Controlled School Choice with Hard Bounds: Existence ...

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Controlled School Choice with Hard Bounds: Existence of Fair and Non-wasteful Assignments Kentaro Tomoeda∗ December 21, 2012

Abstract In the model of controlled school choice employing type speciﬁc constraints with lower bounds and upper bounds, it was shown that the set of feasible and fair assignments may be empty. This paper proposes a suﬃcient condition on a schools’ priority proﬁle for the existence of feasible assignments that are fair and non-wasteful. The condition is that the priority orders for the same type students are common across schools. We introduce an algorithm called Controlled Student Proposing Deferred Acceptance Algorithm with Reproposal (CDAAR) which ﬁnds such an assignment under that suﬃcient condition. For incentives of students, it is shown that CDAAR is not always dominant strategy incentive compatible under the suﬃcient condition.

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Introduction

In the United States, aﬃrmative action is an important method to create a diverse environment for students in schools or workers in ﬁrms. In the context of school choice, controlled school choice programs which try to balance the racial or socioeconomic status at schools as well as to expand students’ choices are implemented. There are many examples of school boards that implemented controlled school choice programs such as Boston Public Schools, Educational Option in New York City high school match, Miami-Dade County Public Schools, or Chicago Public Schools. In the literature of aﬃrmative actions in school choice problems, Abdulkadiro˘glu and S¨onmez (2003) and Abdulkadiro˘glu (2005) considered type-speciﬁc quotas as maximum numbers of students from each type who can be admitted to the schools. ∗

Harvard University, [email protected].

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Kojima (2010) considers a controlled school choice problem by incorporating the quota for majority students, and Hafalir, Yenmez and Yildirim (2011) analyzes it by employing both the majority quota and minority reserves. Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011) analyze the problem by introducing lower bounds of type speciﬁc constraints that schools must admit students of a certain type and upper bounds of type speciﬁc constraints that schools cannot admit more students of a certain type than the bounds. When we look at real-life instances of controlled school choice programs, there are some cases which require not only maximum quotas but the range of numbers or percentages of certain type students for schools. Educational Option in New York City and the Jeﬀerson County School District are the examples. Therefore, the model studied in Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011) is applicable and captures important features of controlled school choice problems. However, there is a diﬃculty in the model which has both lower and upper bounds for type speciﬁc constraints. By naturally extending the fairness notion to the model, they ﬁnd that there may not exist feasible and fair assignments. In order to cope with this problem, Abdulkadiro˘glu and Ehlers (2008) weakened the fairness and non-wasteful concepts to show the existence of the desirable assignments. Ehlers, Hafalir, Yenmez and Yildirim (2011) give another solution; they interpret the constraints as soft bounds and ﬁnd fair and non-wasteful assignments. In this research, I will analyze the same model as Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011) and give a suﬃcient condition on a schools’ priority proﬁle for the existence of feasible, fair and non-wasteful assignments. The suﬃcient condition I propose is consistency of priority orders for the same type students across schools. As I discuss in Section 3, while this condition does not seem to hold in some situations, it can be satisﬁed if school districts can alter the classiﬁcation of types of students or priority structures. Therefore, the main result suggests that even without weakening the solution concepts or changing the interpretation of constraints, it may be possible to implement fair and non-wasteful assignments. The rest of the paper proceeds as follows. Section 2 introduces the model. Section 3 presents the main results and section 4 considers the incentive properties of the algorithm I propose. Section 5 concludes.

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Model: Controlled School Choice

I will consider a controlled school choice problem which was studied by Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011). A con-

