Strategy-proofness and Stability of the Boston Mechanism: An Almost Impossibility Result∗ Taro Kumano† Yokohama National University

February, 2013

Abstract Public school systems generally use one of the three competing mechanisms – the Boston mechanism, the deferred acceptance mechanism and the top trading cycle mechanism – for assigning students to specific schools. Although the literature generally claims that the Boston mechanism is Pareto efficient but neither stable nor strategy-proof, this study delineates a subset of school priority structures for which it fulfills all three criteria. We show that the Boston mechanism is stable if and only if it is strategy-proof if and only if the priority structure is strongly acyclic. However, we find that the condition of strong acyclicity is nearly impossible to satisfy: any priority structure is quasi-cyclic whenever there are two schools whose admission quotas are less than the number of students seeking admission. Keywords: Boston mechanism, stability, strategy-proofness, quasi-cyclicity, strong acyclicity JEL: C71, C78, D71, D78, J44 ∗

I am grateful to Haluk Ergin, Fuhito Kojima, John Nachbar, Kang Rong, Alvin Roth, Ryoko

Susukida, Yosuke Yasuda and the seminar participants at Kyushu University, Washington University in St. Louis and Yokohama National University for the comments and discussions. I especially thank the co-editor and two anonymous referees for elaborating the Discussion section. All errors are mine. † Email: [email protected]

1

1

Introduction

Many school districts worldwide assign students to schools through a central clearinghouse. Each student submits a ranking of schools he or she prefers to attend, each school sets a priority ranking of students, and some mechanism matches a student to a school. Three competing mechanisms are commonly used – the Boston mechanism, the deferred acceptance mechanism and the top trading cycle mechanism – each of which can be easily applied and is endowed with desirable properties. This study examines the Boston mechanism, currently used in Denver and Minneapolis, and explores whether it works better or worse than the other two1 . We adopt three desiderata in evaluating a school choice mechanism: Pareto efficiency, strategy-proofness, and stability. A mechanism is Pareto efficient if it returns an assignment which is not Pareto–dominated by any other feasible assignment. It is strategy-proof if no student is better off by misrepresenting his/her true preference for a school – i.e., there is no room for gaming a mechanism. It is stable if no student or student-school pairing has an incentive to deviate from an assignment. Given our desiderata, it is reasonable to examine the three widely used assignment mechanisms because of the following negative result. Observation 1 A Pareto-efficient, strategy-proof, and stable mechanism for matching students and schools does not generally exist2 . Even though no assignment mechanism satisfies all three properties simultaneously, each of the mechanisms we examine here exhibits one or two. Table 1 summarizes three properties of the three mechanisms. (“X” means that a mechanism satisfies the corresponding property, and “×” means not necessarily.) Table 1: Properties of the three mechanisms

1

DA

TTC

Boston

Pareto efficiency

×

X

X

Strategy-proofness

X

X

×

Stability

X

×

×

Boston’s public school systems abandoned its namesake Boston mechanism after the 2005–2006

school year. 2 Appendix B discusses Observation 1 in detail.

2

In the literature, because of the negative result, it is common to specify schools’ priority rankings (priority structures) for which a mechanism satisfies all the three properties. Ergin (2002) and Kesten (2006) examined the deferred acceptance and top trading cycle mechanisms and found conditions for which each mechanism is Pareto efficient, strategy-proof, and stable. Those conditions are called Erginacyclicity and Kesten-acyclicity, respectively. This paper characterizes priority structures for which the Boston mechanism is Pareto efficient, strategy-proof, and stable. We find that the Boston mechanism is strategy-proof if and only if it is stable if and only if the priority structure is strongly acyclic. It is important in practical application to know how frequently strong acyclicity occurs. As its primary contribution to the literature, this study demonstrates that no priority structure in practice is strongly acyclic. Stated more formally, if any given school system has at least two schools whose admission quotas are less than the number of students seeking admission, any priority structure is quasi-cyclic (we say that a priority structure is quasi-cyclic if it is not strongly acyclic). In short, we show that the Boston mechanism cannot be strategy-proof and/or stable in practice. Among prior studies examining the Boston mechanism, Hsu (2011) is most related to ours. He proposed a sufficient condition for dominance solvability of the preference revelation game induced by the Boston mechanism. His sufficient condition is both on a preference profile and a priority structure, whereas we consider only the priority structure. Because the Boston mechanism is not strategy-proof in general, one direction to explore is the stability of outcomes under a weaker solution concept, Nash equilibrium. Although the Boston mechanism is not stable under true preferences, Ergin and S¨onmez (2006) showed that the set of stable assignments is equivalent to that of Nash equilibrium outcomes. That is, the set of stable assignments of the Boston mechanism is Nash implementable. Note that a stable assignment obtained by Nash equilibrium is not necessarily Pareto efficient. Haeringer and Klijn (2009) further investigated when a stable assignment induced via Nash equilibrium is Pareto efficient, and it is characterized by a condition on a priority structure, called strong X-acyclicity3 . Among other works concerning the Boston mechanism with incomplete infor3

Section 4 discusses strong X-acyclicity.

