Preference Profiles for Efficiency, Fairness, and Consistency in School Choice Problems Eun Jeong Heo∗† First draft: March 15, 2011 This version: September 23, 2014

Abstract We study the problem of assigning school seats to students. Efficiency and fairness are two central properties of rules, but no rule satisfies both in general. We identify preference domains on which these properties are compatible. We first characterize the structure of preference profiles under which the top-trading cycles rule (TTC) is fair. For these profiles, TTC coincides with both the immediate acceptance rule (IA) and the deferred acceptance rule (DA). We next inquire about the relationships between (i) the domain of preference profiles at which TTC is fair, (ii) the domain of preference profiles at which IA is fair, and (iii) the domain of preference profiles at which DA is efficient. We show how these three domains are related: (i) is non-empty and a proper subset of (ii), which itself is a proper subset of (iii). We present similar results for consistency. From these inclusion relations, we conclude that DA performs better than IA, which in turn performs better than TTC. JEL classification Numbers: C71; C78; D71; D78; J44 Keywords: the top-trading cycles rule; the immediate acceptance rule; the deferred acceptance rule; fairness; efficiency; fairness; consistency

∗

Department of Economics, Vanderbilt University, Nashville TN 37235, USA. (email: [email protected]) This paper was previously circulated under the title “Efficiency and Fairness in School Choice Problem: the Maximal Domain of Preference Profiles”. Most of all, I would like to thank William Thomson for his support and guidance. I benefited from discussions with Bettina Klaus, Youngwoo Koh, Fuhito Kojima, Vikram Manjunath, Kyoungwon Seo, and Shigehiro Serizawa. I am also grateful to seminar audiences at the University of Rochester in 2012, Rice University in 2012, the University of Montreal in 2012, the Catholic University of Louvain in 2013, and the 12th meeting of the society for social choice and welfare at Boston College in 2014 for useful comments and suggestions. All remaining errors are my own responsibility. †

1. Introduction We study the “school choice problem” which has to do with assigning schools to students. Students have strict preferences over the schools and they are asked to report their preferences to a school district. Schools have capacity constraints such as certain numbers of available seats. According to state or local law, schools prioritize students on the basis of several factors, such as walk zone, sibling, or a tie-breaking rule. For each pair of schools’ priorities and students’ reported preferences, a rule determines which school is assigned to whom, subject to the capacity constraints of the schools. Efficiency and fairness are two central properties of rules in this problem. Consider a rule. Efficiency requires that there be no Pareto improvement from each selection that the rule makes. Fairness requires that the priority structure be “respected” as follows: at each selection that the rule makes, whenever a student, say student i, is assigned to a school to which he prefers another school, say school a, the students who are assigned to a should have higher priorities than he does at a. There are three best-known rules for this problem: the deferred acceptance rule (DA), the top-trading cycles rule (TTC), and the immediate acceptance rule (IA).1 Unfortunately, none of them satisfies both properties: TTC and IA are efficient but not fair, and DA is fair but not efficient (Gale and Shapley (1962), Abdulkadiroˇglu and S¨onmez (2003)). These observations generalize: no rule satisfies efficiency and fairness (Balinski and S¨onmez, 1999). Of course, there are some preferences and priorities on which these two properties are compatible (for example, identical preferences). It is natural to ask the following questions then. How often do such preferences and priorities appear? Can we identify the entire sets of such preferences and priorities? Are they plausible and interesting classes? Several recent papers have raised this type of questions, and all of them focus on priority profiles. Ergin (2002) identifies the structure of priority profiles under which DA is efficient or “consistent”2 , no matter what preferences are. Similarly, Kesten (2006) identifies the structure of priority profiles under which TTC is stable or consistent, no matter what preferences are. The structure that Kesten (2006) identifies is more restrictive than that in Ergin (2002). Kumano (2013) asks the same question for IA and identifies another set of priority profiles. Unfortunately, such priority profiles do not exist in most cases.3 We denote these domains of priority profiles by ΣA da (Ergin, 2002): the domain of priority profiles at which DA is efficient, ΣA ttc (Kesten, 2006): the domain of priority profiles at which TTC is fair, and 4 ΣA ia (Kumano, 2013): the domain of priority profiles at which IA is fair. 1

This rule is also called the “Boston” mechanism. We adopt the terminology of Thomson (2013b). Consistency is a robustness property pertaining to variable populations but none of the three rules satisfies this property. We present the formal definition in Section 3.3. For an exhaustive survey on consistency, see Thomson (2013a). 3 Precisely, such priority profiles exist only if n ≤ 2, or |A| ≤ 1, or for each pair of distinct schools a, b ∈ A, qa + qb ≥ n. These restrictions are very demanding, as shown by Kumano (2013). There are other restrictions on priority profiles under which rules satisfy various properties: see Kojima (2011), Haeringer and Klijn (2009), Ehlers and Erdil (2010), Hatfield et al. (2011), Hsu (2013), and Han (2014). 4 The structures of these priority profiles are derived from the conditions on priority-capacity pairs (Thomson, 2

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The relations between these domains are illustrated in (a) of Figure 1. It turns out that these priority restrictions are quite demanding. In practice, there are only a few criteria to prioritize students and there are large indifference classes of students (Erdil and Ehlers, 2010). To induce strict priorities, a random tie-breaking rule is often used. The resulting priority profiles are not necessarily those identified in these papers. In our paper, we turn to the other side of the problem, the preference profiles. It is true that preferences are not controlled by a school district in reality. However, it is very often the case that students’ preferences have certain structures. For example, students may have correlated preferences: they may all agree on the set of first-tier schools, the set of second-tier schools, and so on; but they may have different preferences within each tier. We thereby search for restricted domains of preference profiles guaranteeing desirable properties of the aforementioned rules. They are not only interesting by themselves, but also useful to evaluate the “strength” of the baseline impossibility result or the performance of the rules. That is, if the restrictions are very mild, then the impossibility result can be viewed as an “almost” possibility result. If the restrictions for a rule to satisfy certain properties is more demanding than those for another rule, we can say that the former performs worse than the latter. We first characterize the structure of preference profiles under which TTC is fair, no matter what priorities are (Theorem 1). Each profile in this set can be described as a mixture of identical preferences and heterogenous preferences as follows: for some non-negative integer k, all students have the same preferences from their most preferred schools down to their k-th schools, but their (k + 1)-th most preferred schools have to differ to a certain extent. For each preference profile in this set, we also obtain coincidence of these three rules, achieving both fairness and efficiency (Corollary 1). We can similarly define the domains of preference profiles for the remaining rules to be fair, no matter what priorities are.5 Altogether, let us define PN ttc : the domain of preference profiles at which TTC is fair, PN ia : the domain of preference profiles at which IA if fair, and PN da : the domain of preference profiles at which DA is efficient. Our next result shows that these domains are related (Theorem 2) and how. There are conN is a proper subset of P N , which itself is a proper subset of P N ((b) of tainment relations: Pttc ia da Figure 1). This implies that DA performs better than IA, which itself performs better than TTC.6

The inclusion relation between the first and second domains, in particular, contrasts with the results pertaining two priority profiles (Kesten (2006) and Kumano (2013)).7 2013b). Here we simply assume that the capacity constraints are fixed and keep them implicit. 5 N We also figured out the structures of the other two domains. Unlike Pttc , however, these domains can only be identified economy by economy, heavily relying on how the IA and DA algorithms work. The same argument applies to the domain of preference profiles at which the rules satisfy these properties for a given priority profile or for some restricted priority profiles. We do not include these results in this paper. 6 Such comparisons are valid especially when priority profiles are not fully controlled, since we consider all priority profiles in defining these domains. 7 A list of papers on “maximal domain” study other types of preference restrictions (Barber` a et al., 1991). Most

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ΣA da

N Pda N Pia

ΣA ttc

ΣA ia N Pttc

(a) Figure 1.

(b)

(a) The domain of priority profiles for efficiency and fairness (the area with the dotted boundary represents an empty set) (b) The domain of preference profiles for efficiency and fairness

In Section 3, we turn to consistency. As above, we identify the domain of preference profiles at which each of these rules is consistent. We show that (i) the domain of preference profiles at which TTC is consistent is the same as the domain of preference profiles at which TTC is fair and (ii) there are proper inclusion relations for these domains (Theorem 3). Again, these results stand in sharp contrast with the results pertaining two priority profiles (Kesten, 2006). Although preferences and priorities seem to play symmetric roles in the definition of a school choice problem, the implications of our results are different from the results concerning priorities for the following reasons. First, in evaluating the efficiency of an assignment, only the students’ preferences, not the schools’ priorities, are taken into account. Second, the two aforementioned rules are defined by algorithms in which the students first “propose” and the schools make (tentative or final) decisions. That is, students and schools play different roles in these algorithms. Lastly and obviously, each school may be assigned to several students, but each student is assigned to exactly one school. This paper is organized as follows: Section 2 introduces the model, axioms, and the rules. Section 3 presents our main results. Section 4 includes concluding remarks.

2. Model Let N ≡ {1, 2, · · · , n} be the set of students (assume that n ≥ 3). Let A be the set of schools (assume that |A| ≥ 3). For each a ∈ A, let qa ∈ N+ be the capacity (the number of seats) of P school a, and q ≡ (qa )a∈A the capacity profile. For each B ⊆ A, qB ≡ a∈B qa . Each a ∈ A has a of these papers focus on restrictions of individual preferences to guarantee relational properties such as “strategyproofness”. Moreover, they require that a minimally plausible subdomain be included in the resulting domain (for maximal domain result in the matching context, see Kojima (2007)). We have no such minimal subdomain and rather focus on properties of rules that do not involve students’ preference changes. Note also that we work on restrictions on “preference profiles”, not on “individual preferences”. Therefore, the domain of preference profiles we identify includes the Cartesian products of individual preference restrictions achieving the same goal.

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strict ordering over students. We call it the priority at a and denote it by ≺a . Student i has a higher priority than j at a if and only if i ≺a j. Let Σ be the set of all priorities over N . For each S ⊆ N , ≺a |S is the priority ≺a restricted to the students in S. Let ≺≡ (≺a )a∈A be the priority profile and ΣA the set of all priority profiles. P We assume that n ≤ a∈A qa and that each i ∈ N has to be assigned to exactly one school.8 Each i ∈ N has a complete, transitive, and strict preference Pi over A. Let P be the set of all such preferences. For each P0 ∈ P and each a ∈ A, let U (P0 , a) ≡ {b ∈ A : b P0 a or a = b} be the weak upper contour set of a at P0 . For each B ⊆ A, let P0 |B be the preference P0 restricted to the schools in B. Let P ≡ (Pi )i∈N be the preference profile and P N the set of all preference profiles. An economy is a list e ≡ (A, N, q, ≺, P ). Unless otherwise specified, we fix A, N , and q. Then, an economy is a pair (≺, P ). A feasible assignment, or an assignment, is a list x ≡ (xi )i∈N such that (i) for each i ∈ N , xi ∈ A and (ii) for each a ∈ A, |{i ∈ N : xi = a}| ≤ qa . Let X be the set of all assignments. A rule, ϕ, is a function that maps each economy to an assignment.

2.1. Axioms We introduce two central properties of rules. Let P ∈ P N . We say that x ∈ X is efficient at P if no other assignment makes each agent at least as well off as in x. Formally, there is no x0 ∈ X \ {x} such that for each i ∈ N , either x0i Pi xi or x0i = xi . A rule ϕ is efficient if for each ≺∈ ΣA and each P ∈ P N , ϕ(≺, P ) is efficient at P . Next is fairness. Let ≺∈ ΣA , P ∈ P N , and x ∈ X. Suppose that a student, say i, prefers another school, say school a, to his assignment xi . Then, we require that each of the students who are assigned to a should have a higher priority than him at a. If this holds for each student, we say that x is fair at (≺, P ). Formally, for each pair i, j ∈ N , xj Pi xi implies j ≺xj i. A rule ϕ is fair if for each ≺∈ ΣA and each P ∈ P N , ϕ(≺, P ) is fair at (≺, P ). Remark 1. (Balinski and S¨ onmez, 1999) No rule is efficient and fair.

2.2. Rules We introduce three rules. 2.2.1. The (student-proposing) deferred-acceptance rule For each economy, the student-proposing deferred-acceptance rule assigns schools by means of the following algorithm. Note that acceptance at each step is not final, but tentative (Gale and Shapley, 1962). Step 1 Each student applies to his most preferred school. If the number of applicants to a school, say a, does not exceed qa , then all of these students are tentatively accepted. Otherwise, those 8 It is usual that every student is required to be registered in a school by law. We can drop this assumption by introducing an “opting out” option, ∅, and setting its capacity equal to n.

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with at least qa -th priority among them are tentatively accepted: the remaining applicants are rejected. Step t(≥ 2) Each student who is rejected at Step (t − 1) applies to his next most preferred school. For each school a, if the number of students who had been tentatively accepted at Step (t − 1) and the new applicants does not exceed qa , then all of these students are tentatively accepted. Otherwise, those with at least qa -th priority among them are tentatively accepted: the remaining applicants are rejected. The algorithm terminates when no students are rejected. We call this rule the deferred acceptance rule, or DA for short. DA is fair but not efficient. However, it is “fairness-constrained efficient” in the following sense; Remark 2. (Balinski and S¨ onmez, 1999) The selection that DA makes for each (≺, P ) Pareto dominates 9 each other fair assignment at (≺, P ). Moreover, when priority profiles have a structure identified by Ergin (2002), DA is efficient, no matter what preferences are. 2.2.2. Immediate acceptance rule For each economy, the student-proposing immediate acceptance rule assigns schools to students by means of the following algorithm. Note that acceptance at each step is final. Step 1 Each student applies to his most preferred school. If the number of applicants to a school, say a, does not exceed qa , then all of these students are accepted. Otherwise, those with at least qa th priority among them are accepted; the remaining applicants are rejected. The capacity of each school decreases by the number of applicants accepted by the school at this step. Denote it by q 1 ≡ (qo1 )o∈A . Step t(≥ 2) Each student who is rejected at Step (t−1) applies to his t-th most preferred school.10 If the number of applicants to a school, say a, does not exceed qat−1 , then all of these students are accepted. Otherwise, the students with at least qat−1 -th priority among them are accepted; the remaining applicants are rejected. The capacity of each school decreases by the number of students accepted by the school at this step. Denote it by qt ≡ (qot )o∈A . The algorithm terminates when no students are rejected. We call this rule the immediate acceptance rule, or IA for short.11 IA is efficient but not fair. When priority profiles have the structure identified by Kumano (2013), however, IA is fair, We say that x Pareto dominates x0 (6= x) if for each i ∈ N , xi Pi x0i or xi = x0i . Note that according to this algorithm, students may have to apply to schools with no remaining seat at some steps. A plausible reformation of this algorithm is for them to “skip” such schools. Alcalde (1996) first proposes to do so in the context of “marriage” problems and he calls resulting rule the “now-or-never” solution. In the context of school choice problems, Harless (2014) calls it the “immediate acceptance rule with skips”. He compares its properties with those of IA. 11 This terminology is adopted from Thomson (2013b). 9

10

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no matter what preferences are. Unfortunately, such priority profiles do not exist in most cases (see Footnote 3). 2.2.3. Top-trading cycles rule For each economy, the student-proposing top-trading cycles rule assigns schools to student by means of the following algorithm.12 Step 1 Each school a points to the student with the highest priority at ≺a and each student points to his most preferred school. Since there are finite numbers of students and schools, there is at least one cycle. Each student in each cycle is assigned the school that he points to and leave with his assignment; the capacity of each school in each cycle decreases by one. Step t(≥ 2) We proceed with the schools with available seats and students who are not assigned any school by Step (t − 1). Each school a points to the student with the highest priority at ≺a among the remaining students and each student points to his most preferred school among the remaining schools. Since there are finite numbers of students and schools, there is at least one cycle. Each student in each cycle is assigned the school that he points to and leave with his assignment; the capacity of each school in each cycle decreases by one. The algorithm terminates when there is no student left. We call this rule the top trading cycles rule, or TTC for short. TTC is efficient but not fair. When priority profiles have a structure identified by Kesten (2002), however, TTC is fair, no matter what preferences are. The domain of such priority profiles is a subset of what Ergin (2002) identifies.

3. Main Results In this section, we first introduce a family of restricted preference profiles and then present our main results.

3.1. Restriction on preference profiles Consider two particular types of preference profiles: (i) profiles at which for each pair i, j ∈ N , Pi = Pj ; (ii) profiles at which for each school a, the number of students who rank a on top does not exceed qa . We say that profiles of type (i) exhibit maximal conflict, since students have the same most preferred school, the same second most preferred school, and so on. We say that profiles of type (ii) exhibit minimal conflict (or “no conflict”), since all students can be simultaneously assigned their most preferred schools. It is easy to check that efficiency and fairness are satisfied by the three rules for these preference profiles, no matter what priorities are. 12

The definition above is the “TTC algorithm with inheritance” that Kesten (2006) introduces. It is equivalent to the original TTC algorithm proposed by Abdulkadiroˇ glu and S¨ onmez (2003).

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We now define a composite of these two types as follows: for a non-negative integer k, all students have the same preferences from their most preferred school down to their k-th school (maximal conflict); but they have diversified preferences over their (k + 1)-th schools so that each student, except for one, can be assigned a school that he finds at least as desirable as his (k + 1)-th school (“almost” no conflict). The formal definition is as follows. Let P ∈ P N and k ∈ {1, · · · , |A|}. Let Ak (P ) ≡

S

i∈N {a

∈

A : |U (Pi , a)| = k} be the collection of schools that are ranked at the k-th level by some student (without loss of generality, let A0 (P ) ≡ ∅). Note that all students have the same k-th most S preferred school under P if and only if |Ak (P )| = 1. Let Ak0 (P ) ≡ kl=0 Al (P ). Recall that P qAk (P ) = a∈Ak (P ) qa . 0

0

Composite of max-min conflict profiles: for each k ∈ {0, 1, · · · , |A| − 1} with |A0 (P )|, |A1 (P )|, · · · , |Ak (P )| ≤ 1, either (i) |Ak+1 (P )| = 1, or (ii) for each a ∈ Ak+1 (P ) with qa < n − qAk (P ) − 1, |{i ∈ N : |U (Pi , a)| = k + 1}| ≤ qa . 0

Note that for each P ∈

PN ,

there is k ∈ {0, 1, · · · , |A|} such that |A0 (P )|, · · · , |Ak (P )| ≤ 1

(for example, if all students have the same preferences from their most preferred school down to their third school, then k = 0, 1, 2, 3). Let k¯ be the largest among them. Then all students have ¯ the same preferences from their most preferred school down to the k-th school, but at least one ¯ student has a different (k + 1)-th most preferred school from others. ¯ Condition (i) is satisfied by the rankings from the most preferred school down to the k-th school: all students have the same preferences over these schools. Condition (ii) is satisfied by students’ P ¯ ¯ (k+1)-th most preferred schools: if a school in Ak+1 (P ) has a capacity less than n− qb −1, k b∈A0 (P )

then the number of students who rank this school in the (k¯ + 1)-th position should not exceed its capacity. Otherwise, no restriction applies. Denote by P N ttc the collection of all composites of max-min conflict profiles. Here are some examples. Example 1. Preference profiles in P N ttc Let N ≡ {1, 2, 3, 4, 5}, A ≡ {a, b, c, d, e}, and q = (1, 1, 1, 2, 2). P1 P2 P3 P4 P5 a b c d d .. .. .. .. .. . . . . .

P1 P2 P3 P4 P5 a a a a a b b b b b c d d d e .. .. .. .. .. . . . . .

P1 P2 P3 P4 P5 a a a a a b c d d e .. .. .. .. .. . . . . .

P1 P2 P3 P4 P5 a a a a a d d d d d b b c c c .. .. .. .. .. . . . . .

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P1 P2 P3 P4 P5 a a a a a b b b b b c c c c c d d d d d e e e e e

The first profile exhibits no conflict, since all students can be simultaneously assigned their favorite schools. The second profile exhibits maximal conflict over a, but there is no conflict over the second most preferred schools. The third profile exhibits maximal conflict over a and b. Additionally, there is some conflict over d, because three students rank d in the third position but qd = 2. However, we regard this as almost no conflict, because there is always an assignment at which each student, except for at most one, is assigned a school that he finds at least as desirable as his third most preferred school. For example, when students 1 and 5 are assigned a and b, then the remaining students compete for d. Only one of them ends up with a school less preferred than d. A similar argument applies to the fourth profile. There is some conflict over b and c in addition to the maximal conflicts over the top two. However, there is an assignment at which each student, except for at most one, is assigned a school that he finds at least as desirable as his third most preferred school. For example, when students 1, 2, and 3 are assigned a and d collectively, then students 4 and 5 compete for c. Only one of them ends up with an even less preferred school. The last profile exhibits maximal conflict over all schools. 

3.2. Results on Fairness and Efficiency N , then TTC and IA are fair, no matter what We first show that if a preference profile is in Pttc

priorities are. N and each ≺∈ ΣA , TTC(≺, P ) and IA(≺, P ) are fair at (≺, P ). Proposition 1. For each P ∈ Pttc N and ≺∈ ΣN . Proof. Let P ∈ Pttc

We prove that TTC(≺, P ) is fair at (≺, P ).

Let k ∈

{0, 1, · · · , |A|} be the largest number such that |A0 (P )|, |A1 (P )|, · · · , |Ak (P )| ≤ 1. Then, all students have the same preferences from their most preferred school down to their k-th school, but at least one student has a different (k + 1)-th most preferred school from the others. For each t ∈ {1, · · · , k}, let at be the t-th most preferred school of all students (that is, At (P ) = {at }). Apply TTC. At Step 1, a1 is assigned to qa1 students with at least qa1 -th priority among the applicants of a1 . At Step 2, a2 is assigned to qa2 students with at least qa2 -th priority among the remaining students at ≺a2 , and so on. If all students are assigned schools by Step k, then the algorithm terminates by Step k. Otherwise, the algorithm proceeds to Step (k + 1). P Let n ¯ be the number of students remaining at Step (k + 1) (that is, n ¯ ≡ n − kt=1 qat ). These students point to their (k + 1)-th most preferred schools. For each of these schools, say a, there are two possibilities. N , the number of students who point to a does not exceed q . Case 1: qa < n ¯ − 1. Since P ∈ Pttc a

Therefore, all applicants of a are assigned a. Case 2: qa ≥ n ¯ − 1. Suppose that the number of students who point to a does not exceed qa . Then, all applicants are assigned a. Suppose that the number of applicants exceeds qa . The number of those students is at least qa + 1 and at most n ¯ . Since n ¯ ≤ qa + 1, we conclude that n ¯ = qa + 1. Therefore, we now know that all the students remaining at Step (k + 1) point to a. By the definition of TTC, all except for one applicant with the lowest priority at ≺a are assigned a. 8

At Step (k + 2), the one remaining student points (or applies) to his (k + 2)-th most preferred school and he is assigned this school. Therefore, whenever student i prefers another school, say b, to his TTC assignment, all the seats of b have been assigned to students with higher priority according to ≺b . Therefore, the TTC assignment is fair at (≺, P ). The same arguments, except that students point to the schools at N and each ≺∈ ΣN , IA(≺, P ) is stable at each step of IA algorithm, prove that for each P ∈ Pttc (≺, P ). The coincidence of the three rules follows from Proposition 1, Remark 2, and the fact that TTC and IA are efficient. N and each ≺∈ ΣA , DA(≺, P ) = IA(≺, P ) = T T C(≺, P ). Corollary 1. For each P ∈ Pttc N is quite restrictive, but we next show that, for TTC to be fair The structure of profiles in Pttc for all priority profiles, a preference profile should have this structure. That is, if a preference N , there is a priority profile under which the TTC assignment is not profile does not belong to Pttc N the domain of preference profiles at which TTC is fair. fair. We denote by Pttc N. Theorem 1. Let P ∈ P N . If for each ≺∈ ΣA , TTC(≺, P ) is fair at (≺, P ), then P ∈ Pttc N . Let k ∈ {0, · · · , |A| − 1} be the largest number such that |A0 (P )|, Proof. Suppose that P ∈ / Pttc

|A1 (P )|, · · · , |Ak (P )| ≤ 1.13 Then, all students have the same preferences from their most preferred school down to their k-th school, but at least one student has a different (k + 1)-th most preferred school from others. For each t ∈ {1, · · · , k}, let at be the t-th most preferred school of all students N , there is a ∈ Ak+1 (P ) such that q < n − q (that is, At (P ) = {at }). Since P ∈ / Pttc a Ak0 (P ) − 1 and there are more than qa students who rank a as their (k + 1)-th most preferred school. Without loss

of generality, let a be the (k + 1)-th most preferred school of students in {1, . . . , (qa + 1)}. Since |Ak+1 (P )| ≥ 2, there is b ∈ Ak+1 (P ) \ {a}. Without loss of generality, let b be the (k + 1)-th most preferred school of student (qa + 2). That is, P is such that   P1 : a1 , a2 , · · · , ak , a, · · · Pqa +2 : a1 , a2 , · · · , ak , b, · · ·   P2 : a1 , a2 , · · · , ak , a, · · · Pqa +3 : a1 , a2 , · · · , ak , · · ·     .. Pqa +4 : a1 , a2 , · · · , ak , · · ·   . .. P : a , a , · · · , a , a, · · · . qa +1

1

2

k

We now construct ≺∈ ΣA such that T T C(≺, P ) is not fair at (≺, P ). For notational simplicity, we represent this priority profile as follows:   ≺a : 1, 2, · · · , (qa − 1), (qa + 2), (qa ), (qa + 1), · · ·   ≺b : (qa + 1), · · · for each o 6= a, b, ≺o : n, (n − 1), · · · , 2, 1 13

N If k = |A|, all students have the same preferences and P ∈ Pttc .

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That is, ≺ is constructed as follows; • Schools a and b: the first (qa − 1) students have at least (qa − 1)-th priority at ≺a , and they are immediately followed by students (qa + 2), (qa ), and (qa + 1). At ≺b , student (qa + 1) has the highest priority. • Each other school: students are prioritized in decreasing order of their labels. Apply TTC to this economy. We have that P (1) students in {n, n − 1, · · · , n − kl=1 qal + 1} are assigned schools in {a1 , · · · , ak }, (2) students in {1, · · · , qa − 1, qa + 1} are assigned a, (3) student (qa + 2) is assigned b and student qa is assigned a school less preferred than a. Student qa prefers a to his assignment, but he has a higher priority at ≺a than student (qa + 1), in violation of fairness. We similarly define the sets for the other two rules: let P N ia be the domain of preference profiles at which IA is fair, no matter what priorities are; let P N da be the domain of preference profiles N if and only if for each at which DA is efficient, no matter what priorities are. That is, P ∈ Pia N is defined similarly. By Remark 2, any efficient ≺∈ ΣA , IA(≺, P ) is fair at (≺, P ). The set Pda N is also the domain of preference profiles on which and fair rule coincides with DA. Therefore, Pda

efficiency and fairness are compatible. We now show how these sets are related. Theorem 2. N ⊆ P N , and the inclusion is proper if there is a ∈ A such that q ≤ n − 3, and (1) Pttc a ia N ⊆ P N , and the inclusion is proper if there is a pair a, b ∈ A such that q + q ≤ n − 1. (2) Pia a b da

Proof. Inclusion relation (1) follows from Proposition 1. Inclusion relation (2) follows from ReN ( P N . Let P ∈ P N be such mark 2. We show if there is a ¯ ∈ A such that qa¯ ≤ n − 3, then Pttc ia that       

 P1 : a ¯, c, d, · · · , b P2 : a ¯, c, d, · · · , b    ..  .  Pn−1 : a ¯, c, d, · · · , b  Pn : b, a ¯, · · ·

That is, P is constructed as follows: • Students in {1, · · ·, n−1}: a ¯ is the most preferred school and all students in {1, · · · , n − 1} have the same preferences; let b be the least preferred school; • Student n: b is the most preferred school and a ¯ is the second most preferred school. N . We show P ∈ P N . Choose any Since qa¯ ≤ n − 3, we also have qa¯ ≤ n − 2. Thus, P ∈ / Pttc ia

≺∈ ΣA and apply IA to (≺, P ). At Step 1, student n is assigned b and the students with at least qa¯ -th priority among {1, · · · , n− 1} at ≺a¯ are assigned a ¯. There are (n−1−qa¯ ) students who are rejected by a ¯. All of these students have the same preferences. 10

At Step 2, these students apply to their next most preferred school, c. Among them, seats are assigned to the students with at least qc -th priority according to ≺c . The students rejected at Step 2, if any, apply to their next most preferred school, and so on. It follows that whenever a student is rejected by a school, all seats at the school are assigned to students with higher priority. Therefore, the assignment is fair. We next show that if there is a pair of distinct schools a, b ∈ A such that qa + qb ≤ n − 1, then N Pia

N . Let P ∈ P N be such that ( Pda  P1 : a, c, d, · · · , b  P2 : a, c, d, · · · , b   ..  .   Pqa : a, c, d, · · · , b

 Pqa +1 : b, c, d, · · · , a Pqa +2 : b, c, d, · · · , a    ..  .  Pn−1 : b, c, d, · · · , a  Pn : b, a, · · ·

That is, P is constructed as follows: • All students have the same preferences over the schools except for {a, b}; • Students in {1, · · ·, qa }: a is the most preferred school, b is the least preferred school; • Students in {qa + 1, · · ·, n − 1}: b is the most preferred school, a is the least preferred school; and • Student n: b is the most preferred school, a is the second preferred school. N . Choose any ≺∈ ΣA and apply DA to (≺, P ). There are two possibilities. We show that P ∈ Pda

Case 1: student n has at least the qb -th highest priority among the students in {qa + 1, · · · , n} at ≺b . At Step 1, all students with at least qb -th priority among {qa + 1, · · · , n} at ≺b (including student n) are assigned b and all students in {1, · · · , qa } are assigned a. The tentative assignment at a and b at Step 1 does not change until the last step of the algorithm. At Step 2, students in {qa + 1, · · · , n} who are rejected by b apply to c and the seats of c are assigned to the students with at least qc -th priority among them. At Step 3, students who are rejected by c, if any, apply to d and the seats of d are assigned to the students with at least qd -th priority among them, and so on. It is easy to check that no other assignment makes all students at least as well off as in this DA assignment. Therefore, DA(≺, P ) is efficient. Case 2: student n does not have at least the qb -th highest priority among the students in {qa + 1, · · · , n} at ≺b . At Step 1, all students with at least qb -th priority at ≺b (not including student n) are assigned b and all students in {1, · · · , qa } are assigned a. The tentative assignment at b at Step 1 does not change until the last step of the algorithm. At Step 2, the students rejected at Step 1, except for student n, apply to c and student n applies to a. There are two subcases.

11

Subcase 2.1: student n has a lower priority than all students in {1, · · · , qa } at ≺a . Student n is rejected by a and applies to his next most preferred school. The tentative assignment at a at Step 1 does not change until the last step of the algorithm. Note that all the students rejected at Step 1 (including student n) apply to the schools in A \ {a, b} in the same decreasing order of their preferences, say c, d, and so on. The seats of schools in A \ {a, b} are allocated to these students in order of priorities subject to the capacity constraints. It is easy to check that no other assignment makes all students at least as well off as in the DA assignment. Therefore, DA(≺, P ) is efficient at P . Subcase 2.2: student n has a higher priority than someone in {1, · · · , qa } at ≺a . At Step 2, student n is tentatively assigned a, and the student with the lowest priority among {1, · · · , qa }, say student i∗ , is rejected. The tentative assignment at a at Step 2 does not change until the last step of the algorithm. Note that student i∗ and all students rejected at Step 1 apply to the schools in A \ {a, b} in the same decreasing order of their preferences, say c, d, and so on. The same argument in Subcase 2.1 applies and we conclude that DA(≺, P ) is efficient at P . N . Let ≺∈ ΣA be such that We show that P ∈ / Pia   ≺a : n, (n − 1), (n − 2), · · · , 2, 1 ≺b : 1, 2, · · · , qa , (qa + 1), (qa + 2), · · · , n

That is, ≺ is constructed so that (i) ≺a prioritizes students in decreasing order of their labels, and (ii) ≺b prioritizes students in increasing order of their labels. Then, the IA algorithm applied to (≺, P ) is as follows. At Step 1, students in {1, · · · , qa } are assigned a and students in {qa + 1, · · · , qa + qb + 1} are assigned b. Since qa + qb ≤ n − 1, some students, including student n are rejected by b. At Step 2, these students, including student n applies to their next preferred school. Student n is assigned a school less preferred than a. Since student n has the highest priority at a than any student in {1, · · · , qa }, IA(≺, P ) is not fair. One implication of Theorem 2 is as follows. When a planner has to adopt a rule among others, several criteria can be considered. Given that efficiency and fairness are incompatible, one possible criterion is the domain of preference profiles on which a rule satisfies both properties: as the domain for a rule enlarges, it is more likely that the rule satisfies the properties. Proposition 2 says that DA performs better than IA in this respect, which itself performs better than TTC. It is important that priority profiles are not controlled, as we discussed in Introduction. Recall that there are three sets of priority profiles at which the aforementioned rules are efficient and fair ; 14 ΣA da (Ergin, 2002): the domain of priority profiles at which DA is efficient;

ΣA ttc (Kesten, 2006): the domain of priority profiles at which TTC is fair ; ΣA ia (Kumano, 2013): the domain of priority profiles at which IA is fair. 14

N That is, if ≺∈ / ΣA such that DA(≺, P ) is not efficient at P . The other da , there is a preference profile P ∈ P two sets are defined similarly.

12

As illustrated in Figure 1, the results concerning priority profiles are not symmetric to those concerning preference profiles, confirming that priorities and preferences play different roles in this N problem. Moreover, ΣA ia = ∅ for most economies (Kumano, 2013), but Pia 6= ∅ for all economies.