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trolled school choice problem is written as (S, C, (qc )c∈C , PS , ÂC , T, τ, (q Tc , q Tc )c∈C ) which consists of the following elements. 1. a ﬁnite set of students S = {s1 , ..., sn }; 2. a ﬁnite set of schools C = {c1 , ..., cm }; 3. a capacity vector q = (qc1 , ..., qcm ), where qc is the capacity of school c ∈ C or the number of seats in c ∈ C; 4. a students’ preference proﬁle PS = (Ps1 , ..., Psn ), where Ps is the strict preference relation of student s ∈ S over C, i.e. cPs c0 means that student s strictly prefers school c to school c0 ; 5. a schools’ priority proﬁle ÂC = (Âc1 , ..., Âcm ), where Âc is the strict priority ranking of school c ∈ C over S; s Âc s0 means that student s has higher priority than student s0 to be enrolled at school c; 6. a type space T = {t1 , ..., tk }; 7. a type function τ : S → T , where τ (s) is the type of student s; 8. for each school c, two vectors of type speciﬁc constraints q Tc = (q tc1 , ..., q tck ) P and q Tc = (q tc1 , ..., q tck ) such that q tc ≤ q tc ≤ qc for all t ∈ T , and t∈T q tc ≤ P qc ≤ t∈T q tc . The diﬀerence from other speciﬁcations of controlled school choice models comes from the last element. q tc is the minimal number of slots that school c must allocate to students of type t and q tc is the maximal number of slots that school c is allowed to allocate to students of type t. We call (q tc , q tc ) the ﬂoor and the ceiling for type t at school c. An assignment µ is a function from the set C ∪ S to the set of all subsets of C ∪ S such that 1. µ(s) ∈ C for every student s; 2. |µ(c)| ≤ qc and µ(c) ⊆ S for every school c; 3. µ(s) = c if and only if s ∈ µ(c). Here we have another notation: µt (c) denotes the students of type t that are assigned to school c, i.e., µt (c) = µ(c) ∩ St with St ≡ {s ∈ S|τ (s) = t}. A set of students S 0 ⊆ S respects (capacity and controlled choice) constraints at school c if |S 0 | ≤ qc and for every type t ∈ T , q tc ≤ |{s ∈ S 0 |τ (s) = t}| ≤ q tc . An assignment µ respects constraints if for every school c, µ(c) respects constraints at c, i.e., |µ(c)| ≤ qc holds and q tc ≤ |µt (c)| ≤ q tc holds for every type t ∈ T . An assignment µ is legally feasible if µ respects constraints and every student is assigned to a school. Ehlers, Hafalir, Yenmez and Yildirim (2011) propose another way to interpret the type speciﬁc constraints as soft bounds; schools can admit

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fewer students than their ﬂoors or more than their ceilings but they give highest priority to types who do not ﬁll their ﬂoors, medium priority to types who ﬁll their ﬂoors but not ceilings, and lowest priorities to types who ﬁll their ceilings. Since positive results are already shown under the idea of soft bounds, we will stick to the interpretation of constraints as hard bounds, i.e. an assignment µ is feasible only if µ respects constraints. Even when there are enough seats for students in total, a controlled school choice problem (with hard bounds) may not have a feasible solution when there are less students of a certain type than the sum of the ﬂoors for that type for all schools or there are more students of a certain type than the sum of the ceilings for that type for all school. Therefore, as in Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011), I will only analyze problems whose legal constraints ensures the existence of a legally feasible assignment. Next, let us introduce desirable properties of feasible assignments in the controlled school choice problem. The ﬁrst property in Deﬁnition 1 is about nonwastefulness as in Balinski and S¨onmez (1999), but since we have type speciﬁc constraints, students are allowed to claim an empty slot only when the legal constraints will not be violated when the student is assigned a slot. Deﬁnition 1. Student s justiﬁably claims an empty slot at school c under the feasible assignment µ if (nw1) cPs µ(s) and |µ(c)| < qc , τ (s)

(nw2) q µ(s) < |µτ (s) (µ(s))|, and τ (s)

(nw3) |µτ (s) (c)| < q c

.

A feasible assignment µ is non-wasteful if there is no student who justiﬁably claims an empty slot at any school. (nw1) means student s prefers an empty slot at school c to her own assignment, and (nw2) and (nw3) mean that legal constraints are not violated when s is assigned the empty slot without changing other students’ assignments. The second property is about no-envy, which is also widely used in the context of school choice. But due to the structure of controlled school choice, as in Deﬁnition 1, even when a student prefers a school to her own and there is a student with lower priority in the school, the envy is not justiﬁed if the student’s move violates the legal constraints. Deﬁnition 2 formally states the condition for a student to have justiﬁed envy. Deﬁnition 2. Student s justiﬁably envies student s0 at school c under the feasible assignment µ if there exists another feasible assignment µ0 such that (f1) µ(s0 ) = c, cPs µ(s) and s Âc s0 ,

