Preference Profiles for Efficiency, Fairness, and Consistency in School Choice Problems Eun Jeong Heo∗† November 5, 2017

Abstract We study the school choice problem in which a school district assigns school seats to students. There has been a long debate over the three best-known rules for this problem: the deferred acceptance rule (DA), the top-trading cycles rule (TTC), and the immediate acceptance rule (IA). We evaluate these rules by investigating how often they satisfy three central requirements, efficiency, fairness, and consistency. We compare the restricted domains of students’ preferences on which each rule satisfies these requirements. From the containment relations between them, we show that DA performs better than IA, which itself performs better than TTC in terms of efficiency and fairness. If we consider consistency instead, IA performs better than DA, which itself 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; efficiency; fairness; consistency

∗

Department of Economics, Vanderbilt University, Nashville TN 37235, USA ([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 thank William Thomson for his support and guidance. I also appreciate the associate editor and the two anonymous referees giving me constructive comments and suggestions. I benefited from discussions with Anna Bogomolnaia, Bettina Klaus, Youngwoo Koh, Fuhito Kojima, Patrick Harless, Vikram Manjunath, ¨ ¨ ur Yılmaz. I am also grateful to seminar Herv´e Moulin, Kyoungwon Seo, Shigehiro Serizawa, Utku Unver, and Ozg¨ 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 helpful comments. All remaining errors are my own responsibility. †

1. Introduction We study the “school choice problem” in which a school district assigns school seats to students. Students have strict preferences over the schools and they are asked to report their preferences to the district. Schools have capacity constraints as there are fixed 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’ preferences, a rule determines which school is assigned to whom, subject to the capacity constraints. Efficiency and fairness are two central requirements of rules for this problem. Consider a rule. Efficiency requires that there be no Pareto improvement from each selection that the rule makes. Fairness requires that the rule “respect” the priority structure 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 toptrading cycles rule (TTC), and the immediate acceptance rule (IA).1 Unfortunately, none of them satisfies both requirements: TTC and IA are efficient but not fair, and DA is fair but not efficient (Gale and Shapley 1962; Abdulkadiroˇ glu and S¨onmez 2003). These observations generalize: no rule satisfies efficiency and fairness (Balinski and S¨ onmez, 1999). Even in large economies, this conflict may still persist (Che and Tercieux, 2017). This impossibility is obtained, however, as we consider all possible preferences and priorities. We easily find that there are some preferences and priorities at which these two requirements are compatible (for example, identical preferences/priorities). Given this, it is natural to ask the following questions. How often do such preferences and priorities appear? Can we identify such preferences and priorities? If so, can we compare rules based on the “frequency” of these preferences and priorities? Several recent papers have raised this type of questions and most of them focus on priority profiles. Ergin (2002) identifies the structure of priority profiles under which DA is efficient or “consistent”2 for all preferences. Similarly, Kesten (2006) identifies the structure of priority profiles under which TTC is fair or consistent for all preferences. 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, which unfortunately is empty in many 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 (2013). We present the formal definition in Section 2.2. For an exhaustive survey on consistency, see Thomson (2015). 3 Precisely, such priority profiles exist only if there are at most two students, or there is only one school, or the sum of capacities of each pair of distinct schools is no smaller than the total number of students. 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. (2016), Hsu (2013), and Han (2015). 4 The structures of these priority profiles are derived from conditions on priority-capacity pairs. Here we simply assume that the capacity constraints are fixed and keep them implicit. 2

1

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

The relations between these domains are illustrated in Figure 1(a). Unfortunately, these domains turn out to be quite small relative to the unrestricted one: even if the school district wants to pick a “right” priority profile, the choice has to be made within a very limited range. As can be seen in Figure 1(a), however, there is more flexibility in choosing these priorities for DA than for TTC. The figure also shows that it is almost impossible to achieve fairness of IA by “controlling” priorities. Although these results can help the school district set up the school choice system, there still exists a random factor in determining priorities when there are ties among students. Students are prioritized by only a few criteria in reality, which result in large indifference classes of students (Erdil and Ergin 2008; Ehlers and Erdil 2010). To induce strict priorities, a random tie-breaking rule is commonly used and the resulting priority profiles are not necessarily those identified above. In this paper, we turn to the other side of the problem, the preference profiles, which are not observed ex ante either, nor are they under the control of the school district. However, we very often observe that students’ preferences have certain structures in reality. For example, students have correlated preferences over different tiers of schools: 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. Note that we work on restrictions on preference profiles, not on individual preferences, as in earlier papers on priority profiles. The domain of preference profiles we identify therefore includes the Cartesian products of individual preference restrictions achieving efficiency and fairness. We ask whether we can achieve desirable requirements for some restricted preference profiles and if so, how often we do. This is a “quantitative” analysis on these requirements, in contrast to the standard “qualitative” analysis where we only inquire about whether a rule satisfies the requirements or not.5 With this approach, we can evaluate the “strength” of the baseline impossibility result or 5 This explanation is borrowed from Che and Tercieux (2017) who also make an “quantitative” analysis on the conflict between efficiency and fairness in a large matching problem. While we analyze the frequency of inefficient/unstable incidences in terms of preference profiles, they analyze it in terms of the fraction of students (or student-school pairs) involved in those incidences. As the size of the problem grows, they keep track of the change of this fraction and study “asymptotic” notions of efficiency and fairness. In their analysis, the number of school types increases as the matching market grows, while we keep it fixed.