3.3. Consistency We next consider “consistency”, a robustness axiom pertaining to variable populations. It confirms the desirability of selections that a rule makes, by comparing the selections across economies in the following way. First, consider an economy and the assignment selected by a rule for it. Suppose that a group of students is left with the sum of what they were to receive, while the other students leave the economy. Apply the rule to this new economy. Consistency requires that the rule recommend the same assignment for them as it did in the initial economy (Thomson, 2013a). To define this property formally, we need to define a set of economies consisting of subsets of schools and students. Consider a rule ϕ and an economy e ≡ (A, N, q, ≺, P ). Let B ⊆ A, P 0 S ⊆ N , and q 0 ∈ NB be such that for each a ∈ B, 0 < qa0 ≤ qa and a∈B qa ≥ |S|. We call (B, S, q 0 , (≺a |S )a∈B , (Pi |B )i∈S ) a subeconomy of e. An assignment at this subeconomy is defined as in Section 2. Consider a rule ϕ. An extension of ϕ to all subeconomies is a function ϕ that maps each subeconomy of e to an assignment of the subeconomy. Define TTC, IA, and DA rules to be the extensions of TTC, IA, and DA to all subeconomies. Now, we define our robustness axiom. Let e ≡ (A, N, q, ≺, P ) and x ∈ X. For each S ⊆ N , let xS ≡ (|{i ∈ S : xi = a}|)a∈A be the assignments that students in S collectively received at x. Note that there may be schools that no student in S received at x. Let A(xS ) ≡ {a ∈ A : xSa > 0}. We call e(x, S) ≡ (A(xS ), S, (xSa , ≺a |S )a∈A(xS ) , (Pi |A(xS ) )i∈S ) the reduced economy of e at (x, S). A rule ϕ is consistent at (≺, P ) if for each S ⊆ N , (ϕi (e))i∈S = ϕ(e(ϕ(e), S)). A rule ϕ is consistent if for each ≺∈ ΣA and each P ∈ P N , ϕ is consistent at (≺, P ).15 None of DA, IA, TTC are consistent (Ergin (2002), Kesten (2006), and Kumano (2013)).16 As in the previous section, we define the sets of preference profiles on which the aforementioned rules are consistent, no matter what priorities are; N

P ttc : the domain of preference profiles at which TTC is consistent, N

P ia : the domain of preference profiles at which IA is consistent, and N P da : the domain of preference profiles at which DA is consistent. N

That is, P ∈ P ttc if and only if for each ≺∈ ΣA , TTC is consistent at (P, ≺). Similarly, we denote the sets of priority profiles by 15

In defining consistency, we fix an original economy and then check the selections that a rule makes only for its subeconomies. This notion of consistency is introduced by Thomson and Zhou (1993). 16 Note that there are two ways of formulating reduced economies of e. The difference comes from the fact that there can be schools with no available seats at xS . We find it reasonable to update students’ preferences to be defined over the schools with available seats in the reduced economy. IA is not consistent based on our formulation. Alternatively, we may keep A as the set of schools, even if some schools have no seats at xS (Bu (2012) and Kojima ¨ and Unver (2014)). IA is consistent based on their formulation. On the other hand, the extension of the immediate acceptance rule with skips, discussed in Footnote 10, satisfies both notions of consistency.

13

N

N =P Pda da

A

ΣA da = Σda

N Pia

A Σttc

ΣA ttc

ΣA ia

=

N

A Σia

N = Pttc

(a) Figure 2.

N P ttc

P ia

(b)

Summary: relations between these sets (a) The domain of priority profiles for efficiency, fairness, and consistency (b) The domain of preference profiles for efficiency, fairness, and consistency

A

Σttc (Kesten, 2006): the domain of priority profiles at which TTC is consistent, A Σia (Kumano, 2013): the domain of priority profiles at which IA is consistent, and A

Σda (Ergin, 2002): the domain of priority profiles at which DA is consistent. The following theorem shows that these sets are related and how. As illustrated in Figure 2, the results pertaining priority profiles in (a) stand in sharp contrast to ours in (b). Theorem 3. N

N

N ⊆ P , where the inclusion is proper if there are a, b ∈ A with q + q ≤ n − 1, and (1) P ttc = Pttc a b ia N N , where the inclusion is proper if there are a, b, c ∈ A with q + q + q ≤ n − 1, and (2) P ia ⊆ Pia a c b N

N. (3) P da = Pda N

N

N . Suppose otherwise. Then, there is P ∈ P \P N . Proof. (1) We start by showing that P ttc ⊆ Pttc ttc ttc

Such P can be represented as in the proof of Theorem 1. Also, consider the priority profile ≺ and the resulting TTC assignment x provided in the proof of that theorem. Consider now the reduced economy of (A, N, q, ≺, P ) at (x, N \ {qa + 2}). According to the TTC algorithm applied to e(x, S), student qa is assigned a and student (qa + 1) is assigned a school less preferred than a, a violation of consistency. N

N ⊆ P . Let P ∈ P N , ≺∈ ΣA , and x ≡ TTC(A, N, q, ≺, P ). Let S ⊆ N and We show that Pttc ttc ttc

x0 ≡ TTC(e(x, S)). We show that (xi )i∈S = x0 . Let k ∈ {0, · · · , |A|−1} be the largest number such that |A0 (P )|, · · · , |Ak (P )| ≤ 1. For each t ∈ {1, · · · , k}, let at be the t-th most preferred school of all students (that is, At (P ) ≡ {at }). If k = 0, then Ak0 (P ) = ∅. Otherwise, the assignment of Ak0 (P ) is given as follows. Assignments of Ak0 (P ): As seen in the proof of Proposition 1, the TTC algorithm applied to (A, N, q, ≺, P ) is as follows. At Step 1, a1 is assigned to the students with at least qa1 -th priority according to ≺a1 (denote these students by N1 ); 14