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(f2) µ0 (s) = c, µ0 (s0 ) 6= c, and µ0 (ˆ s) = µ(ˆ s) for all sˆ ∈ S \ {s, s0 }. A feasible assignment µ is fair across types (or fair) if there is no student who justiﬁably envies any other student. (f1) is the normal condition which says that there is a student who has lower priority than s at school c which s prefers to her own assignment. (f2) means that (µ(c) \ {s0 }) ∪ {s} respects the controlled choice constraints at school c and student s0 can be assigned to school c0 = µ0 (s0 ) such that (µ(c0 ) \ {s}) ∪ {s0 } respects the controlled choice constraints at c0 . There is a weaker version of fairness where envy is justiﬁed only if both the envying student and the envied student are of the same type. Deﬁnition 3. Student s justiﬁably envies student s0 of the same type at school c under the feasible assignment µ if (f1*) µ(s0 ) = c, cPs µ(s) and s Âc s0 , and (f2*) τ (s) = τ (s0 ). A feasible assignment µ is fair for same types if there is no student who justiﬁably envies any other same type student. (f1*) is the same condition as (f1). Since student s and student s0 are of the same type by (f2*), the assignment is still feasible when students s and s0 exchange their slots. Namely, we can make µ0 in a way that µ0 (s) = µ(s0 ), µ0 (s0 ) = µ(s), and µ0 (ˆ s) = µ(ˆ s) for all sˆ ∈ S \ {s, s0 }.

3 Existence of Fair and Non-wasteful Assignments Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011) showed two impossibility results on compatibility of legal constraints and desirable properties such as fairness and non-wastefulness. Theorem 1. (Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011)) (1) The set of feasible assignments that are fair across types may be empty in a controlled school choice problem. (2) The set of feasible assignments that are both fair for same types and nonwasteful may be empty in a controlled school choice problem. I will give counterexamples for each of (1) and (2) used in the proof of the theorems in Abdulkadiro˘glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011).

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Example 1. (Ehlers, Hafalir, Yenmez and Yildirim (2011)) Consider the following problem consisting of schools {c1 , c2 , c3 } and {s1 , s2 , s3 }. Students s1 and s2 are of type t1 and student s3 is of type t2 . The schools’ priorities and students’ preferences are in the following table.

capacities ceiling for t1 ﬂoor for t1 ceiling for t2 ﬂoor for t2

Âc1 s2 s1 s3 1 1 1 1 0

Âc2 s2 s1 s3 1 1 0 1 0

Âc3 s1 s2 s3 1 1 0 1 0

Ps1 c2 c3 c1

Ps2 c3 c2 c1

Ps3 c2 c3 c1

There are four feasible assignments in Example 1: Ã ! Ã ! c1 c2 c3 c1 c2 c3 µ1 = , µ2 = , s1 s2 s3 s1 s3 s2 Ã µ3 =

c1 c2 c3 s2 s3 s1

!

Ã , µ4 =

c1 c2 c3 s2 s1 s3

! ,

but there is a student who justiﬁably envies another student in each assignment. Example 2. (Abdulkadiro˘glu and Ehlers (2008)) Consider the following problem consisting of schools {c1 , c2 , c3 } and {s1 , s2 }. Students s1 and s2 are of the same type t. The schools’ priorities and students’ preferences are in the following table.

capacities ceiling for t ﬂoor for t

Âc1 s2 s1

Âc2 s2 s1

Âc3 s1 s2

2 2 1

2 2 0

2 2 0

Ps1 c2 c3 c1

Ps2 c3 c2 c1

There are four feasible assignments in Example 2: Ã ! Ã ! c1 c2 c3 c1 c2 c3 µ1 = , µ2 = , s1 s2 ∅ s1 ∅ s2 Ã µ3 =

c1 c2 c3 s2 ∅ s1

!

Ã , µ4 =

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c1 c2 c3 s2 s1 ∅

! ,

but there is a student who justiﬁably envies another student or justiﬁably claims an empty slot in each assignment. As is seen in above examples, the non-existence of such assignments seems to stem from the properties of schools’ priority orderings in which the orders for two students of the same type are not consistent across schools. In both examples, s2 has a higher priority than s1 at c2 while s1 has a higher priority than s2 at c3 . If s2 had a higher priority than s1 at c3 , then µ2 would be an assignment which is both fair across types and non-wasteful in both examples. From this point of view, we could conjecture that consistency of priority orders for the same type students across schools might resolve the emptiness of the set of desirable assignments. In order to capture this idea, deﬁne a condition on a school’s priority proﬁle. Deﬁnition 4. A schools’ priority proﬁle ÂC has a common priority order for type t ∈ T if s Âc s0 ⇔ s Âc0 s0 for any c, c0 ∈ C and s, s0 such that τ (s) = τ (s0 ) = t. The main theorem shows that the condition that a schools’ priority proﬁle ÂC has a common priority order for every type t ∈ T is suﬃcient for the existence of feasible assignments which are both fair and non-wasteful. This condition may be strong and hard to be satisﬁed when the classiﬁcation of types is coarse. For instance, if the type set is {high income, low income} and there is a priority for students who live in each school’s walk zone, priority orders for high income students will diﬀer across schools in general. However, this can be modiﬁed by making a ﬁner type classiﬁcation, {high income, low income} × {c1 ’s walk zone, c2 ’s walk zone,...}. In this case, if there are no other priority requirements, it is natural to use a single tie breaking among students with the same priority and the condition of having a common priority order for every type is satisﬁed. Another possibility is changing priority structures. If the walk zone priority is abolished or replaced by priority that is given to (a subset of) high or low income students, the condition is satisﬁed. Therefore, with the modiﬁcation of type division rules or priority structures, this condition is not diﬃcult to be satisﬁed in reality. This existence is proved by a constructive way. Abdulkadiro˘glu and Ehlers (2008) proposed an algorithm Controlled Student Proposing Deferred Acceptance Algorithm (CDAA) to show that the set of feasible assignments that are fair for the same type and constrained non-wasteful is not empty. I will propose an algorithm called Controlled Student Proposing Deferred Acceptance Algorithm with Reproposal (CDAAR), which adds reproposal steps to the original CDAA.

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Controlled Student Proposing Deferred Acceptance Algorithm with Reproposal Start: Construct a strict ordering on the set of students f : S → {1, ..., n}. Students are allowed to make proposals to schools in the order of f −1 (1), f −1 (2),..., f −1 (n). We will always deﬁne a tentative assignment ν which allows students to be unassigned. The tentative assignment is such that it is possible to allocate the unassigned students to schools such that the resulting assignment is feasible. Let F denote the set of all feasible assignments and ν0 be the empty assignment, i.e. ν0 (s) = s for all s ∈ S. For each school c and each student s, s is either in the rejected status for c or the non-rejected status for c. At the start of the algorithm, every student is in the non-rejected status for every school c. 1: Let student f −1 (1) apply to the school which is ranked ﬁrst under Pf −1 (1) , say c1 . If there is some µ ∈ F such that µ(f −1 (1)) = c1 , then set ν1 (f −1 (1)) = c1 and ν1 (s) = ν0 (s) = s for all s ∈ S \ {f −1 (1)}. Otherwise f −1 (1)’s status for c1 changes to the rejected status and we set ν1 = ν0 . .. . k: If there is a student who is unassigned or whose assigned school is not the most preferable one among the schools for which she is in the non-rejected status, let s be the student with minimal number f (s) among those students. Let c be the school which is most preferred under Ps among the schools for which s is in the non-rejected status. (1) If there is µ ∈ F such that µ(s) = c and µ(s0 ) = νk−1 (s0 ) for all students s0 such that νk−1 (s0 ) 6= s0 , then student s justiﬁably claims an empty slot at school c under νk−1 . Then we set νk (s) = c and νk (s0 ) = νk−1 (s0 ) for all s0 ∈ S \ {s}. (2) If (1) is not true but there is a student such that student s justiﬁably envies that student at school c under νk−1 , then let s0 be the student who has the lowest priority under Âc among students who are justiﬁably envied by student s at school c under νk−1 . Then we set νk (s) = c, νk (s0 ) = s0 and νk (s00 ) = νk−1 (s00 ) for all s00 ∈ S \ {s, s0 }, i.e. school c rejects s0 and tentatively admits s. (3) If (1) and (2) are not true, we set νk (s) = νk−1 (s) and student s’s status for c changes to the rejected status. Not only student s who proposed to a school in step k but other students may

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change their status for some schools. We say a student sˆ is free at µ if ¯ ¯ ¯ ¯ s) and µ(s) = s} ∪ {ˆ s}¯ ¯{s ∈ Sτ (ˆs) |f (s) > f (ˆ n o X max 0, q τc (ˆs) − |µτ (ˆs) (c) \ {ˆ > s}| c∈C

Otherwise, sˆ is unfree at µ. A student sˆ’s status for school cˆ changes from the rejected status to the non-rejected status at the end of step k if sˆ was in the rejected status for school cˆ at νk−1 and one of the following conditions holds (we call them reproposal conditions): (i) sˆ ∈ Sτ (s) , f (ˆ s) > f (s) and sˆ was unfree at νk−1 but became free at νk . (ii) sˆ ∈ Sτ (s) , f (ˆ s) < f (s) and cˆ = νk (s). (iii) |νk (ˆ c)| < qcˆ and sˆ has the highest priority for cˆ among students who justiﬁably claim an empty slot under νk . End: The algorithm ends when there is no student who is unassigned or whose assigned school is not the most preferable one among the schools for which she is in the non-rejected status. Then the tentative assignments become ﬁnal. Obviously, this algorithm may not end in ﬁnite steps because of complex reproposal steps. However, owing to these reproposal steps, we allow students to repropose to schools at which they might justiﬁably claim an empty slot or justiﬁably envy some student if they did not repropose to. Therefore, though this algorithm may not work well in general, under some conditions it succeeds in producing an assignment and reducing justiﬁed envy and claims. First, we will show that CDAAR ends in ﬁnite steps when ÂC has a common priority order for every type and an ordering f of CDAAR is constructed in a certain way. Proposition 1. Suppose that the schools’ priority proﬁle ÂC has a common priority order for every type t ∈ T and that an ordering f of students in the CDAAR algorithm satisﬁes f (s) < f (s0 ) ⇔ s Âc s0 for some c ∈ C and for any s, s0 such that τ (s) = τ (s0 ). Then, the CDAAR algorithm ends in ﬁnite steps. Proof: Suppose that the CDAAR algorithm does not end for some problem where ÂC has a common priority order for every type, i.e. there is a step t such that there is a previous step t0 < t in which νt is equivalent to νt0 . Let t be the smallest step among such steps and t0 be the previous step with t0 < t and νt = νt0 . Consider a cycle from step t0 + 1 to step t. Suppose any student who is free (unfree) in some step in the cycle is always free (unfree) in other steps in the cycle. If there is a student who is always unfree in

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the cycle, then let sˆ be such a student. Since ¯ ¯ X n o ¯ ¯ τ (ˆ s) s) and νl (s) = s} ∪ {ˆ s}¯ = max 0, q τc (ˆs) − |νl (c) \ {ˆ s}| ¯{s ∈ Sτ (ˆs) |f (s) > f (ˆ c∈C τ (ˆ s)

holds for any step l in the cycle, at any school c which sˆ is admitted, q τc (ˆs) = |νl (c)| holds for any step l in the cycle. Hence, sˆ’s rejected status for any school which sˆ once proposed to cannot change to the non-rejected status by reproposal conditions (i) or (iii). Moreover, since a student s such that s ∈ Sτ (ˆs) and f (ˆ s) < f (s) cannot be assigned to a school which sˆ prefers to her own assignment, reproposal condition (ii) does not apply to sˆ. Therefore, sˆ is always assigned to the same school in the cycle. Now we can separate the set of students who are always unfree because they do not change their assignments in the cycle. With the set of students who are always free in the cycle, only the reproposal condition (iii) could apply and it is when there was a reproposal before step t0 . But a reproposal based on (iii) gives a empty slot to a student and does not drop a student from a school. Therefore, when only the set of students who are always free move, the algorithm ends and does not form a cycle. This is a contradiction. Then we know that there is at least one student who is free at some step in the cycle and unfree at some another step in the cycle. Let si be such a student, t00 ∈ [t0 , t] be a step when si becomes free from unfree, and cb be a school such that νt00 (si ) = cb . There are two types of changes at step t00 . 1. A student sj such that τ (sj ) = τ (si ) and f (sj ) < f (si ) dropped from a school ca at step t000 ∈ [t0 , t] and was assigned to a school cb such that q τc (si ) = b

i |νt00 −1 (cb )|. Let sk be the student who was assigned to ca at step t000 and dropped sj . Since sk was not assigned to ca at step t000 − 1 and this is a cycle, sk must leave ca after step t000 . There are two cases on how sk leaves ca after step t000 .