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the performance of the rules. That is, if the identified restrictions are very mild, then the underlying impossibility result can be viewed as an “almost” possibility result. Likewise, if the restrictions for a rule to satisfy certain requirements is more demanding than those for another rule, we can say that the former performs worse than the latter.6 Pathak and S¨ onmez (2013) have also adopted this approach to evaluate the rules on the basis of students’ incentive to misreport their preferences. They compare domains of preference profiles on which a strategic condition is satisfied (or violated): a rule performs better than another if any preference profile that leads to profitable manipulations under the former rule, also leads to profitable manipulations under the latter. We focus on other central requirements, efficiency and fairness, to evaluate the aforementioned rules. The domains of preference profiles of our attention are defined as follows: PN ttc : the domain of preference profiles at which TTC is fair, PN ia : the domain of preference profiles at which IA is fair, and PN da : the domain of preference profiles at which DA is efficient. Our first main result shows that these domains are related (Theorem 1) and how. There are N is a proper subset of P N , which itself is a proper containment relations as shown in Figure 1(b): Pttc ia N . That is, any preference profile that leads to efficient and fair TTC assignments for all subset of Pda

priority profiles, also leads to efficient and fair IA assignments for all priority profiles. Likewise, any preference profile that leads to efficient and fair IA assignments for all priority profiles, also leads to efficient and fair DA assignments for all priority profiles. We therefore conclude that in terms of these two requirements together, DA performs better than IA, which itself performs better than TTC. The inclusion relation between the first and second domains, in particular, contrasts with the results on priority profiles (Kesten 2006; Kumano 2013). We next turn to consider consistency, a standard robustness requirement, to evaluate these rules again. This requirement has been extensively explored in resource allocation problem as well as in ¨ school choice problems (Ergin 2002; Kesten 2006; Kojima and Unver 2014; Bu 2015; Harless 2016; and Jaramillo 2017). 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, 2015). We again identify the domain of preference profiles on which each rule is consistent. Our second main result shows that in terms of consistency, IA performs better than DA, which itself performs better than TTC, a different conclusion from the above. In fact, the domain to achieve N . Similarly, the domain to achieve consistency of DA coincides consistency of TTC coincides with Pttc 6

It is very common to investigate preference restrictions, given an underlying impossibility result. A best-known example is the single-peaked preference domain in the social choice model. A list of papers on “maximal domain” also study preference restrictions (Barber` a et al., 1991). Most of these papers focus on restrictions on individual preferences to achieve some relational properties, such as “strategy-proofness”. They also require that a minimally plausible subdomain be included in the resulting domain. In this paper, however, we do not consider such a minimal subdomain and we consider non-relational properties of rules. For a maximal domain approach in the matching context, see Kojima (2007).

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N . The domain for IA is unrestricted, on the other hand, because IA is consistent for all priority with Pda ¨ profiles (Bu 2015; Kojima and Unver 2014; Harless 2016).

Our two main results exemplify how to evaluate rules by using the “frequency” of desirable assignments chosen by a rule. Our approach can be applied to other requirements that the school district would consider. This paper is organized as follows: Section 2 introduces the model, requirements, and the rules. Section 3 presents our main results. We then conclude in Section 4.

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 be a non-negative integer representing the capacity (the number P of seats) of school a, and q ≡ (qa )a∈A the capacity profile. For each B ⊆ A, qB ≡ a∈B qa . Each a ∈ A has a 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.7 Each i ∈ N has a complete, transitive, and strict preference Pi over A. For each pair a, b ∈ A, a Pi b if and only if i prefers a to b. Let P be the set of all such preferences. For each Pi ∈ P and each a ∈ A, let U (Pi , a) ≡ {b ∈ A : b Pi a or a = b} be the weak upper contour set of a at Pi . For each B ⊆ A, let Pi |B be the preference Pi 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 ).

2.1. Economies with Fixed Populations When we fix A, N , and q, an economy is simply 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. We now define two central requirements of rules. Let P ∈ P N . We say that x ∈ X is efficient at P if no other assignment makes each student 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. 7

In case n >

P

a∈A

qa , we can introduce an outside option ∅ of “home-schooling” and set its capacity n.

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We now introduce three rules that are most commonly used in practice. 2.1.1. The (student-proposing) deferred-acceptance rule For each economy e = (A, N, q, ≺, P ), 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 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. For each x, x0 ∈ X, we say that x weakly Pareto dominates x0 at P if for each i ∈ N , xi Pi x0i or xi = x0i . Remark 2. (Balinski and S¨ onmez, 1999) The selection that DA makes for each (≺, P ) weakly Pareto dominates each other fair assignment at P . Moreover, when priority profiles have a structure identified by Ergin (2002), DA is efficient for all preferences. 2.1.2. Immediate acceptance rule For each economy e = (A, N, q, ≺, P ), 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. 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.

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We call this rule the immediate acceptance rule, or IA for short.8 IA is efficient but not fair. When priority profiles have the structure identified by Kumano (2013), however, IA is fair for all preferences. Unfortunately, such priority profiles do not exist in most cases (see Footnote 3). 2.1.3. Top-trading cycles rule For each economy e = (A, N, q, ≺, P ), the student-proposing top-trading cycles rule assigns schools to students by means of the following algorithm.9 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 leaves 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 leaves 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 (2006), however, TTC is fair for all preferences. The domain of such priority profiles is a subset of what Ergin (2002) identifies.

2.2. Economies with Variable Populations Consider an economy e ≡ (A, N, q, ≺, P ). We define a set of economies consisting of subsets of schools and students in e. Let B ⊆ A, S ⊆ N , and q 0 ∈ NB be such that for each a ∈ B, 0 ≤ qa0 ≤ qa and P 0 0 a∈B qa ≥ |S|. We call (B, S, q , (≺a |S )a∈B , (Pi |B )i∈S ) a subeconomy of e. An assignment at this subeconomy is defined as in Section 2.1. 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. For each subeconomy, efficiency and fairness are defined as in Section 2.1. We next consider consistency, a robustness requirement pertaining to the set of subeconomies of e. 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, 2015). 8

This terminology is adopted from Thomson (2013). 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). 9

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This requirement is formally defined as follows. 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. We call e(x, S) ≡ (A, S, (xSa , ≺a |S )a∈A , (Pi )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 ).10

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3. Main Results We evaluate the aforementioned rules by comparing the domains of preference profiles on which the rules satisfy efficiency and fairness together, and/or consistency.

3.1. Result on Efficiency and Fairness We first define the following domains of preference profiles. PN ttc : the domain of preference profiles at which TTC is fair for all priorities; PN ia : the domain of preference profiles at which IA is fair for all priorities; PN da : the domain of preference profiles at which DA is efficient for all priorities. N if and only if for each ≺∈ ΣA , T T C(≺, P ) is fair at (≺, P ). The sets P N and P N That is, P ∈ Pttc ia da

are defined similarly. N is the largest domain of preference profiles on which efficiency and fairness Remark 3. Note that Pda are compatible for all priorities. This follows from Remark 2: any efficient and fair assignment for a

problem coincides with what DA selects for the problem. We now show whether these sets are related and how. Theorem 1. N ⊆ P N , and the inclusion is proper if there is a ∈ A such that q ≤ n − 2, 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

Theorem 1 says that any preference profile that leads to efficient and fair TTC assignments for all priority profiles, also leads to efficient and fair IA assignments for all priority profiles. Similarly, any preference profile that leads to efficient and fair IA assignments for all priority profiles, also leads to efficient and fair DA assignments for all priority profiles. Therefore, we can say that DA performs better than IA, which itself performs better than TTC in terms of these two requirements. 10

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). 11 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 keep A as the set of schools, even if some schools have no seats at xS . This is ¨ an adaptation of consistency in Bu (2015) and Kojima and Unver (2014) to the variable-population setting, which is also referred to as “pre-assignment invariance” in Harless (2016). On the other hand, it is also possible to update students’ preferences to be defined only over the schools with available seats in the reduced economy. Jaramillo (2017) formulates this notion of consistency and figures out “minimal consistent enlargements” of the rules.

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Proof. We divide the proof into two parts. N N N . Consider two particular types of preference Part 1. We define Pmax-min and show that Pmax-min = Pttc

profiles: (a) profiles at which for each pair i, j ∈ N , Pi = Pj ; (b) 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 (a) 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 (b) 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 for all priorities. 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); as for their (k + 1)-th schools, there are two possibilities. (1) They can have diversified preferences over their (k + 1)-th schools so that all students can be assigned to their (k + 1)-th most preferred schools or better (“almost” no conflict). (2) Otherwise, all students, who are not assigned to their k-th most preferred schools or better, have the same (k + 1)-th most preferred school and all of them except for one can be assigned to that school (again, maximal conflict). The formal definition is as follows. Let P0 ∈ P and P ∈ P N and k ∈ {1, · · · , |A|}. Let Ak (P0 ) ≡ {a ∈ A : |U (P0 , a)| = k} be the k-th most preferred school at P0 and let Ak (P ) ≡ ∪i∈N Ak (Pi ) be the collection of k-th most preferred schools. Without loss of generality, let A0 (P ) ≡ ∅. Note that all students have the same k-th most preferred school under P if and only if |Ak (P )| = 1. Let Ak0 (P ) ≡ P Sk l a∈Ak (P ) qa . l=0 A (P ). Recall that qAk (P ) = 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) each a ∈ Ak+1 (P ) satisfies qa ≥ n − qAk (P ) − 1 or |{i ∈ N : Ak+1 (Pi ) = a}| ≤ qa . 0

Note that for each P ∈ P N , 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 some other students. ¯ Condition (i) is relevant to 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 relevant to the students’ P ¯ (k¯ + 1)-th most preferred schools: if a school in Ak+1 (P ) has a capacity less than n − b∈Ak (P ) qb − 1, 0 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 max-min the collection of all composites of max-min conflict profiles.12 12

Here are some examples.

Example 1. Preference profiles in P N max-min Let N ≡ {1, 2, 3, 4, 5}, A ≡ {a, b, c, d, e}, and q = (1, 1, 1, 2, 2).

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N We first show that if a preference profile is in Pmax-min , then TTC and IA are fair for all priorities. N Proposition 1. For each P ∈ Pmax-min and each ≺∈ ΣA , TTC(≺, P ) and IA(≺, P ) are fair at (≺, P ). N Proof. Let P ∈ Pmax-min and ≺∈ ΣA .

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

{0, 1, · · · , |A|} be the largest number such that

|A0 (P )|, |A1 (P )|, · · ·

, |Ak (P )|

Let k ∈

≤ 1. Then, all stu-

dents 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 some other students. 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, all students point to a1 and then a1 is assigned to the student with the highest priority at ≺a1 ; at Step 2, a1 is assigned to the student with the highest priority at ≺a1 among the remaining students; and so on. Until Step qA10 (P ) , a1 is assigned to qa1 students with at least qa1 -th priority at ≺a1 . From the next step down to Step qA20 (P ) , 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 in Ak0 (P ), then the algorithm terminates before Ak+1 (P ) is considered. Otherwise, the algorithm proceeds to the next step to assign schools in Ak+1 (P ). Let n ¯ be the number of students who still remain in the algorithm Pk (that is, n ¯ ≡ n − t=1 qat ). These students point to their (k + 1)-th most preferred schools. There are two possibilities. Case 1: For some a ∈ Ak+1 (P ), qa ≥ n ¯ − 1 and the number of students pointing to a 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 this step point to a. By the definition of TTC, all except for one applicant with the lowest priority at ≺a are assigned a in the following (¯ n − 1) steps. At the last step, the one remaining student points (or applies) to his (k + 2)-th most preferred school and he is assigned this school.

P1 a b c .. .

P2 a b d .. .

P3 a b d .. .

P1 a .. .

P2 b .. .

P4 a b d .. .

P5 a b e .. .

P3 c .. .

P4 d .. .

P5 d .. .

P1 a d b .. .

P2 a d b .. .

P3 a d c .. .

P1 a b .. .

P2 a c .. .

P4 a d c .. .

P5 a d c .. .

P3 a d .. .

P4 a d .. .

P5 a e .. .

P1 a b c d e

P2 a b c d e

P3 a b c d e

P4 a b c d e

P5 a b c d 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. There are two cases: (1) All students are assigned to their third most preferred schools or better (almost no conflict). This is when a or b is assigned to one of students 2, 3, and 4. (2) Not all students are assigned their third most preferred schools or better. This is when a and b are assigned to students 1 and 5. The remaining students compete for d and one of them is eventually assigned a school worse 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. Again, there are two cases: (1) All students are assigned to their third most preferred schools or better (almost no conflict). This is when a and d are assigned to one of students 1 and 2 and two of students 3, 4, and 5. (2) Not all students are assigned to their third most preferred schools or better. This is when a and d are assigned either to students 3, 4, and 5 or to students 1, 2, and one of students 3, 4, and 5. The two remaining students compete for the same school and one of them is eventually assigned to a school worse than this school. The last profile exhibits maximal conflict over all schools. 

9

Case 2: For each a ∈ Ak+1 (P ), either (i) qa < n ¯ − 1 or (ii) qa ≥ n ¯ − 1 and the number of students N pointing to a does not exceed qa . Consider a ∈ Ak+1 (P ) satisfying (i). As P ∈ Pmax-min , the number of students pointing to a does not exceed qa . Together with other schools satisfying (ii), all students

remaining at this step will be assigned their (k + 1)-th preferred schools in the end, respectively. 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 N is fair at (≺, P ). The same arguments apply to prove that for each P ∈ Pmax-min and each ≺∈ ΣA ,

IA(≺, P ) is fair at (≺, P ). We omit it. The coincidence of the three rules follows from Proposition 1, Remark 2, and the fact that TTC and IA are efficient. N Corollary 1. For each P ∈ Pmax-min and each ≺∈ ΣA , DA(≺, P ) = IA(≺, P ) = T T C(≺, P ). N The structure of profiles in Pmax-min is quite restrictive, but we next show that, for TTC to be fair

for all priority profiles, a preference profile should have this structure. That is, if a preference profile N does not belong to Pmax-min , there is a priority profile under which the TTC assignment is not fair. N Proposition 2. Let P ∈ P N . If for each ≺∈ ΣA , TTC(≺, P ) is fair at (≺, P ), then P ∈ Pmax-min . N Proof. Suppose that P ∈ / Pmax-min . Let k ∈ {0, · · · , |A| − 1} be the largest number such that |A0 (P )|, |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 some other students. For each t ∈ {1, · · · , k}, let at be the t-th most preferred school of all N students (that is, At (P ) = {at }). Since P ∈ / Pmax-min , there is a ∈ Ak+1 (P ) such that qa < n−qAk (P ) −1 0

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 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. 13

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

10

• Each other school: students are prioritized in decreasing order of their labels. Apply TTC to this economy. From the definition of TTC, it follows 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. N Remark 4. From the definition of Pmax-min , it is easy to see that identical preference profiles – identical down to the colleges that are actually allocated to students – are the only Cartesian products N. of individual preference restrictions that belong to Pttc N N. By Proposition 1 and Proposition 2, we conclude that Pmax-min = Pttc N ⊆ P N ⊆ P N . The first inclusion relation follows from Proposition 1 and Part 2. We show that Pttc ia da

the second inclusion relation follows from Remark 2. N ( P N . Let P ∈ P N be Part 2.1. We now show if there is a ¯ ∈ A such that qa¯ ≤ n − 2, then Pttc ia

such 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 ≺∈ ΣA and apply IA to (≺, P ). Since qa¯ ≤ n − 2, we have P ∈ / Pttc ia

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. 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. Part 2.2. We next show that if there is a pair of distinct schools a, b ∈ A such that qa + qb ≤ n − 1, N ( P N . Let P ∈ P N be such that then Pia da

11



P1 : a,  P2 : a,      Pqa : a,

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

c, d, · · · , b c, d, · · · , b .. . c, d, · · · , b

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. 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 12

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 } 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 higher priority at a than any student in {1, · · · , qa }, IA(≺, P ) is not fair. 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 for all preferences; ΣA ttc (Kesten, 2006): the domain of priority profiles at which TTC is fair for all preferences;

ΣA ia (Kumano, 2013): the domain of priority profiles at which IA is fair for all preferences. As illustrated in Figure 1, the containment relations between priority profiles are quite different from those between preference profiles. Also notice that ΣA ia = ∅ for many economies (Kumano, 2013), N 6= ∅ for all economies. but Pia

Remark 5. Although preferences and priorities seem to play symmetric roles in the definition of a school choice problem, they are not symmetric components of the problem for the following reasons. First, in evaluating efficiency of an assignment, only the students’ preferences, not the schools’ priorities, are taken into account. Second, DA and IA are defined by algorithms in which the students first “propose” and then schools make (tentative or final) decisions. That is, students and schools (and their preferences and priorities) play different roles in these algorithms. Note, however, that TTC assigns symmetric roles to students and schools when they point to the other side, even if TTC is also based on a student-proposing algorithm. Lastly and obviously, each school may be assigned to several students, but each student is assigned to exactly one school.

3.2. Result on Consistency As in the previous section, we define the sets of preference profiles on which the aforementioned rules are consistent for all priorities; 14

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

13

N

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

P ia : the domain of preference profiles at which IA is consistent for all priorities, and N

P da : the domain of preference profiles at which DA is consistent for all priorities. N

That is, P ∈ P ttc if and only if for each ≺∈ ΣA , TTC is consistent at (P, ≺). The other two domains are defined similarly. Here is our second main result. Theorem 2. N N (1) P ttc ⊆ P da and the inclusion is proper if there is a ∈ A such that qa ≤ n − 2 or there is a pair a, b ∈ A such that qa + qb ≤ n − 1, and N

N

(2) P da ⊆ P ia = P N and the inclusion is proper if there is a pair a, b ∈ A such that qa + qb ≤ n − 1. Theorem 2 says that in terms of consistency, IA performs better than DA, which itself performs better than TTC. This is a different conclusion from Theorem 1, which is not surprising. Depending on the requirements we consider, the rules may well be evaluated differently. Theorems 1 and 2 exemplify how to apply our approach. Proof. We divide the proof into three parts. N

N

N . We start by showing that P N Part 1. We prove P ttc = Pttc ttc ⊆ Pttc . Suppose otherwise: then, N

N . Since P ∈ N , there is k ∈ {0, · · · , |A| − 1}, the largest number such that there exists P ∈ P ttc \ Pttc / Pttc |A0 (P )|, |A1 (P )|, · · · , |Ak (P )| ≤ 1 (that is, k cannot be |A|). There also exists a ∈ Ak+1 (P ) such that

qa < n − qAk (P ) − 1 and there are more than qa students who rank a as their (k + 1)-th most preferred 0

school. Next, consider the priority profile ≺ and the resulting TTC assignment x provided in the proof of Proposition 2. Consider now the reduced economy of (A, N, q, ≺, P ) at (x, N \ {qa + 2}). According to the TTC algorithm applied to e(x, N \ {qa + 2}), student qa is assigned a and student (qa + 1) is assigned a school less preferred than a, a violation of consistency. N ⊆ P N . Let P ∈ P N , ≺∈ ΣA , and x ≡ TTC(A, N, q, ≺, P ). Let S ⊆ N We next show that Pttc ttc ttc and x0 ≡ TTC(e(x, S)). We show that (xi )i∈S = x0 . Let k ∈ {0, · · · , |A|} be the largest number such that |A0 (P )|, · · · , |Ak (P )| ≤ 1. For each t ∈ {0, 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 ) = ∅ and without loss of generality, let q∅ ≡ 0. 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 ) runs as follows. In the first qa1 steps, a1 is assigned to the students with at least qa1 -th priority according to ≺a1 (denote these students by N1 ); In the following qa2 steps, a2 is assigned to the students with at least qa2 -th priority among the remaining students according to ≺a2 (denote these students by N2 ); and so on. Therefore, for each pair l, m ∈ {1, · · · , k} with l < m, all students in Nl have higher priorities at ≺al than those in Nm in the TTC algorithm for the original economy (denote this statement by (∗)). 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 .15 For each i, j ∈ S and each a ∈ A, i ≺a j if and only 15

That is, a1 is assigned to the students with at least xS a1 -th priority according to ≺a1 |S ; a2 is assigned to the students with at least xS a2 -th priority among the remaining students according to ≺a2 |S ; and so on.

14

if i ≺a |S j. By (∗), we conclude that the assignments of Ak0 (P ) are the same as in x for the students in S. Now we work with the remaining students. Assignments of Ak+1 (P ): Since we know that the TTC assignments of Ak0 (P ) in the original economy and the reduced economy are equal, the sets of students who are not assigned any school in Ak0 (P ) are also equal. Now, consider the steps of TTC algorithms for the original economy and the reduced economy where each of these students applies to his (k + 1)-th most preferred school. N , the TTC assignment for the original economy is such that all students present at this Since P ∈ Pttc step, except for at most one, are assigned schools in Ak+1 (P ), as shown in the proof of Proposition 1.

Therefore, we have two cases to consider. Case 1: All students present at this step are assigned their (k + 1)-th most preferred schools in the original economy. Apply TTC to the reduced economy. Note that the number of available seats at each school and the number of applicants to the school are equal. Therefore, TTC also assigns each student his (k + 1)-th most preferred school. Therefore, x0 = (xi )i∈S . Case 2: All students present at this step, except for one, are assigned their (k+1)-th most preferred schools in the original economy. 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 this step, (i) qa∗ = n − qAk (P ) − 1 should hold and 0

(ii) all students present at this step should apply to a∗ .16 Moreover, i∗ has the lowest priority at a∗ than all other students present at this step. 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 schools 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 other students in S are assigned a∗ . N

N

N = P . We first show that P N ⊆ P . Suppose otherwise. Then, there Part 2. We now prove Pda da da da

are P ∈ P N and ≺∈ ΣA 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 the assignment such that for each i ∈ S, x00i ≡ x0i and for each i ∈ / S, x00i ≡ xi . Since x00 Pareto dominates x at P , x is not efficient at P , a contradiction. N

N . Suppose otherwise. Then, there are P ∈ P N and ≺∈ ΣA such We next prove that P da ⊆ Pda

that DA is consistent at (≺, P ), but DA(≺, P ) ≡ x 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 assignment y. Choose y ∈ X with the smallest |S|. Note that {xi : i ∈ S} = {yi : i ∈ S} and that N We assume that i∗ is rejected by a∗ . If qa∗ < n − qAk (P ) − 1, then, since P ∈ Pttc , the number of applicants to a∗ at 0 ∗ this step does not exceed qa . Then, all of them are accepted, contradicting the assumption. If qa∗ > n − qAk (P ) − 1, then 0 school a∗ can accommodate all applicants at this step, because the number of all students present at this step is n−qAk (P ) . 0 This again contradicts the assumption. Therefore, qa∗ = n − qAk (P ) − 1. On the other hand, suppose that any student 0 present at this step applies to a school other than a∗ . Then, the number of applicants of a∗ is at most (n − qAk (P ) − 1), 0 which is exactly qa∗ . Therefore, a∗ can accommodate all applicants at this step, contradicting the assumption. 16

15

(yi )i∈S Pareto dominates (xi )i∈S in the reduced economy e(x, S).17 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, P i2 .. .

P i1 .. .

···

P i3 .. .

P is .. .

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

xis−1 (≡ yis ) .. .

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 some step, say Step Mt , of the DA algorithm.18 Choose a student it∗ so that 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 (t∗ 6= s∗ ). 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. Altogether, we have xit∗ −1 Pis∗ xis∗ −1 Pis∗ xis∗ , which also implies that s∗ 6= t∗ − 1. We now have ···

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 can be made at x by reallocating the seats. We now show that S 0 ( S = {i1 , · · · , is }. Since s∗ , t∗ ∈ {1, · · · , s} and s∗ 6= t∗ −1, we have three possibilities. First, s∗ < t∗ −1 and t∗ 6= 1. Then, 1 ≤ s∗ < t∗ −1 < s and therefore, is ∈ / S0. Second, s∗ < t∗ − 1 and t∗ = 1.19 Since s∗ 6= t∗ = 1, we should have 2 ≤ s∗ . Therefore, i1 ∈ / S 0 . Third, s∗ > t∗ − 1. Since s∗ 6= t∗ , we have t∗ − 1 < s∗ − 1 < s∗ and S 0 = {is∗ , is∗ +1 , · · · , is , i1 , i2 , · · · , it∗ −1 }. We therefore have is∗ −1 ∈ / S 0 . In all cases, we have S 0 ( S that contradicts the assumption that S was chosen to have the smallest cardinality. Part 3. We lastly prove the proper inclusion relations in (1) and (2). The proper inclusion in (1) N

follows from Theorem 1 and Parts 1 and 2 of the proof above. Next, since IA is consistent, P ia = P N 17

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}. 18 Let i0 ≡ is . 19 Recall that the statement is modulo s.

16

A

N

Σia = ΣA

P ia = P N N

N =P Pda da

A

ΣA da = Σda ΣA ttc

N Pia

A Σttc

N

N =P Pttc ttc

ΣA ia

(a) Figure 2.

(b)

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

(Harless, 2016). We lastly show that there are P ∈ P N and ≺∈ ΣA such that DA(P, ≺) is not efficient if there is a pair a, b ∈ A such that qa + qb ≤ n − 1.20 Let P ∈ P N be such that   P1 : a, b, · · · Pqa +1 : b, a, · · ·  P2 : a, b, · · · Pqa +2 : b, a · · ·      .. ..   . .    Pqa : a, b, · · · Pqa +qb : b, a, · · ·  Pqa +qb +1 : a, c, · · · while all other students can have any preferences in P. Note that student (qa + qb + 1) exists because qa + qb ≤ n − 1. Let ≺∈ ΣA be such that   ≺a : 1, 2, 3, · · · , (qa − 1), (qa + qb ), (qa + qb + 1), qa , · · · ≺b : qa , (qa + 1), · · · , (qa + qb ), 1, 2, 3, · · · , n while all other schools can have any priorities in Σ. When DA is applied to this problem, student (qa +qb ) is assigned school a and student qa is assigned school b, a violation of efficiency. This result is summarized in Figure 2(b). Similarly, we can also define the sets of priority profiles to achieve consistency: A

Σttc (Kesten, 2006): the domain of priority profiles at which TTC is consistent for all preferences, A

Σia : the domain of priority profiles at which IA is consistent for all preferences, and A Σda (Ergin, 2002): the domain of priority profiles at which DA is consistent for all preferences, which are illustrated in Figure 2(a). Again, we see that the inclusion relations among these priority profiles are quite different from what we obtain for preference profiles. We conclude with the complete relations between all preference domains that follow from Theorems 1 and 2. N

N

N

N ⊆ PN ⊆ P N Corollary 2. P ttc = Pttc da = Pda ⊆ P ia . ia 20

N Since DA is not efficient, Pda ( PN .

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4. Concluding Remark We now conclude by discussing a few relevant issues.

4.1. Restrictions on Both Sides: Priorities and Preferences 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. That is, 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 requirements 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 requirements of the rule, or as the domain we define with a “conservative” criterion. Our conclusion from Theorem 1 and Theorem 2 should also be viewed as one conservative way of evaluating these rules. We may take a less conservative approach instead, by assuming that priorities also have certain structures. We then have to work with combinations of restrictions both on priorities and preferences. This approach can be more practical and interesting in applications, but we come up with the following question. What are the 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 flexibility to some extent. We find that this question has to be answered on the grounds of 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.

4.2. Other Preference Domains In this paper, we introduced the domains of preference profiles to compare the performance of the three N and P N , we find that there are alternative ways to rules. Although we could not fully characterize Pia da describe these domains: N if and only if for each ≺, the IA algorithm for (≺, P ) • A preference profile P belongs to Pia is such that, in every step, no student is rejected by a school that already accepted some students in an earlier step.21

The domain for DA can be described by means of a well-known notion of “rejection chain” (Kojima and Manea, 2010): N if and only if for each ≺, the DA algorithm for • A preference profile P belongs to Pda

(≺, P ) does not make a sequence of rejection chains such that (i) student i forms the first 21 Suppose that for some ≺, a student, say i, is rejected by a school at a step, but the school already accepted another student, say j, in an earlier step. Modify the priority of this school such that all students who are accepted to the school, except for j, are placed above j and student i is now placed just above j. It is easy to see that the IA assignment resulting from this new priority profile and the initial preference profile is not fair.

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and the last rejection chains of the sequence and (ii) all students in the sequence, except for i, are assigned to the schools that they apply in these rejection chains.22 In a similar manner, we can describe the domain for TTC: N if and only if for each ≺, the TTC algorithm for • A preference profile P belongs to Pttc

(≺, P ) is such that whenever there is a trading cycle involving school a, everyone should be assigned a school that is weakly better than a.23

4.3. Other Requirements In this paper, we focused on three standard requirements, efficiency, fairness, and consistency in evaluating rules in school choice problems. However, it is possible that the planner finds other requirements more desirable and important. We can still apply our approach by replacing efficiency, fairness, and consistency with them and figure out the corresponding preference domains. Examples include the following requirements: Strategy-proofness: no matter what other students’ preferences are, a student should not be better off by misreporting his preference. Capacity monotonicity: when capacities of all schools weakly decrease, then all student should be weakly worse off. Population monotonicity: when some students leave without being assigned any school seats, all remaining students should be weakly better off. Note that DA and TTC satisfy strategy-proofness, but IA does not. Note also that IA and DA satisfy the two monotonicity requirements, but TTC does not (Harless, 2016). Therefore, the comparisons between preference domains for each of these requirements are rather trivial: in terms of strategyproofness, TTC and DA perform as well as each other, but they do better than IA; in terms of each monotonicity requirement, IA and DA perform as well as each other, but they do better than TTC.24 Instead of working with a single requirement, it is also possible to consider combinations of several desirable requirements and look for the corresponding preference domains, just as we did for efficiency and fairness. This is another 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. 22

Student i forms a rejection chain if he applies to school a in the DA algorithm and school a accepts i temporarily but rejects another student j. Suppose that the rejected student j forms another rejection chain, and so on, until student i forms a rejection chain again. If all students, other than i, are assigned the schools that they apply in these rejection chains, these students can make a Pareto improvement from their final allocations. 23 Suppose that there are a trading cycle involving a and a student k who is assigned to a school worse than a in the TTC algorithm. Let student i yield a to student j along the trading cycle. This implies that i is place above j and k in ≺a . Now, modify ≺a such that j is placed below k. Then, the algorithm runs the same way, but stability is violated. 24 N We can also compare Pia and the domain of preference profiles at which no student can benefit from misrepresenting N N N his preferences. If we denote the latter domain by Pˆia , it is easy to check that Pˆia ( Pia (the proper inclusion can be shown with identical preferences). The similar comparisons can be made on monotonicity requirements for TTC.

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[21] Kumano, T. 2013. “Strategy-proofness and stability of the Boston mechanism: an almost impossibility result,” Journal of Public Economics 105, 23-29. [22] Pathak, P.A., and T. S¨ onmez, 2013. “School admissions reform in Chicago and England: comparing mechanisms by their vulnerability to manipulation,” American Economic Review 103, 80-106. [23] Thomson, W., 2013, “Strategy-proof allocation rules,” mimeo. [24] Thomson, W., 2015. “Consistent allocation rules,” mimeo. [25] Thomson, W. and L. Zhou, 1993, “Consistent solutions in atomless economies,” Econometrica 61, 575-587.

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