At Step 2, a2 is assigned to the students with at least qa2 -th priority among the remaining students according to ≺aw (denote these students by N2 ); and so on. For each pair l, m ∈ {1, · · · , k} with l < m, all students in Nl have higher priorities at ≺al than those in Nm (denote this statement by (∗)). The schools in Ak0 are assigned to students by Step k of the TTC algorithm. Now, apply TTC to e(x, S). Note that all students in S have the same preferences from their most preferred school down to their k-th school. Therefore, a1 down to ak are assigned in order of students priorities subject to the capacity constraint xS .17 For each i, j ∈ S and each a ∈ A, i ≺a j if and only if i ≺a |S j. By (∗), we conclude that the assignments of Ak0 are the same as in x for the students in S. Now we work with the remaining students. Assignments of Ak+1 (P ): In the TTC algorithms applied to the original economy and the reduced economy, the remaining students at Step (k+1) are equal, and each of these students applies N , as shown in the proof of Proposition 1, to his (k + 1)-th most preferred school. Since P ∈ Pttc each student, except for at most one, is assigned a school that he finds at least as desirable as his (k + 1)-th most preferred school at x. Therefore, we have two cases to consider in the original economy. Case 1: Each student present at Step (k + 1) is assigned his (k + 1)-th most preferred school. Apply TTC to the reduced economy. At Step (k + 1), the number of available seats at each school and the number of applicants to the school are equal. Therefore, TTC assigns each student his (k + 1)-th most preferred school. Therefore, x0 = (xi )i∈S . Case 2: Each student present at Step (k + 1), except for one, is assigned his (k + 1)-th most preferred school. Let i∗ be the one who is rejected by his (k + 1)-th most preferred school, say a∗ . To have one student rejected by a∗ at Step (k + 1), (i) qa∗ = n − qAk (P ) − 1 should hold and (ii) all 1 students present at Step (k + 1) should apply to a∗ .18 Moreover, i∗ has the lowest priority at a∗ than all other students present at Step (k + 1). Now, apply TTC to the reduced economy. Suppose that i∗ ∈ / S. Then, all students in S are assigned their (k + 1)-th most preferred school. By the argument used in Case 1, they are assigned their (k + 1)-th most preferred school in the reduced economy. Suppose that i∗ ∈ S. Since i∗ has the lowest priority at a∗ than all other students in S, he is rejected by a∗ again in the reduced economy. All the other students in S are assigned a∗ . N

N ⊆P Inclusion relation Pttc ia can be shown by the same argument as above, so we omit it. 17

That is, a1 is assigned to the students with at least xS a1 -th priority according to ≺a1 |S ; at Step 2, a2 is assigned to the students with at least xS a2 -th priority among the remaining students according to ≺a2 |S at Step 2 of the TTC algorithm; and so on. 18 N Recall that P ∈ Pttc . We assume that i∗ is rejected by a∗ . If qa∗ < n − qAk (P ) − 1, then, the number of 1 ∗ applicants to a at Step (k + 1) does not exceed qa∗ . Then, all of them are accepted, contradicting the assumption. ∗ ∗ If qa > n − qAk (P ) − 1, then school a can accommodate all applicants at Step (k + 1), because the number of all 1 students present at Step (k +1) is n−qAk (P ) , This again contradicts the assumption. Therefore, qa∗ = n−qAk (P ) −1. 1 1 On the other hand, suppose that any student present at Step (k + 1) applies to a school other than a∗ . Then, the ∗ ∗ number of applicants ofl a is at most (n − qAk (P ) − 1), which is exactly qa∗ . Therefore, a can accommodate all 1 applicants at Step (k + 1), contradicting the assumption.

15

The proper inclusion relation, when there are a, b ∈ A such that qa + qb ≤ n − 1, can be shown with the following preference profile, P : (i) all students except for one, say student i, have the same preferences, ranking a as the most preferred school and b as the least preferred school; (ii) student i ranks b as the most preferred school. It is easy to check that for all ≺∈ ΣA , IA is N. consistent at (≺, P ), but P ∈ / Pttc N

N . Suppose otherwise. Then, there are P ∈ P N and ≺∈ ΣA such that IA (2) We prove P ia ⊆ Pia is consistent at (≺, P ), but IA(≺, P ) is not fair at (≺, P ). Let x ≡ IA(≺, P ). That is, there is a

pair of students i, j ∈ N such that xj Pi xj and i ≺xj j. Let S ≡ {i, j}. Then, IAi (e(x, S)) = xj , violating consistency of IA at (≺, P ). The proper inclusion relation, when there are a, b, c ∈ A with qa + qb + qc ≤ n − 1, can be shown with the following profile P :   P1 : a, c, · · · , b Pqa +2 : b, c, · · · , a  P2 : a, c, · · · , b Pqa +3 : b, c, · · · , a      .. ..   . . Pqa +1 : a, c,

··· , b

Pn : b, c,

··· , a

For each ≺∈ ΣA , at Step 1 of the IA algorithm applied to (≺, P ), all the seats of a and b are distributed to students with at least qa -th and at least qb -th priorities among the applicants according to ≺a and ≺b , respectively. Since those who are rejected at Step 1 have the same preferences over the remaining schools, all the seats of these schools are allocated in order of priorities subject to the capacity constraints. Therefore, IA(≺, P ) is fair at (≺, P ). We show that there is ≺∈ ΣA such that IA is not consistent at (≺, P ). Let ≺∈ ΣA be such that   ≺c : n, (n − 1), (n − 2), · · · , 2, 1 for each o ∈ A \ {c}, ≺o : 1, 2, · · · , (n − 1), n Let x ≡ IA(≺, P ). Then, for each i ∈ {1, · · · , qa }, xi = a; for each i ∈ {qa + 2, · · · , qa + qb + 1}, xi = b; xn = c; and student (qa + 1) is assigned a school less preferred than c. Let S ≡ {qa + 1, qa + 2, · · · , qa + qb + 1, n}. Let x0 ≡ IA(e(x, S)). At Step 1 of the IA algorithm applied to this reduced economy, student (qa + 1) applies to c and all the other students apply to b. Student (qa + 1) is assigned c, violating consistency. N

N ⊆ P . Suppose otherwise. Then, there are P ∈ P N and ≺∈ ΣA (3) We first show that Pda da

such that DA(≺, P ) is efficient at P , but DA is not consistent at (≺, P ). Let x ≡ DA(≺, P ). There is S ⊆ N such that (xi )i∈S 6= DA(e(x, S)). Let x0 ≡ DA(e(x, S)). Since x is fair at (≺, P ), it follows that (xi )i∈S is fair in the reduced economy e(x, S). Note that x0 is fair and Pareto dominates (xi )i∈S in the reduced economy (Remark 2). Lastly, let x00 be an assignment such that / S, x00i ≡ xi . Since x00 Pareto dominates x at P , x is not for each i ∈ S, x00i ≡ x0i and for each i ∈ efficient at P , a contradiction. N

N

N . Suppose otherwise. Then, there are P ∈ P Now, we prove that P da ⊆ Pda and ≺∈ ΣA such that DA is consistent at (≺, P ), but DA(≺, P ) is not efficient at P . There is y ∈ X \ {x} that

Pareto dominates x at P . Let S ≡ {i ∈ N : yi 6= xi }. Note that there may be more than one such

16

assignment y. Choose y ∈ X with the smallest |S|. Note that {xi : i ∈ S} = {yi : i ∈ S} and that (yi )i∈S Pareto dominates (xi )i∈S in the reduced economy e(x, S).19 Altogether, for each i ∈ S, yi Pi xi and there is j ∈ S \ {i} such that yi = xj . Without loss of generality, relabel students in S as i1 , · · · , is in such a way that for each t ∈ {1, · · · , s}, xit−1 = yit and xit−1 Pit xit (all statements henceforth are modulo s). That is, ···

Pi3 .. .

P i2 .. .

P i1 .. .

P is .. .

xis (≡ yi1 ) xi1 (≡ yi2 ) xi2 (≡ yi3 ) · · · .. .. .. . . .

xis (≡ yis−1 ) .. .

xi1 (≡ yi2 ) xi2 (≡ yi3 ) xi3 (≡ yi4 ) · · · .. .. .. . . .

xis (≡ yi1 ) .. .

Apply DA to e(x, S). Since DA is consistent at (≺, P ), we have (xi )i∈S = DA(e(x, S)). For each t ∈ {1, · · · , s}, student it applies to schools in the decreasing order of his preferences. Therefore, for it to be allocated xit , he should have applied to xit−1 and he is rejected at the same step, say Step Mt , of the DA algorithm.20 Choose a student it∗ with Mt∗ ≤ Mt , for all t ∈ {1, · · · , s}. At Step Mt∗ , student it∗ applies to xit∗ −1 from which he is rejected. Thus, there should be at least one other student who is tentatively assigned xit∗ −1 at the same step. Let is∗ be such a student. We claim that is∗ ranks xit∗ −1 above xis∗ −1 . Suppose otherwise. Then, is∗ ranks xis∗ −1 above xit∗ −1 and he is rejected by xis∗ −1 at an earlier step, contradicting the assumption that Mt∗ is the smallest. Therefore, ···

Pis∗ .. .

Pis∗ +1 .. .

Pis∗ +2 .. .

xit∗ −1 .. .

x i s∗ .. .

xis∗ +1 .. .

xit∗ −2 .. .

xis∗ .. .

xis∗ +1 .. .

xis∗ +2 .. .

xit∗ −1 .. .

Pit∗ −1 .. .

Let S 0 ≡ {is∗ , is∗ +1 , · · · , it∗ −1 }. Among students in S 0 , a Pareto improvement from x can be made as above. Since S 0 ( S, we obtain a contradiction that S was chosen to have the smallest cardinality. From Theorems 2 and 3, we have the following relations described in Figure 2 (b). N

N

N

N ⊆P N N Corollary 2. P ttc = Pttc ia ⊆ Pia ⊆ P da = Pda .

The implication of Theorem 3 is analogous to that of Theorem 2: comparing the sets of preference profiles guaranteeing consistency, DA performs better than IA, which itself performs better than TTC. Moreover, we have shown that (i) it is as hard to restore the consistency of TTC 19

Since it is well-known that DA is “non-wasteful ” (Kojima and Manea, 2010), it is not possible at y to assign some seats that were not assigned to anyone at x. Therefore, {xi : i ∈ S} = {yi : i ∈ S}. 20 Let i0 ≡ is .

17

as to restore its fairness, but (ii) it is harder to restore the consistency of IA than to restore its fairness.

4. Concluding Remark Given the two components of economies, preference profiles and priority profiles, we focused on restrictions on the first component to guarantee efficiency and fairness (or consistency) of TTC, IA, and DA, respectively. The preference profiles that we identify make TTC (or IA, or DA) efficient and fair (or consistent) for all priority profiles. If we want to satisfy these properties for some priority profiles, we should obtain a larger domain of preference profiles. In this sense, we regard the domain that we identify as the “smallest” domain we can define to guarantee these properties of the rule, or the domain we define with a “conservative” criterion. Our conclusion from Theorems 2 and 3 should also be viewed one conservative way of evaluating these rules. What happens then if we require these rules to satisfy the properties for some priority profiles? Even if priorities are not fully controlled by a school district, they may also have a certain structure in reality. It is reasonable then to figure out combinations of restrictions on both components. For example, we may search for preference profiles at which TTC is stable for all priority profiles that Ergin (2002) identifies.21 This may be more practical and interesting question. To work with such combinations, however, we first have to define a certain structure of priority profiles, so that we identify the corresponding structure of preference profiles. What are interesting and plausible restrictions on priority profiles to start with? They should be structured in an acceptable manner, but in presence of a priority tie-breaking rule, they should also have much flexibility. It is not obvious to figure out such profiles. We find that this question should be answered in relation to some empirical evidence of school choice problems. As it goes beyond the scope of our paper, we leave it as an open question for future research.

References [1] Abdulkadiroˇ glu, A. and T. S¨ onmez, 2003. “School Choice: A Mechanism Design Approach,” American Economic Review 93, 729-747. [2] Alcalde, J., 1996. “Implementation of stable solutions to marriage problems,” Journal of Economic Theory 69, 240-254. [3] Balinski, M. and T. S¨ onmez, 1999. “A Tale of Two Mechanisms: Student Placement,” Journal of Economic Theory 84, 73-94. [4] Barbera, S., H., Sonnenschein, and L. Zhou, 1991, “Voting by committees”, Econometrica 59, 595-609. 21

We identified some classes of these preference profiles, but not all.

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[5] Bu, N., 2012. “Two new fairness notions in the assignment of indivisible resources,” mimeo. [6] Ehlers, L. and A. Erdil, 2010. “Efficient assignment respecting priorities,” Journal of Economic Theory 145, 1269-1282. [7] Erdil, A. and H.I. Ergin, 2008. “What’s the Matter with Tie-Breaking? Improving Efficiency in School Choice,” American Economic Review 98, 669-689. [8] Ergin, H.I., 2002. “Efficient Resource Allocation on the Basis of Priorities,” Econometrica 70, 2489-2497. [9] Gale, D. and L.S. Shapley, 1962. “College admissions and the stability of marriage,” American Mathematical Monthly 69, 9-15. [10] Haeringer, G. and F. Klijn, 2009. “Constrained school choice,” Journal of Economic Theory 144, 1921-1947. [11] Han, X. 2014. “House allocation with weak priority orders,” mimeo. [12] Harless, P. 2014. “A school choice compromise: between immediate and deferred acceptance,” mimeo. [13] Hatfield, J.W., F. Kojima, Y. Narita, 2011. “Promoting school competition through school choice: a market design approach,” mimeo. [14] Hsu, C.L. 2013. “When is the Boston Mechanism Dominance-Solvable?,” mimeo. [15] Kesten, O., 2006. “On two competing mechanisms for priority-based assignment problems,” Journal of Economic Theory 127, 155-171. [16] Kojima, F. 2007. “When can manipulations be avoided in two-sided matching markets? – maximal domain results” The B.E. Journal of Theoretical Economics 7, 1-18. [17] Kojima, F. and M. Manea, 2010. “Axioms for deferred acceptance,” Econometrica 78, 633-653. [18] Kojima, F. 2011. “Robust stability in matching markets,” Theoretical Economics 6, 257-267. ¨ [19] Kojima, F. and M. Unver, 2014. “The ‘Boston’ school-choice mechanism: an axiomatic approach,” Economic Theory 55, 515-544. [20] Kumano, T. 2013. “Strategy-proofness and stability of the Boston mechanism: an almost impossibility result,” Journal of Public Economics 105, 23-29. [21] Thomson, W., 2013a. “Consistent allocation rules,” mimeo. [22] Thomson, W., 2013b, “Strategy-proof allocation rules,” mimeo. [23] Thomson, W. and L. Zhou, 1993, “Consistent solutions in atomless economies,” Econometrica 61, 575-587.

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