τ (s )

sk leaves ca by reproposing to cc with cc Psk ca : Since sk couldn’t be enrolled in cc at step t000 but is assigned to cc later, there is a school cd (which can be ca or cc ) which had a student sl who became free from unfree when another student sm such that τ (sm ) = τ (sl ) and f (sm ) < f (sl ) came to cd and dropped a student. In order to have this situation, sm must have been dropped from the previous school ce by the proposal of another student sn . Note that sn has the parallel structure as sk and sn cannot be si , sj , sk , sl or sm . sk is rejected by ca by the proposal of another student sp : One possibility is that sp leaves ca by reproposing to cc with cc Psp ca . If sp is rejected by ca , then there is another student sq who proposed to ca at that step. In this way, since this rejection doesn’t continue inﬁnite times, there is a student sr who was enrolled in ca after step t000 and left ca by reproposing to more preferable

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school. Then, since there is a student sr who left ca by reproposing to more preferable school and comes back to it later, there is a school cd (which can be ca or cc ) which had a student sl who became free from unfree when another student sm such that τ (sm ) = τ (sl ) and f (sm ) < f (sl ) came to cd and dropped a student. In order to have this situation, sm must have been dropped from the previous school ce by the proposal of another student sn . Note that sn has the parallel structure as sk and sn cannot be si , sj , sk , sl , sm , sr . According to these arguments, in any case, in order for sn to leave ce and again come back to it, we need another student so . In this way, there should be inﬁnitely many students and this is a contradiction. 2. When si was unfree at cb , a student sj such that τ (sj ) = τ (si ) and f (sj ) > f (si ) was assigned to ca with ca Psi cb before step t00 , and si became free by reproposing to ca based on condition (ii). In order to have this situation, si must have been rejected by ca at step t000 ∈ [t0 , t] and was assigned to school cb . Then, since sj can be admitted to ca later, there is a student sk who was assigned to ca at step t000 but must leave ca after step t000 . The argument is the same as case 1 by replacing the indices of si and sj . We have shown that there is no cycle and the CDAAR algorithm ends in ﬁnite steps. Because CDAAR has reproposal steps, it eliminates some of envy and claims to empty slots which could occur if it didn’t have reproposal steps. Actually it is shown that whenever CDAAR ends in ﬁnite steps, the assignment produced by CDAAR satisﬁes feasibility, fairness across types, and non-wastefulness. Proposition 2. When the CDAAR algorithm ends in ﬁnite steps, it produces a feasible assignment which is fair across types and non-wasteful. Proof: By the nature of the algorithm, every student is assigned to the most preferable school among schools for which her status is non-rejected. Let µ be the assignment produced by the CDAAR algorithm and consider a student s and a school c such that cPs µ(s). Think about the steps from when s proposed to c last time to the ﬁnal step. During these steps, s was always in the rejected status for c. Suppose s was free when she proposed to c. Since there was no reproposal condition (ii) which applied to s for c during those steps, s does not justiﬁably envy a student of the same type in µ(c). Since there was no reproposal condition (iii) which applied to s for c during those steps, students assigned to c were replaced by those who have even higher priorities and even when there was an empty slot, the student who justiﬁably claimed the empty slot with the highest priority was not s. Therefore, s cannot justiﬁably envy some student at c or claim an empty slot at c at the produced assignment µ.

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Suppose s was unfree when she proposed to c. Since there was no reproposal condition (i) which applied to s during those steps, s was unfree through the steps. This means that q τc (s) = |µτ (s) (c)| holds and it is not possible for s to justiﬁably envy some student of a diﬀerent type at c or claim an empty slot at c at µ. Moreover, because there was no reproposal condition (ii) which applied to s for c during those steps, s does not justiﬁably envy a student of the same type. Henceforth, the assignment produced by the CDAAR algorithm is fair across types and non-wasteful. Finally, I will formally state the main result and it is directly proved by the previous two propositions. Theorem 2. Suppose that the schools’ priority proﬁle ÂC has a common priority order for every type t ∈ T . Then there exists a feasible assignment that is fair across types and non-wasteful. Proof: Consider the CDAAR algorithm whose ordering f satisﬁes f (s) < f (s0 ) ⇔ s Âc s0 for some c ∈ C and for any s, s0 such that τ (s) = τ (s0 ). This is well-deﬁned since ÂC has a common priority order for every type t ∈ T . Then, it is straightforward from Proposition 1 and 2; the CDAAR algorithm ends in ﬁnite steps by Proposition 1 and it produces a feasible assignment which is fair across types and non-wasteful by Proposition 2.

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Incentives of CDAAR

Under the assumption of common priority orders for every type, CDAAR works well to produce a feasible assignment which is fair across types and non-wasteful. Since students’ preferences are usually private information, we need to consider their incentives for CDAAR under the assumption. A mechanism which makes truthful revelation of preferences a dominant strategy is called (dominant strategy) incentive compatible. Here, we state a negative result on the incentives of CDAAR. Theorem 3. Suppose that the schools’ priority proﬁle ÂC has a common priority order for every type t ∈ T and that an ordering f of students in the CDAAR algorithm satisﬁes f (s) < f (s0 ) ⇔ s Âc s0 for some c ∈ C and for any s, s0 such that τ (s) = τ (s0 ). Then, there is an ordering f with which the CDAAR mechanism is not dominant strategy incentive compatible.

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Proof: Consider the following problem consisting of schools {c1 , c2 , c3 , c4 } and {s1 , s2 , s3 , s4 , s5 }. Students s1 and s2 are of type t1 , student s3 and s4 are of type t2 and student s5 is of type t3 . The schools’ priorities and students’ preferences are in the following table.

capacities ceiling for t1 ﬂoor for t1 ceiling for t2 ﬂoor for t2 ceiling for t3 ﬂoor for t3

Âc 1 s1 s2 s3

Âc2 s5 s1

Âc3 s3 s4

Âc 4

2 2 1 2 0 2 0

1 1 0 1 0 1 0

1 1 0 1 1 1 0

2 2 0 2 0 2 0

Ps1 c2 c1

Ps2 c4 c1

Ps3 c1 c3 c4 c2

Ps4 c3 c4

Ps5 c2

Ordering for omitted schools and students in the table are arbitrary as long as s1 has a higher priority than s2 and s3 has a higher priority than s4 at any school. Consider an ordering f such that (f (1), f (2), f (3), f (4), f (5)) = (s1 , s3 , s4 , s2 , s5 ) which satisﬁes the assumption. Then, the assignment produced by CDAAR when all students truthfully represented their preferences is Ã ! c1 c2 c3 c4 µ= . s1 s5 s3 {s2 , s4 } Suppose student s3 misrepresents her preference to Ps03 such that c1 Ps03 c4 Ps03 c3 Ps03 c2 . Then, the assignment produced by CDAAR would be Ã ! c c c c 1 2 3 4 µ0 = . {s1 , s3 } s5 s4 s2 Since s3 does better oﬀ by stating Ps03 , there is an ordering f with which the CDAAR mechanism is not dominant strategy incentive compatible.

5

Concluding Remark

This paper proposed a suﬃcient condition for the existence of feasible, fair across types and non-wasteful assignments in controlled school choice problems. The

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CDAAR algorithm ﬁnds such an assignment when the suﬃcient condition that priority orders are common for the same type students across schools is satisﬁed. The implication to the real-life controlled school choice programs is that school districts may be able to ensure the condition to hold by modifying the type division rules or priority structures. Another result is that although CDAAR ﬁnds such a desirable assignment, it is shown not always to have a good incentive property. For further research, it will be possible to identify the necessary and suﬃcient condition for the existence by relaxing our strong condition. The condition could be more complex than the suﬃcient condition, but it is helpful if such a condition is derived. As for the incentive property, we have not examined whether CDAAR is not incentive compatible if a certain type of an ordering f is used. If we restrict the ordering f used in CDAAR, there might be a positive result for CDAAR with some type of orderings f .

References [1] Abdulkadiro˘glu, A. (2005): “College Admission with Aﬃrmative Action,” International Journal of Game Theory, 33, 535-549. [2] Abdulkadiro˘glu, A. and E. Ehlers (2008): “Controlled School Choice,” mimeo. [3] Abdulkadiro˘glu, A., and T. S¨onmez (2003): “School Choice: A Mechanism Design Approach,” American Economic Review, 93, 729-747. [4] Balinski, M., and T. S¨onmez (1999): “A Tale of Two Mechanisms: Student Placement,” Journal of Economic Theory, 84, 73-94. [5] Ehlers, E., I.E. Hafalir, M.B. Yenmez, and M.A. Yildirim, (2011): “School Choice with Controlled Choice Constraints: Hard Bounds versus Soft Bounds,” mimeo. [6] Hafalir, I.E., M.B. Yenmez, and M.A. Yildirim, (2011): “Eﬀective Aﬃrmative Action in School Choice,” mimeo. [7] Kojima, F (2010): “School Choice: Impossibilities for Aﬃrmative Action.”

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