3

mation about students’ utilities, Abdulkadiro˘glu, Che and Yasuda (2011) showed that when all students’ ordinal preferences are identical and all schools rank all students equally, the Boston mechanism ex ante Pareto-dominates the deferred acceptance mechanism. In characterizing the Boston mechanism, Kojima and ¨ Unver (2011) provided axioms: respecting preference rankings, consistency, resource monotonicity, and an auxiliary invariance property, and showed that a mechanism satisfies these axioms if and only if it is the Boston mechanism for some priority.

1.1

Deferred acceptance and top trading cycle mech-

anisms Gale and Shapley (1962) introduced the deferred acceptance mechanism used in Boston, New York and other cities. In 2005-06, Boston abandoned its eponymous Boston mechanism and adopted the student-proposing deferred acceptance mechanism4 . The deferred acceptance mechanism is stable and strategy-proof (Dubins and Freedman (1981) and Roth (1982)). Although not generally Pareto efficient, it is constrained efficient. That is, assignments obtained by the deferred acceptance mechanism are Pareto dominant among stable assignments. Ergin (2002) showed that the deferred acceptance mechanism is Pareto efficient if and only if a priority structure is Ergin-acyclic5 . Roughly speaking, a priority structure is Ergin-acyclic if all schools have similar priority rankings over students. Abdulkadiro˘glu and S¨onmez (2003) introduced the top trading cycle mechanism, as an adaptation of a trading mechanism proposed by Shapley and Scarf (1974) in the context of school choice. It is Pareto efficient and strategy-proof but not stable. Kesten (2006) characterized the stable top trading cycle mechanism by a condition on priority structures called Kesten-acyclicity6 . Kesten’s (2006) main theorem partially asserts that the top trading cycle mechanism is stable if and only if a priority structure is Kesten-acyclic. Note that Kesten-acyclicity implies Ergin-acyclcity, but the converse is not implied.

4

Hereafter, we refer to the student-proposing deferred acceptance mechanism as “the deferred ac-

ceptance mechanism.” 5 Section 4 discusses Ergin-acyclicity. 6 We discuss Kesten-acyclicity in Section 4.

4

2

Model

The number of students and schools is finite. The sets of students and schools are denoted by N and X, respectively. Each student i has a complete, transitive, and antisymmetric preference Ri over X ∪ {i}. A student i ranks both schools and him/herself, {i}, because it is usual in the United States that a student prefers home-schooling over attending schools. The strict part of Ri is written by Pi . Each school x has qx (≥ 1) seats and a complete, transitive, and antisymmetric priority x over the set of students. We denote a preference profile by R = (Ri )i∈N and a priority structure by = (x )x∈X . An assignment µ is a mapping from N to N ∪ X with (1) for all i ∈ N , µ(i) ∈ X ∪ {i} and (2) for all x ∈ X, |µ−1 (x)| ≤ qx . Let M denote the set of assignments. An assignment µ is individually rational if for all i ∈ N , µ(i)Ri i. A blocking pair is defined by a pairing of a student i and a school x such that xPi µ(i) and i x j for some j ∈ µ−1 (x). An assignment is non-wasteful if there is no pairing of a student i and a school x such that xPi µ(i) and |µ−1 (x)| < qx . An assignment is stable if it is individually rational and non-wasteful and there is no blocking pair. An assignment µ is Pareto efficient if there is no ν ∈ M such that, for all i, ν(i)Ri µ(i), and for some j, ν(i)Pj µ(j). Although school priority structures function similar to preferences, we disregard schools with respect to efficiency because they are thought of as a public good. Given  and q, a mechanism f is a mapping from the set of preference profiles to an assignment. The assignment of a student i under R is denoted by fi (R). A mechanism is stable or Pareto efficient if each outcome is stable or Pareto efficient, respectively. A mechanism f is said to be strategy-proof if there is no R and Ri0 such that fi (Ri0 , R−i )Pi fi (R). Hereafter, we indicate the Boston mechanism by f B.

2.1

Description of the Boston mechanism

Given  and q, the Boston mechanism f B works as follows: Step 1: Each student i applies to his/her first choice school, if any. Each school x accepts students on the basis of its priority ranking until all positions are filled, and then rejects all other applicants. .. .

5

Step t: Each student rejected at step t − 1 applies to his/her first choice school among schools the student has not been rejected yet, if any. Each school x accepts students on the basis of its priority ranking until its remaining positions are filled, and then rejects all others. The algorithm terminates when no student applies to a school.

?

Once students are accepted, their assignment is finalized. Observation 2 Given any  and q, f B is Pareto efficient. Generally, however, f B is neither stable nor strategy-proof. Example 1 Suppose there are three students, N = {i, j, k}, and two schools, X = {x, y}, with one seat apiece. Consider the following preferences and priorities:7 Ri :

x : i

x y

j k

y : k j i

Rj : y x Rk : x y (

Then

i

f B (R) =

j k

)

x y k

.

First, f B (R) is not stable since yPk fkB (R) = k and k y

(

)−1 f B (R) (y) = j.

Second, a student k has an incentive to misreport her true preference. Consider Rk0 : y. Then

( f

B

(Rk0 , R−k )

=

i

j k

x j y

) , ♦

and a student k becomes better off.

The Boston mechanism differs from the other two mechanisms in lacking the two properties, which might be the reason why it was replaced in several districts. However, it remains in use because its operation is easay for students and parents to understand. The condition concerning priority structures in Section 3 characterizes when the Boston mechanism is indeed Pareto efficient, strategy-proof, and stable. 7

We abbreviate Ri or x by listing schools or students in hierarchy of preferences or priorities. For

each student, only schools that are preferred to home-schooling are listed. The ranking of other schools is irrelevant in the mechanism. For example, Ri : xPi yPi i is written as Ri : x y.

6

3

Results

The Boston mechanism generally is neither strategy-proof nor stable, but those properties are sometimes satisfied when schools have a specific interdependence in their priorities. We determine when that is the case by a condition on priority structures. We first introduce our acyclic condition. Definition 1 We say that  is quasi-cyclic if there are distinct i, j, k ∈ N and x, y ∈ X such that (C) i x j y k, (S) there are two distinct sets Sx , Sy ⊆ N \{i, j, k} such that |Sx | = qx − 1 and |Sy | = qy − 1 and ∀` ∈ Sx , ` x j and ∀` ∈ Sy , ` y k. We say that  is strongly acyclic if it is not quasi-cyclic. We are now ready to state a characterization theorem. Theorem 1 For any  and q, the following are equivalent: (1) f B is strategy-proof (2) f B is stable (3)  is strongly acyclic Proof : Appendix A. When a priority structure is strongly acyclic, no student has the incentive to misrepresent his/her preferences, and the outcome of the Boston mechanism is Pareto efficient and stable. One surprising finding is that the strategy-proofness of the Boston mechanism is equivalent to its stability in our setting.

3.1

Impossibility Result

However, our strong acyclicity condition leaves almost no possiblity for the Boston mechanism to be stable and/or strategy-proof, as the following proposition explains: Proposition 1 If |N | ≥ 3, |X| ≥ 2 and there are two schools, x and y, such that qx + qy ≤ |N | − 1, then any  is quasi-cyclic.

7

Proof : Appendix A. Proposition 1 leads to a new impossibility for the Boston mechanism:  is strongly acyclic and the Boston mechanism is stable and strategy-proof only if, given that one school has |N |/2 seats, all other schools have at least |N |/2 seats8 . However, in practice, no school has |N |/2 or more seats. Therefore, the Boston mechanism is never stable and/or strategy-proof.

4

Discussion

4.1

Relationship among acyclicities

We analyze the relationship among the Boston mechanism, the deferred acceptance mechanism, and the top trading cycle mechanism. As noted, none is simultaneously Pareto efficient, strategy-proof, and stable. We compare the flexibility of the three mechanisms’ priority structures. The deferred acceptance mechanism, denoted as f DA , is stable and strategyproof but not Pareto efficient. Ergin (2002) characterized its Pareto efficiency by the following acyclic condition on : Definition 2 Ergin (2002)  is Ergin-cyclic if there are distinct i, j, k ∈ N and x, y ∈ X such that (C) i x j x k y i, (S) there are two distinct sets Sx , Sy ⊆ N \{i, j, k} such that |Sx | = qx − 1 and |Sy | = qy − 1 and ∀` ∈ Sx , ` x j and ∀` ∈ Sy , ` y i.  is said to be Ergin-acyclic if it is not Ergin-cyclic. Ergin’s main theorem partially asserts that the deferred acceptance mechanism is Pareto efficient, strategy-proof, and stable if and only if  is Ergin-acyclic. Observation 3 If  is Ergin-cyclic, it is quasi-cyclic.

8

More precisely, if a school x has qx < |N |/2 seats, then all other schools have more than |N | − qx

seats. Otherwise all schools have |N |/2 or more seats.

8

If a priority structure is Ergin-cyclic, then j x k y i and there is Sy such that |Sy | = qy − 1 and for all ` ∈ Sy , ` y i. Furthermore, there is Sx such that |Sx | = qx − 1 and for all ` ∈ Sx , ` x j so ` x k. Hence, it is quasi-cyclic. The top trading cycle mechanism, denoted as f T T C , is Pareto efficient and strategy-proof but not stable. Kesten (2006) characterized the stability of f T T C by the following acyclic condition on : Definition 3 Kesten (2006)  is Kesten-cyclic if there are distinct i, j, k ∈ N and x, y ∈ X such that (C) i x j x k and k y i y j (S) there is Sx ⊆ N \{i, j, k} such that |Sx | = qx −1 and Sx ⊆ Ux (i)∪[Ux (j)\Uy (k)] where Uz (`) = {m ∈ N |m x `}.  is Kesten-acyclic if it is not Kesten-cyclic. Kesten’s main theorem partially states that the top trading cycle mechanism is Pareto efficient, strategy-proof, and stable if and only if  is Kesten-acyclic. Note that a Kesten-acyclic priority structure is also Ergin-acyclic. Observation 4 Kesten (2006) If  is Ergin-cyclic, it is Kesten-cyclic. No logical relationship exists between Kesten-acyclicity and our strong acyclicity, as Figure 1 depicts. The set of stable assignments is Nash implementable in the Boston mechanism (Ergin and S¨onmez (2006)), but a stable assignment is not necessarily Pareto efficient. Haeringer and Klijn (2009) proposed strong X-acyclicity and proved that it is equivalent to that the stable assignment induced by some Nash equilibrium is Pareto efficient9 . On the other hand, we characterize that a Pareto efficient and stable assignment is a dominant strategy equilibrium outcome if and only if  is strongly acyclic. Definition 4 Haeringer and Klijn (2009)  is weakly X-cyclic if there are distinct i, j ∈ N and x, y ∈ X such that 9

They also show that strong X-acyclicity ensures that the set of stable assignments is singleton.

9

(C) i x j and j y i (S) There are distinct Sx ⊆ N \{i} and Sy ⊆ N \{j} such that |Sx | = qx − 1 and |Sy | = qy − 1 and ∀` ∈ Sx , ` x j and ∀` ∈ Sy , ` y i.  is strongly X-acyclic if it is not weakly X-cyclic. As with Kesten-acyclicity and strong acyclicity, strong X-acyclicity implies Erginacyclicity. Observation 5 Haeringer and Klijn (2009) If  is Ergin-cyclic, it is weakly X-cyclic. Figure 1 summarizes the relationship among the four acyclic priority structures. (Corresponding examples appear in Appendix C.) '

$

Ergin-acyclic

'

$

Kesten-acyclic $

' '

$

&

Strongly X-acyclic

%

Strongly acyclic

&

&

% %

&

%

Figure 1: Relationship among the four acyclic priority structures. There are two other notions of acyclicity – essential homogeneity (Kojima (2011a)) and virtual homogeneity (Hatfield, Kojima and Narita (2011)) – which characterize properties of mechanisms not discussed in this paper. Hatfield, Kojima and Narita (2011) showed that virtual homogeneity implies essential homogeneity and essential homogeneity implies strong X-acyclcitiy. To set our findings

10

within the context of the literature, let us note the relationship to them. Formal definitions are as follows: Definition 5 Kojima (2011a)  is essentially homogeneous if there do not exist x, y ∈ X and i, j ∈ N such that (C) i x j and j y i (S) There are distinct Sx , Sy ⊂ N \{i, j} such that |Sx | = qx − 1, |Sy | = qy − 1, for all ` ∈ Sx , ` x j, and for all ` ∈ Sy , ` y i. Definition 6 Hatfield, Kojima and Narita (2011)  is virtually homogeneous if r` (x) = r` (y) for all x, y ∈ X, and ` > min{qx |x ∈ X} where r` (x) means the `-th ranked student at school x. Figure 2 summarizes the relationship to strong acyclicity. (Corresponding examples are in Appendix C.) We conclude that strong acyclicity is logically unrelated to essential and virtual homogeneity. '

$

Strongly acyclic

' ' '

$ $ $

Virtually Homogeneous

&

Essentially Homogeneous &

% %

Strongly X-acyclic

&

% &

%

Figure 2: Relationship to essential and virtual homogeneity

11

4.2

Other properties of the Boston mechanism char-

acterized by strong acyclicity Other desirable properties of the deferred acceptance and the top trading cycle mechanisms are characterized by Ergin-acyclicity and Kesten-acyclicity, respectively. For instance, Ergin-acyclicity is equivalent to the deferred acceptance mechanism being group-strategy-proof, consistent (Ergin (2002)), or robustly stable (Kojima (2011b)). This section shows that when a priority structure is strongly acyclic, the Boston mechanism possesses further desirable properties – specifically, group strategyproofness, consistency, Maskin monotonicity, and robust stability, as well as strategyproofness and stability. We begin the demonstration by introducing formal definitions of those properties. 0 f is group strategy-proof if there is no N 0 ⊆ N with N 0 6= ∅, R, RN 0 = 0 ,R 0 0 Πi∈N 0 Ri0 such that fi (RN 0 −N 0 )Ri fi (R) for all i ∈ N and fj (RN 0 , R−N 0 )Pj fj (R)

for some j ∈ N 0 . Given  and q, an economy is written by E = (N, R, q, ). A smaller economy for E is denoted by E 0 = (N 0 , RN 0 , q 0 ,  |N 0 ) where N 0 ⊂ N with N 0 6= ∅, qx0 ≤ qx for all x ∈ X and  |N 0 = (x |N 0 )x∈X . The extended mechanism f¯ is defined as a map from any smaller economy E 0 for E to the outcome obtained by the mechanism for E 0 . Given E = (N, R, q, ), an assignment µ = f¯(E), and N 0 ⊂ N with N 0 6= ∅, the reduced problem r(E, µ, N 0 ) = (N 0 , RN 0 , q 0 ,  |N 0 ) where for all x ∈ X, q 0 = qx − |µ−1 (x)\N 0 |. f¯ is consistent if µ|N 0 = f¯(r(E, µ, N 0 )). We x

indicate the extended Boston mechanism as f¯B . A preference profile R0 is a Makin monotonic transformation of R at x if for all i ∈ N , y, z ∈ X with yRi0 x and zRi0 x, yRi0 z implies yRi z. f is Maskin monotonic if R0 is a Maskin monotonic transformation of R at f (R), then f (R) = f (R0 ). f is robustly stable if (1) f is stable, (2) f is strategy-proof, and (3) there are no i ∈ N , x ∈ X, R and Ri0 such that (a) xPi fi (R) and (b) i x j for some j ∈ ν −1 (x) or |ν −1 (x)| < qx , where ν = f (Ri0 , R−i ). Corollary 1 In addition to Theorem 1, for any  and q, the following are further equivalent: (3)  is strongly acyclic.

12

(4) f B is group strategy-proof. (5) f¯B is consistent. (6) f B is Maskin monotonic. (7) f B is robustly stable. Proof: Appendix A.

Appendix A

Proofs

A.1

Proof of Theorem 1

We prove (1) ⇒ (3) and (2) ⇒ (3), and then (3) ⇒ (2) and (3) ⇒ (1). (1) ⇒ (3) & (2) ⇒ (3) By way of contradiction. Suppose there is a quasi-cycle. Then consider the following preference profile R: for all ` ∈ Sx , x is their top choice, for all ` ∈ Sy , y is their top choice, Ri :

x i

Rj : x y j , Rk : y k and for any others, N \[{i, j, k} ∪ Sx ∪ Sy ], their top choice is not being matched. Then   f B (R) =  i

Sx

Sy

x...x

y...y

z }| { z }| { j k `1 . . . `qx −1 `01 . . . `0qy −1

x j y

N \[{i,j,k}∪Sx ∪Sy ]

z }| { `001 . . . `00n

  

`001 . . . `00n

Then yPj fjB (R) and j y k so that a pair j and y blocks the assignment. Hence f B is not stable. If j misrepresents Rj0 as y is his top choice. Then 

Sy

 f B (Rj0 , R−j ) =  i

Sx z }| { z }| { j k `1 . . . `qx −1 `01 . . . `0qy −1

x y k

x...x

13

y...y

N \[{i,j,k}∪Sx ∪Sy ]

z }| { `001 . . . `00n `001 . . . `00n

  

Therefore, y = fjB (Rj0 , R−j )Pj fjB (R) = j, which contradicts the assumption that f B is strategy-proof. (3) ⇒ (2) Suppose f B is unstable. Then there exists R such that f B (R) is unstable. Let f B (R) be µ. We will show that there is a quasi-cycle. Since µ is unstable, Pareto efficiency of f B implies that there is a blocking pair j and y such that yPj µ(j) and j y ` for some ` ∈ µ−1 (y). Take k such that ` y k for all ` ∈ µ−1 (y)\{k}. Then there are qy − 1 students who have higher priority than k in µ−1 (y). Let them be Sy . Since j is not assigned y under R, when j applies to y, there are already qy students accepted at y and hence it is not the first step. Because j applies to y under R, j should be rejected in all the previous steps, especially in the first step. In the first step, j applies to a school different from y, say x, and j is rejected because there are at least qx students who are of higher priority than j. Consider the top qx higher ranked students within the applicants at x in the first step. Let one of them be i and the other of them be Sx . Note that they are accepted in the first step, so their assignment is x at µ. Hence, i, j, k, Sx and Sy are distinct and, together with x and y, they consist a quasi-cycle. (3) ⇒ (1) Suppose a priority structure is strongly acyclic, but f B is not strategyproof. Then there are k, R and Rk0 such that fkB (Rk0 , R−k )Pk fkB (R). Let fkB (Rk0 , R−k ) and f B (R) be y and µ, respectively. From (3) ⇒ (2), f B is stable so that for all ` ∈ µ−1 (y), ` y k under R. Under (Rk0 , R−k ), k is assigned y, which implies, together with the procedure of the Boston mechanism, that there is a step t > 1 such that a school y is not filled until step t under R. Otherwise, since R−k is the same across R and (Rk0 , R−k ), if all µ−1 (y) apply to y in the first step, then it contradicts that k is assigned y under (Rk0 , R−k ). Note that applications of N \{k} in the first step is exactly the same both under R and (Rk0 , R−k ). Let j be the student who is assigned y under R and does not apply to y in the first step. Let µ−1 (y)\{j} be Sy . Note that j y k and ∀` ∈ Sy , ` y k by stability of f B (R). Then under R, j should be rejected by some school, say x, in

14

the first step. It is because there are at least qx students who is of higher priority than j and apply to x in the first step. Choose the top qx higher ranked students within them, and split those qx students into one student and the others, and let each of them be i and Sx . Clearly, i x j and |Sx | = qx − 1 and ∀` ∈ Sx , ` x j. Overall, i, j, k, Sx and Sy are all distinct, and together with x and y, they consist 

a quasi-cycle.

A.2

Proof of Proposition 1

By supposition, there are two schools x and y such that qx + qy ≤ |N | − 1. Fix  arbitrary. Rename students in each ranking of a school x and y as follows: x

j1 x . . . x jn

y

k1 y . . . y kn

where |N | = n. Note that the same student is named differently in those two schools. We will show that  is quasi-cyclic. (Case1) jn 6= kn . Since jn is the least ranked at x, kn is ranked better than jn at x. Then there are the other |N | − 2 students listed at x. Choose one student from |N |−2 students, and let her be i. Since qy ≥ 1 and by supposition, qx −1 ≤ |N |−3 so that it is possible to find qx − 1 students who are ranked higher than jn from N \{i, jn , kn } (possibly empty). Let them be Sx . For a school y, it is possible to choose qy − 1 students who are distinct from Sx ∪ {i, jn , kn } because |N | − |{z} 3 −(qx − 1) ≥ |N | − 3 − (|N | − qy − 2) | {z } i,jn ,kn

Sx

= qy − 1 Therefore, since i x jn y kn , i, jn , kn satisfy the condition (C), and both Sx and Sy follows the condition (S) of quasi-cyclicity. (Case2) jn = kn . At x, there are |N | − 2 students who are ranked higher than jn−1 . Choose i and qx − 1 students distinctly among them. Let the qx − 1 students be Sx . It is possible because qx ≤ |N | − 3. At y, since jn = kn , i and jn−1 are ranked higher than kn , furthermore, there are |N | − 3 students other than i and

15

jn−1 who are ranked higher than kn . As similar to Case 1, it is possible to find qy − 1 students who are ranked higher than kn and distinct from i, jn−1 and Sx . Let them be Sy . Now i x jn−1 y kn so that i, jn−1 , kn satisfy the condition of (C), and Sx and Sy satisfy the condition of (S) of quasi-cyclicity.

A.3



Proof of Corollary 1

We first show (3) ⇔ (4). Then (4) ⇔ (5) ⇔ (6) straightforwardly follows from Ergin (2002) and Kojima and Manea (2010). We then show the equivalence between (3) and (7). Lemma 1 If  is strongly acyclic, then f B = f DA . Proof: It is obvious for the cases that |N | ≤ 2 or |X| = 1. So we assume that |N | ≥ 3 and |X| ≥ 2. From Proposition 1, a strongly acyclic priority structure requires that if one school x with the least seats has qx < |N |/2 then all the other schools have |N | − qx seats or more, otherwise every school has |N |/2 or more seats. Then under both mechanisms there is at most one school whose seats are over-demanded. If there is no school whose seats are over-demanded at the first step, then both mechanisms work identically, and hence f B = f DA . Otherwise, there is only one school whose seats are over-demanded. Then when students who are rejected at the first step apply to their second preferred school, all other schools have more remaining seats than the number of students who are rejected at the first step, and hence both mechanisms end at the second step and the both outcomes are identical, which implies that f B = f DA .



Lemma 2 Given E, if  is strongly acyclic, then  |N 0 is also strongly acyclic for all r(E, µ, N 0 ). Proof: If not, then for some r(E, µ, N 0 ),  |N 0 is quasi-cyclic. Then there are distinct i, j, k ∈ N 0 , x, y ∈ X, and Sx0 , Sy0 ⊂ N 0 \{i, j, k} such that i x j y k, and |Sx0 | = qx0 − 1 and |Sy0 | = qy0 − 1 such that for all ` ∈ Sx0 , ` x j and for all ` ∈ Sy0 , ` y k. Suppose i∗ ∈ N \N 0 joins back in r(E, µ, N 0 ). If µ(i∗ ) 6= x, y, then  |N 0 ∪{i∗ } is also quasi-cyclic. Suppose without loss of generality µ(i∗ ) = x. If i∗ x j, then i, j, k, x, y, Sx∗ = Sx0 ∪ {i∗ } and Sy0 constitute a quasi-cycle. If j x i∗ , then

16

j, i∗ , k, x, y, Sx∗ = Sx0 ∪ {i} and Sy0 constitute a quasi-cycle. By applying the same argument to N \[N 0 ∪ {i∗ }] · · · N , we conclude that  is quasi-cyclic, a contradic

tion. Proof of Corollary 1: (3) ⇔ (4)

(⇒) When  is strongly acyclic, it is Ergin-acyclic. From Ergin (2002), f DA is group strategy-proof. By Lemma 1, f B is also group strategy-proof. (⇐) Group strategy-proofness of f B implies strategy-proofness of f B and by Theorem 1, strategy-proofness of f B implies strong acyclicity. (4) ⇔ (5) ⇔ (6) Since Ergin (2002) further shows the equivalence between the consistency of f¯DA and the group strategy-proofness of f DA , and Kojima and Manea (2010) show the equivalence between the Maskin monotonicity of f DA and the group strategyproonfness of f DA , when  is strongly acyclic, those equivalences are taken over to f B by Lemma 1 and 210 . (3) ⇔ (7) To compete the proof, we finally show the equivalence between strong acylicity and robust stability of f B . Since strong acyclicity implies Ergin-acyclicity and by Lemma 1, (3) ⇒ (7) follows from Kojima (2011b). To show the other direction, suppose that  is quasi cyclic. From Theorem 1, f B is neither stable nor strategyproof, and it contradicts that f B is robustly stable.

B



On the observation 1

We can show a slightly stronger claim. Claim 1 Ergin-acyclicity is the maximum domain for a mechanism being efficient and stable. Proof: Suppose a priority structure is Ergin-cyclic, then there are distinct i, j, k ∈ N , x, y ∈ X and Sx , Sy ⊂ N \{i, j, k} such that i x j x k y i, 10

Kojima and Manea (2010) show the equivalence of efficiency, Maskin monotonicity and group

strategy-proofness of f DA in a more general environment.

17

and |Sx | = qx − 1 such that ∀` ∈ Sx , ` x j and |Sy | = qy − 1 such that ∀` ∈ Sy , ` y i. Let ϕ be a stable mechanism, and consider the following preference profile: Ri : y x i Rj

: xj

Rk : x y k ∀` ∈ Sx , R` : x ` ∀` ∈ Sy , R` : y ` ∀` ∈ N 0 , R` : ` where N 0 = N \[{i, j, k} ∪ Sx ∪ Sy ]. Since a stable assignment is unique for the above priority structure and preference profile, that  Sy Sx z }| { z }| {  i j k ` ···` 1 qx −1 `qx · · · `qx +qy −2 µ= x j y x···x y...y

is, N0



z }| { `qx +qy −1 · · · `n−3  , `qx +qy −1 · · · `n−3

it must be that ϕ(R) = µ. However, µ is Pareto improved by an exchange of school seats between i and k. 

It is a desired contradiction.

C C.1

Examples Kesten-acyclicity, strong X-acyclicity, and strong

acyclicity We abbreviate Ergin-acyclicity, Kesten-acyclicity, strong X-acyclicity, and strong acyclcity as EA, KA, SXA, and SA, respectively. Example 2 SA but neither KA nor SXA N = {i, j, k} and X = {x, y}, x : i x j x k qx = 1 y : k y i y j

18

qy = 2

Example 3 KA and SA but not SXA (Haeringer and Klijn (2008)) N = {i, j} and X = {x, y}, x : i x j qx = 1 y : j y i qy = 1 Example 4 SXA and SA but not KA (Haeringer and Klijn (2008)) N = {i, j, k} and X = {x, y}, x : i x j x k qx = 1 y : k y i y j

qy = 3

Example 5 KA, SXA, and SA N = {i, j} and X = {x, y}, x : i x j qx = 1  y : i  y j qy = 1 Example 6 KA but neither SXA nor SA N = {i, j, k} and X = {x, y}, x : i x j x k qx = 1 y : i y k y j

qy = 1

Example 7 KA and SXA but not SA N = {i, j, k} and X = {x, y}, x : i x j x k qx = 1 y : i y j y k qy = 1 Example 8 SXA but neither KA nor SA N = {i, j, k} and X = {x, y, z}, x : i x j x k qx = 1 y : i y j y k qy = 1 z : k z i z j

qz = 3

Example 9 EA but neither KA, SXA nor SA N = {i, j, k} and X = {x, y, z}, x : i x j x k qx = 1 y : k y i y j

qy = 2

z : i z j z k

qz = 1

19

C.2

Strong X-acyclicity, essential homogeneity, vir-

tual homogeneity, and strong acyclicity We abbreviate strong acyclicity, strong X-acyclicity, essential homogeneity, and virtual homogeneity as SA, SXA, EH, and VH, respectively. Example 10 SA and SXA, but not EH N = {i, j, k} and X = {x, y}, x : k x i x j qx = 2 y : k x j y i qy = 2 Example 11 SA and EH, but not VH N = {i, j} and X = {x, y}, x : i x j qx = 1 y : j y i qy = 2 Example 12 SA and VH N = {i, j} and X = {x, y}, x : i x j qx = 1  y : i  y j qy = 1 Example 13 SXA but neither EH nor SA N = {i, j, k1 , k2 , k3 } and X = {x, y}, x : k1 x i x j x k2 x k3 qx = 2 y : k1 x j y i y k2 x k3 Example 14 EH but neither SA nor VH N = {i, j, k} and X = {x, y, z}, x : i x j x k qx = 1 y : j y i y k qy = 2 z : i z j z k

qz = 1

Example 15 VH but not SA N = {i, j, k} and X = {x, y}, x : i x j x k qx = 1 y : i y j y k qy = 1

20

qy = 2

References [1] A. Abdulkadiro˘glu, Y-K. Che, Y. Yasuda, Resolving Conflicting Preferences in School Choice: The “Boston Mechanism” Reconsidered, Amer. Econ. Review 101 (2011), 399–410. [2] A. Abdulkadiro˘glu, P. A. Pathak, A. E. Roth, Strategy-proofness versus efficiency in matching with indifferences: redesigning the NYC high school match, Amer. Econ. Review 99 (2009), 1954–1978. [3] A. Abdulkadiro˘glu, P. A. Pathak, A. E. Roth, T. S¨onmez, The Boston public school match, Amer. Econ. Review (Papers and Proceedings) 95 (2005), 368– 371. [4] A. Abdulkadiro˘glu, T. S¨onmez, School choice: a mechanism design approach, Amer. Econ. Review 93 (2003), 729–747. [5] L.E. Dubins, D.A. Freedman, Machiavelli and the Gale-Shapley algorithm, Amer. Math. Monthly 88 (1981), 485-494. [6] H. