b. Intuitively, by not applying to her acceptable but less preferred choice, course a, student 1 can eliminate a “rejection chain”, a chain reaction of applications and rejections in the algorithm that displaces 1 from her first choice, course b. The main proof of the claim that no stable mechanism is strategy-proof if the priority structure violates essential homogeneity generalizes this example, showing that one can always find a “rejection chain” similar to the above one for DA under an appropriate preference profile. The main idea behind the necessity of essential homogeneity for efficiency can be seen by Example 1 as well. In that example, a rejection chain initiated by student 1’s application to a causes more rejections, thus making students worse off. The rejection chain also hurts student 1 because it results in an additional application by student 2 to student 1’s first choice, course b, which displaces student 1. The proof for the general case proceeds similarly, by finding an appropriately chosen preference profile such that a rejection chain causes an efficiency loss. The proofs of the sufficiency of essential homogeneity for efficiency and strategy-proofness are also presented in Appendix A. While these proofs are more complicated than the necessity proofs, the basic idea is fairly intuitive. An essentially homogeneous priority implies that students are ordered by a common priority, except possibly for an irrelevant ordering among qc top students at each course c. Hence the mechanism is the multi-unit version of a serial dictatorship associated with that common priority. Indeed, Theorem 3 in Section 4.2 shows that the mechanism is a serial dictatorship, which leads to strategy-proofness and efficiency as its implications.16 However, we chose to provide the proofs of strategy-proofness and efficiency directly from essential homogeneity of the priority structure. This is because we believe that our direct proofs

16 The sketch of the proof that mechanism DA under any essentially homogeneous priority structure is a serial dictatorship is as follows. We first construct an ordering over students. Then we verify that, for each preference profile, the allocation under DA coincides with the serial dictatorship associated with the constructed order. A challenge in the proof is that the appropriate order over students is not necessarily obvious from the priority structure. This is because supplies of some courses can be larger than one: Across courses with multiple supplies, some variations in the ordering are allowed even under essential homogeneity. In fact, the ordering over students is not necessarily unique. Thus the proof can proceed by finding one, not necessarily unique,

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illuminate the roles of essential homogeneity for these properties clearly. We refer readers who prefer proofs based on a serial dictatorship to Appendix A.3. Theorem 1 suggests that inefficiency and strategic manipulation may be unavoidable unless the priority structure is essentially the same across all courses. Thus almost no room is left for the mechanism designers to achieve policy goals by a judicious choice of priority structures. Thus the practical applicability of DA, or any priority-based mechanism, seems to be severely limited in resource allocation under multi-unit demand. 4. Discussion 4.1. Characterization of essential homogeneity Let r  (c ) be the student who is -th ranked in c . That is, r  (c ) = s if and only if |{s ∈ S | s c s}| = . We now provide a characterization of essential homogeneity in terms of the ranking of students in different courses. Theorem 2. A priority structure (, q C ) is essentially homogeneous if and only if r  (c1 ) = r  (c2 ) for each c 1 , c 2 ∈ C and  > max{qc1 , qc2 }. Proof. See Appendix A.2.

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This theorem formalizes the intuition that essential homogeneity means that all courses have the same priority order over students, except possibly among the very top students. 4.2. Equivalence with serial dictatorship Let  be a strict, complete, and transitive binary relation over S. The (multi-unit demand) serial dictatorship with respect to order  is defined as follows. The first student s1 with respect to  receives her most preferred courses up to her demand q s1 , the second student s2 at  receives his most preferred courses among the remaining ones up to his demand q s2 , and we proceed similarly until the last student at  receives courses. We show that there exists a stable mechanism at (, q C ) that coincides with a serial dictatorship if and only if (, q C ) is essentially homogeneous. Theorem 3. There exists an order over students such that some stable mechanism coincides with a serial dictatorship with respect to that order if and only if (, q C ) is essentially homogeneous. Moreover, if (, q C ) is essentially homogeneous, then DA coincides with the serial dictatorship with respect to c for a course c such that qc = min{qc  | c  ∈ C }. Proof. See Appendix A.3.

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Note that part of Theorem 1 can be obtained as a corollary of this result. More specifically, Theorem 3 implies that if

(, q C ) is essentially homogeneous, then there exists a stable mechanism that is efficient and strategy-proof. 4.3. Consistency The previous section focused on efficiency and strategy-proofness as the basic properties for the stable mechanisms to satisfy. However, there may be other properties of interest. In this section we consider an axiom known as consistency (Thomson, 1988). In the single-unit demand setting, a mechanism is consistent if whenever a student is removed from the problem with her assigned course seat, the assignment for each remaining student is unchanged. Given that in our context each student demands more than one course, we also impose an analogous requirement when only “part of a student” is removed with a corresponding assignment, as formalized below. Since we consider changes in course supply in this section, we explicitly write ϕ ( P ; q C ) to denote the matching under mechanism ϕ when the student preference profile is P and the supplies of courses are given by q C . For each preference P s of s ∈ S and any c ∈ C ∪ {∅}, we denote by P s−c a preference of s such that the demand associated with P s−c is one smaller than that for P s and P s−c coincides with P s over individual courses except that c is declared unacceptable at P s−c if c = ∅. A mechanism is consistent at environment ( S , C , , q C ) if (1) for each c ∈ ϕs ( P ; q C ), ϕ−s ( P ; q C ) = ϕ−s ( P s−c , P −s ; qc − 1, q−c ) and (2) ϕ ( P ; q C ) = ϕ ( P s−∅ , P −s ; q C ) if |ϕs ( P ; q C )| < q s .

ϕs ( P ; qC ) = ϕs ( P s−c , P −s ; qc − 1, q−c ) ∪ {c },17 and

ordering over students and showing that the allocation under DA coincides with that of the serial dictatorship with that ordering. See Appendix A.3 for detail. 17 We denote by q−c the profile of supplies of all courses except for c.

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Note that this definition allows for the possibility that a student only “partially exits” the problem in the sense that a student remains in the problem while reducing her demand by one and possibly taking one course seat out of the problem. Theorem 4. DA is consistent at ( S , C , , q C ) if and only if the priority structure (, q C ) is essentially homogeneous. Proof. See Appendix A.4.

2

4.4. When preferences are not responsive We have assumed that student preferences are responsive. In some applications, however, preferences could exhibit substitutability or complementarity. For instance, in course allocation students may be unable to take multiple courses offered at the same time slot or they may not want to take too many courses in one subject, resulting in substitutable preferences. A complementarity may exist between two courses as well. For instance, if a course in advanced microeconomics builds on concepts and techniques taught in a course in intermediate microeconomics, then these courses may be complements. When preferences exhibit complementarity, even the existence of a stable matching is not guaranteed (Kelso and Crawford, 1982). In fact, Sönmez and Ünver (2010) show that the class of substitutable preferences is a maximal domain of preferences for the existence of a stable matching.18 Thus the use of stable mechanisms such as DA is difficult and perhaps unrealistic in the first place, even in the absence of the efficiency or incentive issues studied in this paper. Thus we do not investigate efficiency or incentive properties under complementarity further. On the other hand, our main result can be obtained even if we allow students to have and express substitutable preferences that are not responsive. In this sense, the implication of our main result does not depend on responsiveness of student preferences. To see why our main result holds in the case of substitutable preferences, first note that a stable matching exists and can be produced by a generalization of DA if preferences of each student are substitutable (see Chapter 6 of Roth and Sotomayor, 1990). Moreover, under substitutable preferences, DA weakly Pareto dominates any other stable mechanism, and any stable mechanism can be profitably manipulated by a student whenever DA can be profitably manipulated by a student. Thus one can investigate incentive or efficiency properties of DA in a manner analogous to the case with responsive preferences. To show that our theorem generalizes, first observe that the necessity of essential homogeneity extends straightforwardly to the case with substitutability, because responsive preferences are substitutable. This implies that both efficiency and strategy-proofness are stronger requirements under substitutability, so necessity is only reinforced. Second, the sufficiency of essential homogeneity also generalizes with little modification. This is because sufficiency is proved by contradiction, and proceeds by finding a sequence of students and colleges that form a cycle. By inspection of the proof, a cycle can be constructed in the same manner as in the proof whether or not student preferences are responsive, and thus the proof still works. 4.5. When student demands are public information We have assumed that preferences of a student are private information of that student, including the demand of the student. The motivation behind this modeling decision is to allow for a full degree of freedom for student preferences within the class of responsive preferences. However, in some applications the demands of students may be specified exogenously or public information. For instance, course administrators in business schools often restrict the number of courses that each student should take in one semester. It turns out that the main result of this paper can be extended to the case in which demands for students are public information. We assume that the demand of each student is at least two.19 To see how our result extends to this case, first note that sufficiency of essential homogeneity clearly extends because any preferences that can be reported under this assumption can also be reported when a student’s demand is private information: in the former, student demands are known and thus cannot be misreported, while in the latter, students can potentially misreport their demands. Second, the necessity of essential homogeneity for both efficiency and strategy-proofness also extends. To see this point, first note that the proof for efficiency is unchanged because no argument there hinges on the informational structure. The proof of strategy-proofness is less obvious because students cannot misreport their demands in a setting where demands are publicly known. To consider this case, note that the proof proceeds by contradiction, first assuming that the priority structure is not essentially homogeneous and then finding a profitable deviation. By inspection, the reported preferences used for a successful manipulation in the proof involve a student misreporting only the preference ordering over individual courses. Thus the proof goes through even if we assume that the student’s demand is publicly known and thus cannot be reported strategically.

18 Hatfield and Kojima (2008) offer an alternative formulation of the necessary and sufficient condition and show that substitutability is necessary and sufficient for the existence of a stable matching in their sense as well. 19 This assumption is motivated by our interest in applications such as course allocation, and leads to a clean characterization. Essential homogeneity does not necessarily give a characterization without this assumption since, for instance, Ergin (2002) shows that acyclicity (which is strictly weaker than essential homogeneity) is necessary and sufficient for the existence of an efficient stable mechanism if the demand of every student is one.

8

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4.6. Weaker versions of incentive compatibility This section considers robustness of our results to restrictions on the class of manipulations in which students can engage. Preference P s is a truncation of preference P s if the associated demands for P s and P s are equal, c 1 P s c 2 implies c 1 P s c 2 for each c 1 , c 2 ∈ C , and ∅ P s c implies ∅ P s c for each c ∈ C . In words, P s is a truncation of P s if some additional courses at the bottom of P s are declared to be unacceptable under P s , but the ranking between acceptable courses, as well as the associated demand, is unchanged between P s and P s . Studies of stable matching have often focused on truncations; see Roth and Vande Vate (1991) and Roth and Rothblum (1999), for instance. A mechanism ϕ is non-manipulable via truncation if ϕs ( P ) R s ϕs ( P s , P −s ) for each s ∈ S, preference profile P , and truncation P s of preference P s . It turns out that stability is compatible with non-manipulability via truncation if and only if the priority structure is essentially homogeneous. The “if ” part is a corollary of our main result because a truncation is a special case of preference misreporting. Meanwhile, the “only if ” part is obtained by noting that the successful preference misreporting used in the proof of “(2) ⇒ (3)” of Theorem 1 uses a truncation. A preference P s is a demand misreporting of preference P s if c 1 P s c 2 if and only if c 1 P s c 2 for each c 1 , c 2 ∈ C ∪ {∅} (while the demand qs associated with P s may be different from q s ). In words, a demand misreporting only changes the number of courses that a student demands, but the ranking between courses (and the option of not taking a course, ∅) remains unchanged from the true preferences. A mechanism ϕ is non-manipulable via demand misreporting (Sönmez, 1997) if ϕs ( P ) R s ϕs ( P s , P −s ) for each s ∈ S, preference profile P , and P s that is a demand misreporting of preference P s .20 As is the case with truncations, DA is non-manipulable via demand misreporting if and only if the priority structure is essentially homogeneous. The “if ” part is a corollary of our main result because a demand misreporting is a special case of preference misreporting. Meanwhile, the “only if ” part is obtained by noting that, in the environment described in the proof of “(1) ⇒ (3)” of Theorem 1, student i can profitably manipulate by declaring that she has a demand of one. 5. Conclusion In resource allocation problems with multi-unit demand, this paper demonstrated that there exists a stable mechanism that is efficient or strategy-proof if and only if the priority structure is essentially homogeneous. Essential homogeneity is violated in most applications, so our result suggests one reason why stable mechanisms are rarely used in practical resource allocation problems under multi-unit demand unlike those under single-unit demand. This paper reinforces the view of some recent studies that designing a satisfactory mechanism under multi-unit demand can be very difficult (Budish and Cantillon, 2012). We conclude by discussing possible directions for future research. One possibility would be to study weak priorities (indifferences in priorities).21 In many real-world resource allocation problems such as school choice and course allocation, there are only coarse classes of priorities. That is, more than one student may be granted the same priority at a course. We did not consider this issue in the current study, but a similar characterization may exist as long as one focuses on deterministic mechanisms. On the other hand, random mechanisms are often employed if priorities are weak, say in the form of random tie-breaking. Studies such as Hylland and Zeckhauser (1979), Bogomolnaia and Moulin (2001), Abdulkadiro˘glu et al. (2011), and Budish and Cantillon (2012) show that randomizations over ex post efficient mechanisms can result in significant ex ante inefficiencies. Extending our analysis to random mechanisms under weak priorities would require very different modeling choices and is beyond the scope of this paper, but it would widen the applicability of the analysis. Another interesting direction of research is to design a practical mechanism in the presence of multi-unit demand. Given the impossibility of exact incentive compatibility or efficiency, achieving approximate versions of these properties may be a fruitful approach. Indeed, such an approach has been taken to study asymptotic properties in two-sided matching (Roth and Peranson, 1999; Immorlica and Mahdian, 2005; Kojima and Pathak, 2009) as well as in resource allocation (Manea, 2009; Kojima and Manea, 2010b; Che and Kojima, 2010). In the context of course allocation, Sönmez and Ünver (2010) present a model in which priorities (or “prices” in their context) are determined by “bids” in terms of bidding points (mock currency), and show that a student reports true preferences in DA under an assumption which they call the “price-taking behavior”.22 As pointed out by Krishna and Ünver (2008), it is in large markets that the assumption of price-taking behavior appears to be the most plausible, thus DA may be difficult to profitably manipulate in large markets. More recently, Budish (2011) proposes a new mechanism motivated by the Walrasian mechanism that provides approximate, rather than exact, incentive compatibility and efficiency. Hashimoto (2012) proposes mechanisms that are incentive compatible and asymptotically efficient. However, the problem of an “optimal mechanism” is yet to be analyzed. More generally, finding (or perhaps even appropriately defining) a “satisfactory” allocation mechanism under multi-unit demand is an interesting open question.

20

The original definition by Sönmez (1997) uses the term non-manipulability via capacities. See Abdulkadiro˘glu et al. (2009) and Erdil and Ergin (2008) for issues related to weak priorities in the school choice context. 22 The concept of price-taking behavior assumes that a student fails to recognize the full influence of her own reporting behavior on prices, so it does not contradict the fact that DA is not strategy-proof. 21

F. Kojima / Games and Economic Behavior 82 (2013) 1–14

9

Appendix A A.1. Proof of Theorem 1 It is well known that DA weakly Pareto dominates any other stable mechanism (Gale and Shapley, 1962). Also any stable mechanism can be profitably manipulated by a student whenever DA can be profitably manipulated by a student (see Pathak and Sönmez, 2013). Thus this theorem is equivalent to the following claim. Claim 1. For environment ( S , C , , q C ), the following conditions are equivalent. (1) DA is efficient. (2) DA is strategy-proof. (3) The priority structure (, q C ) is essentially homogeneous. We will show (1) ⇒ (3), (3) ⇒ (1), (2) ⇒ (3), and (3) ⇒ (2) in the statement of the above claim in this order. Proof of “(1) ⇒ (3)”. We show the claim by contraposition. Suppose that the priority structure is not essentially homogeneous. Then, by definition, there exist a, b ∈ C , i , j ∈ S such that

• i a j and j b i, and • there exist S a , S b ⊂ S \ {i , j } such that | S a | = qa − 1, | S b | = qb − 1, s a j for each s ∈ S a , and s b i for each s ∈ S b . Consider the following preferences:

P i : b, a, ∅;

q i = 2,

P j : a, b, ∅;

q j = 1,

P s : a, ∅;

q s = 1 for each s ∈ S a \ S b ,

P s : b, ∅;

q s = 1 for each s ∈ S b \ S a ,

P s : a, b, ∅; P s : ∅;

qs = 2

for each s ∈ S a ∩ S b ,





q s = 1 for each s ∈ S \ {i , j } ∪ S a ∪ S b .

It is easy to see that

⎧ {a}, ⎪ ⎪ ⎪ ⎪ b}, { ⎪ ⎪ ⎪ ⎨ {a}, DAs ( P ) = ⎪ {b}, ⎪ ⎪ ⎪ ⎪ {a, b}, ⎪ ⎪ ⎩ ∅,

Let

⎧ {b}, ⎪ ⎪ ⎪ ⎪ {a}, ⎪ ⎪ ⎪ ⎨ {a}, μs = ⎪ {b}, ⎪ ⎪ ⎪ ⎪ ⎪ {a, b}, ⎪ ⎩ ∅,

Note that

s = i, s = j, s ∈ Sa \ Sb , s ∈ Sb \ Sa ,

(1)

s ∈ Sa ∩ Sb , s ∈ S \ [{i , j } ∪ S a ∪ S b ].

s = i, s = j, s ∈ Sa \ Sb , s ∈ Sb \ Sa,

(2)

s ∈ Sa ∩ Sb , s ∈ S \ [{i , j } ∪ S a ∪ S b ].

μs R s DAs ( P ) for each s ∈ S. Moreover, μi = {b} P i {a} = DAi ( P ). Thus DA( P ) is not efficient. 2

Proof of “(3) ⇒ (1)”. We show the claim by contradiction. To this end, suppose that (, q C ) is essentially homogeneous but DA is not efficient. Then there exist preference profile P and matching μ such that μs R s DAs ( P ) for each s ∈ S and μs P s DAs ( P ) for at least one s ∈ S. Let s be a student such that μs P s DAs ( P ). This implies that there exists c  ∈ C such that s ∈ μc  \ DAc  ( P ) and either  c P s ∅ and |DAs ( P )| < q s or c  P s c for some c ∈ (DAs ( P ) \ μs ). Because DA( P ) is a stable matching by assumption, this implies that |DAc  ( P )| = qc  and s c  s for each s ∈ DAc  ( P ) (otherwise (s , c  ) is a blocking pair, which is a contradiction to the stability of DA( P )). Since |DAc  ( P )| = qc  while s ∈ μc  \ DAc  ( P ), there exists s ∈ DAc  ( P ) \ μc  . This in particular implies

10

F. Kojima / Games and Economic Behavior 82 (2013) 1–14

μs = DAs ( P ). This fact and μs R s DAs ( P ) (which holds by the assumption that μs R s DAs ( P ) for each s ∈ S) imply that there exists c  ∈ C such that s ∈ μc  \ DAc  ( P ) and c  P s c for some c ∈ {∅} ∪ DAs ( P ) \ μs . Applying this argument repeatedly (and noting that the number of students is finite), we conclude that there exist finite sequences of students s1 , s2 , . . . , sn−1 , sn and courses c 1 , c 2 , . . . , cn−1 , cn such that st ∈ DAct −1 ( P ) \ DAct ( P ),



DAc ( P ) = qc ,

(3)

s ct st

(5)

t

(4)

t

for each s ∈ DAct ( P ),



either ct P st ∅ and DAst ( P ) < q st

or ct P st c for some c ∈ DAst ( P ),

(6)

for each t = 1, 2, . . . , n − 1, n (where indices are defined modulo n, that is, 0 and n are understood to be identical). In particular, relations (3) and (5) imply

st +1 ct st

for each t = 1, 2, . . . , n − 1, n.

(7)

The following claim plays a key role in the remainder of the proof. Claim 2. For each t = 1, 2, . . . , n − 1, n, we have st +1 cn st . Proof. We show this claim by induction. Consider first the case t = n as a base step. In this case, the conclusion holds by relation (7). Then, as an inductive step, we assume that the conclusion of the claim holds for n, n − 1, . . . , t + 1 (where t < n) and show that it holds for t. Suppose, for a contradiction, that

st cn st +1 .

(8)

First note that

st +1 ∈ / DAcn ( P ),

(9)

because otherwise the inductive assumption sn cn st +1 and the properties (3) and (6) imply that (sn , cn ) blocks DA( P ), contradicting the stability of DA. Define

S ct = DAct ( P ) \ {st +1 },



S cn =

DAcn ( P ) \ {st } if st ∈ DAcn ( P ), DAcn ( P ) \ {s1 } otherwise.

Then, by construction and properties (3) and (9),

S ct , S cn ⊆ S \ {st , st +1 },

(10)

and, by relations (3) and (4),

| S c t | = q c t − 1,

| S c n | = q c n − 1.

(11)

Moreover, by (5) and S ct ⊆ DAct ( P ) (the latter property follows from the definition of S ct ),

s ct st

for each s ∈ S ct .

(12)

Further, because (i) s cn sn by relation (5), and (ii) sn cn sn−1 cn · · · cn st +1 by the inductive assumption,

s cn st +1

for each s ∈ S cn .

(13)

Relations (7), (8), (10), (11), (12), and (13) contradict the assumption that the priority structure is essentially homogeneous. 2 To finish the proof, simply observe that Claim 2 implies

s1 cn sn cn · · · cn s2 cn s1 . This is a contradiction.

2

Proof of “(2) ⇒ (3)”. We show the claim by contraposition. Suppose that the priority structure is not essentially homogeneous. Consider the environment and preference profile P in the proof of “(1) ⇒ (3)”. From that proof, DA( P ) is given by

F. Kojima / Games and Economic Behavior 82 (2013) 1–14

11

Eq. (1). In particular, DAi ( P ) = {a}. Now consider a false preference of student i, P i : b, ∅; qi = 1. We write P  = ( P i , P −i ). Then it is easy to show that DA( P  ) = μ as defined by (2). In particular, DAi ( P  ) = {b}. Since DAi ( P  ) = {b} P i {a} = DAi ( P ), we conclude that DA is not strategy-proof. 2 Proof of “(3) ⇒ (2)”. We show the claim by contradiction. To this end, suppose that (, q C ) is essentially homogeneous but DA is not strategy-proof. Then there exist preference profile P , student s ∈ S and her reported preference P s  such that μs P s DAs ( P ), where we denote μ := DA( P s  , P −s ). This implies that there exists c  ∈ C such that s ∈ μc \ DAc ( P ) and either c  P s ∅ and |DAs ( P )| < q s or c  P s c for some c ∈ (DAs ( P ) \ μs ). Because DA( P ) is a stable matching by assumption, this implies that |DAc  ( P )| = qc  and s c  s for each s ∈ DAc  ( P ) (otherwise (s , c  ) is a blocking pair, which is a contradiction / μc , to stability of DA( P )). Since |DAc  ( P )| = qc  while s ∈ μc  \ DAc  ( P ), there exists s ∈ DAc  ( P ) \ μc  . Since s c  s but s ∈ by stability of μ under ( P s  , P −s ) there exists c  ∈ C such that s ∈ μc  \ DAc  ( P ) and c  P s c  . Since the number of students is finite, applying this argument repeatedly we conclude that there exist sequences of students s1 , s2 , . . . , sn−1 , sn and courses c 1 , c 2 , . . . , cn−1 , cn such that

st ∈ DAct −1 ( P ) \ DAct ( P ),



DAc ( P ) = qc , t

s ct st

t

for each s ∈ DAct ( P ),





either ct P st ∅ and DAst ( P ) < q st

or ct P st c for some c ∈ DAst ( P ),

for each t = 1, 2, . . . , n − 1, n (where indices are defined modulo n, that is, 0 and n are understood to be equivalent). These conditions are identical to relations (3)–(6) in the proof of “(3) ⇒ (1)”. Thus, following the same argument as in that proof, we can conclude that the priority structure is not essentially homogeneous. This contradiction completes the proof. 2 A.2. Proof of Theorem 2 Proof of the “if ” direction. Let ρc (s) be the ranking of student s in c . That is, ρc (s) =  if and only if r  (c ) = s. Suppose that (, q C ) satisfies the condition at the end of the statement of the theorem and a, b ∈ C and i , j ∈ S satisfy

i a j ,

j b i ,

(14)

or equivalently ρa (i ) < ρa ( j ) and ρb ( j ) < ρb (i ). Suppose that ρb (i )  ρa ( j ) without loss of generality. Then, since ρa (i ) < ρa ( j )  ρb (i ) by assumption (14), the condition in the statement of the theorem implies that ρb (i )  max{qa , qb }. Consider the following cases. (1) Suppose qb = max{qa , qb }. Then, since ρb ( j ) < ρb (i )  qb , there does not exist S b ⊆ S \ {i , j } such that | S b | = qb − 1 and s b i for each s ∈ S b . (2) Suppose qb = max{qa , qb }. Then qa = max{qa , qb }. Then, since ρa (i ) < ρa ( j )  ρb (i )  max{qa , qb } = qa , there does not exist S a ⊆ S \ {i , j } such that | S a | = qa − 1 and s a j for each s ∈ S a . The above arguments establish essential homogeneity of (, q C ).

2

Proof of the “only if ” direction. We shall prove the contraposition. Thus assume that (, q C ) does not satisfy the condition at the end of the statement of the theorem. This assumption implies that there exist a, b ∈ C such that λ > max{qa , qb } where λ = max{ ∈ N: r  (a ) = r  (b )}. Denote i = r λ (b ) and j = r λ (a ). By maximality of λ, it follows that i a j and j b i. Moreover, since λ > max{qa , qb }, there exist S a , S b ⊆ S \ {i , j } such that | S a | = qa − 1, | S b | = qb − 1, s a j for each s ∈ S a , and s b i for each s ∈ S b . 2 A.3. Proof of Theorem 3 The “only if ” part of the first statement directly follows from Theorem 1. This is because a serial dictatorship at any ordering is known to be efficient and strategy-proof, while by Theorem 1, if the priority structure is not essentially homogeneous, then no stable mechanism is efficient or strategy-proof. Therefore, it suffices to show that when (, q C ) is essentially homogeneous, DA coincides with the serial dictatorship with respect to the agent ordering given by c for a course c such that qc = min{qc  | c  ∈ C }. We will show the result by proving the following claim: at each step t of the serial dictatorship at c , i.e., at the step in which the student st who is ranked as the t-th most preferred according to c chooses courses, if all the seats of a course c  have already been taken by Step (t − 1), say by students s(1) , . . . , s(qc ) , then s(1) c  st , . . . , s(qc ) c  st . This claim implies

12

F. Kojima / Games and Economic Behavior 82 (2013) 1–14

our result, because a version of DA in which students apply one at a time23 according to the order c coincides with the serial dictatorship with respect to c .24 The proof that s(1) c  st , . . . , s(qc ) c  st goes as follows. First note that s(1) c st , . . . , s(qc ) c st by the assumption that s(1) , . . . , s(qc ) have chosen their courses before st in the serial dictatorship with respect to c . This fact implies that ρc (st ) > qc .25 Since qc = min{qc | c  ∈ C }, it follows that ρc (st ) > max{qc , qc }. Because (, qC ) is essentially homogeneous, by Theorem 2 it follows that

ρc (st ) = ρc (st ).

(15)

Now suppose for contradiction that st c  s(k) for some k ∈ {1, . . . , qc  }. Then so by essential homogeneity of (, q C ) and Theorem 2, we obtain







ρc (s(k) ) > ρc (st ) = ρc (st ) > max{qc , qc },



ρc s(k) = ρc s(k) .

(16)

Eqs. (15) and (16) and the assumption st c  s(k) imply st c s(k) , which is a contradiction. A.4. Proof of Theorem 4 Proof of the “if ” direction. We show the claim by contraposition. Suppose that DA is not consistent. Then there exists a preference profile P such that either (1) there exists a course c ∈ DAs ( P ; q C ) such that DA−s ( P ; q C ) = DA−s ( P s−c , P −s ; qc − 1, q−c ) or DAs ( P ; q C ) = DAs ( P s−c , P −s ; qc − 1, q−c ) ∪ {c }, or (2) DA( P ; q C ) = DA( P s−∅ , P −s ; q C ) and |DAs ( P ; q C )| < q s . We consider case (1) only, because the proof for case (2) is analogous. By construction of preference P s−c , the matching μ defined by μ−s = DA−s ( P ; q C ) and μs = DAs ( P ; q C ) \ {c } is stable at ( P s−c , P −s ; qc − 1, q−c ). By assumption we obtain μ = DA( P s−c , P −s ; qc − 1, q−c ) and, because DA is the student-optimal stable mechanism, it follows that DA( P s−c , P −s ; qc − 1, q−c ) Pareto dominates μ at ( P s−c , P −s ), that is, DAs ( P s−c , P −s ; qc − 1, q−c ) R s μs for each s = s c  and DAs ( P s−c , P −s ; qc − 1, q−c ) R − s μs , with at least one relation being strict. However, then matching μ defined by  − c  − c μ−s = DA−s ( P s , P −s ; qc − 1, q−c ) and μs = DAs ( P s , P −s ; qc − 1, q−c ) ∪ {c } Pareto dominates DA( P ; qC ) at P , which implies that DA is not efficient, a contradiction to essential homogeneity of (, q C ) and Theorem 1. 2 Proof of the “only if ” direction. We show the claim by contraposition. Suppose that the priority structure is not essentially homogeneous. Then, by definition, there exist a, b ∈ C , i , j ∈ S such that

• i a j and j b i, and • there exist S a , S b ⊂ S \ {i , j } such that | S a | = qa − 1, | S b | = qb − 1, s a j for each s ∈ S a and s b i for each s ∈ S b . Let

P i : b, a, ∅;

q i = 2,

P j : a, b, ∅;

q j = 1,

P s : a, ∅;

q s = 1 for each s ∈ S a \ S b ,

P s : b, ∅;

q s = 1 for each s ∈ S b \ S a ,

P s : a, b, ∅; P s : ∅;

q s = 2 for each s ∈ S a ∩ S b ,





q s = 1 for each s ∈ S \ {i , j } ∪ S a ∪ S b .

It is easy to see that

23

Note that it is well known that this “sequential” version of DA is equivalent to the “simultaneous” version defined by Gale and Shapley (1962). The following is a formal inductive proof of this statement. Suppose that the first (t − 1) steps of DA have coincided with the first (t − 1) steps of the serial dictatorship under c , and consider Step t of DA, at which student st applies to courses (note that the first step of DA is clearly equivalent to that of serial dictatorship). By the above claim, for each course c  that have already been tentatively matched up to supplies with some of the students s1 , . . . , st −1 , an application by student st cannot be tentatively kept. Thus at Step t of DA, student st applies to, and be tentatively matched with, the set of up to q st courses that she prefers most among those that have not filled all their supplies during Steps 1, . . . , (t − 1). This means that Step t of DA coincides with serial dictatorship under c . To finish the proof, note that DA terminates at the end of Step | S | because no student is displaced at that step, and hence the tentative matching at that step becomes final (and that finalized matching coincides with the matching under serial dictatorship with respect to c ). 25 Recall that ρc (st ) denotes the ranking of student st in c . 24

F. Kojima / Games and Economic Behavior 82 (2013) 1–14

⎧ {a}, ⎪ ⎪ ⎪ ⎪ b}, { ⎪ ⎪ ⎪ ⎨ {a}, DAs ( P ; q C ) = ⎪ {b}, ⎪ ⎪ ⎪ ⎪ {a, b}, ⎪ ⎪ ⎩ ∅,

13

s = i, s = j, s ∈ Sa \ Sb , s ∈ Sb \ Sa , s ∈ Sa ∩ Sb , s ∈ S \ [{i , j } ∪ S a ∪ S b ].

Note that |DAi ( P ; q C )| = 1 < 2 = qi . However,

⎧ {b}, ⎪ ⎪ ⎪ ⎪ a}, { ⎪ ⎪ ⎪  −∅  ⎨ {a}, DA P i , P −i ; q C = ⎪ {b}, ⎪ ⎪ ⎪ ⎪ {a, b}, ⎪ ⎪ ⎩ ∅,

s = i, s = j, s ∈ Sa \ Sb , s ∈ Sb \ Sa , s ∈ Sa ∩ Sb , s ∈ S \ [{i , j } ∪ S a ∪ S b ],

so DAi ( P ; q C ) = DAi ( P i−∅ , P −i ; q C ), which shows that DA is not consistent.

2

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