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 Harvard University

December 2012

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Main Results

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Consider the model of controlled school choice employing type speciﬁc constraints with lower bounds and upper bounds, in which the set of feasible and fair assignments may be empty.

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This research gives a suﬃcient condition on a schools’ priority proﬁle for the existence of feasible assignments that are fair and non-wasteful.

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Also introduces an algorithm which ﬁnds such an assignment under that suﬃcient condition.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Controlled School Choice

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Controlled school choice programs are speciﬁc school choice programs that try to balance the racial or socioeconomic status.

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Some of them introduce lower bounds and upper bounds of type speciﬁc quotas.

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Examples include New York City high school match, Miami-Dade County Public Schools, and Chicago Public Schools.

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With lower bounds and upper bounds, the set of feasible and fair assignments may be empty. ⇒ Need to (1) weaken the solution concept or (2) restrict priority structures.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Model A controlled school choice problem is (S, C , (qc )c∈C , PS , ÂC , T , τ, (q T , qT c )c∈C ) with c 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 ; 4. a students’ preference proﬁle PS = (Ps1 , ..., Psn ), where Ps is the strict preference relation of student s ∈ S over C ; 5. a schools’ priority proﬁle ÂC = (Âc1 , ..., Âcm ), where Âc is the strict priority ranking of school c ∈ C over S; 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 t1 , ..., q tk ) such that = (q tc1 , ..., q tck ) and q T qT c c = (q c ∑ c ∑ q tc ≤ q tc ≤ qc for all t ∈ T , and t∈T q tc ≤ qc ≤ t∈T q tc . Kentaro Tomoeda

Controlled School Choice with Hard Bounds

An Assignment

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). µ is feasible if 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 and every student is assigned to a school.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Fairness and Non-wastefulness Deﬁnition 1. s justiﬁably claims an empty slot at c under µ if 1. cPs µ(s) and |µ(c)| < qc , τ (s)

2. q µ(s) < |µτ (s) (µ(s))|, and τ (s)

3. |µτ (s) (c)| < q c

.

µ is non-wasteful if there is no student who justiﬁably claims an empty slot at any school. Deﬁnition 2. s justiﬁably envies s 0 at c under µ if there exists another µ0 such that 1. µ(s 0 ) = c, cPs µ(s) and s Âc s 0 , 2. µ0 (s) = c, µ0 (s 0 ) 6= c, and µ0 (ˆs ) = µ(ˆs ) for all ˆs ∈ S \ {s, s 0 }. µ is fair across types (or fair) if there is no student who justiﬁably envies any other student. Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Non-existence of Feasible and Fair Assignments [Theorem 1] Theorem 1. (Abdulkadiro˘ glu and Ehlers (2008) and Ehlers, Hafalir, Yenmez and Yildirim (2011)) The set of feasible assignments that are fair across types may be empty in a controlled school choice problem.

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

Kentaro Tomoeda

Âc3 s1 s2 s3 1 1 0 1 0

Ps1 c2 c3 c1

Ps2 c3 c2 c1

Ps3 c2 c3 c1

Controlled School Choice with Hard Bounds

Non-existence of Feasible and Fair Assignments [Theorem 1]

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

Four feasible assignments: ( ) ( c1 c2 c3 c1 µ1 = , µ2 = s1 s2 s3 s1 ( ) ( c1 c2 c3 c1 µ3 = , µ4 = s2 s3 s1 s2

P s2 c3 c2 c1

c2 c3 s3 s 2 c2 c3 s1 s 3

P s3 c2 c3 c1

) , ) .

In each assignment, there is a student who justiﬁably envies another student. Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Solutions in the Literature

Two papers to analyze weaker solution concepts. I

Abdulkadiro˘glu and Ehlers (2008) showed that the set of feasible assignments that are fair for the same type and constrained non-wasteful is not empty.

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Ehlers, Hafalir, Yenmez and Yildirim (2011) give another solution; they interpret the constraints as soft bounds and ﬁnd fair and non-wasteful assignments.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

A Suﬃcient Condition: Common Priority Order for Every Type

In this paper, I don’t weaken the solution concept but give a suﬃcient condition for the existence. Deﬁnition 4. A schools’ priority proﬁle ÂC has a common priority order for type t ∈ T if s Âc s 0 ⇔ s Âc 0 s 0 for any c, c 0 ∈ C and s, s 0 such that τ (s) = τ (s 0 ) = t. This is satisﬁed if priority structures are more coarse than the type classiﬁcation and the single tie-breaking is used.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Main Theorem

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.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

An Algorithm: CDAAR Under the suﬃcient condition, Controlled Student Proposing Deferred Acceptance Algorithm with Reproposal (CDAAR) achieves such an assignment. [Start] Construct an ordering f such that f (s) < f (s 0 ) ⇔ s Âc s 0 and students make proposals to schools in this order. Start from the empty assignment. 1. Student s with f (s) = 1 proposes to the most preferable school and s is assigned to that school if there is a feasible assignment. If rejected, s is in the rejected status at c. k. Student s with minimum f −1 (s) who is unassigned or whose assigned school is not the most preferable one among the schools with non-rejected status makes a proposal. If s justiﬁably claims an empty slot or envies someone, s is admitted, and if there was no empty slot, a student with the lowest priority is rejected. Otherwise, s is rejected. Kentaro Tomoeda

Controlled School Choice with Hard Bounds

An Algorithm: CDAAR

k. (contd) ˆs ’s status at cˆ changes from the rejected status to the non-rejected status if ˆs was in the rejected status and one of the following conditions holds (reproposal conditions): (i) ˆs ∈ Sτ (s) , f (ˆs ) > f (s) and ˆs was unfree at step k-1 but became free at step k. (ii) ˆs ∈ Sτ (s) , f (ˆs ) < f (s) and cˆ = νk (s). (iii) There is an empty slot at cˆ and ˆs has the highest priority for cˆ among students who justiﬁably claim an empty slot. [End] The algorithm ends when there is no student to make a proposal.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Sketch of Proof [Theorem 2]

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First, show that the CDAAR algorithm ends in ﬁnite steps under the condition. I

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Can show that there is no cycle of proposals.

Next, show that the achieved assignment is feasible, fair across types and non-wasteful. I

Can show that the enough reproposals are made.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Incentives

There is a negative result on the incentives of students. 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 (s 0 ) ⇔ s Âc s 0 for some c ∈ C and for any s, s 0 such that τ (s) = τ (s 0 ). Then, there is an ordering f with which the CDAAR mechanism is not dominant strategy incentive compatible.

Kentaro Tomoeda

Controlled School Choice with Hard Bounds

Summary

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Introduced a suﬃcient condition “common priority for the same type students” for the existence of feasible assignments that are fair and non-wasteful in the controlled school choice.

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The proof is done by an algorithm CDAAR, which ﬁnds such an assignment under that suﬃcient condition.

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CDAAR is not always incentive compatible with any possible ordering f .

Kentaro Tomoeda

Controlled School Choice with Hard Bounds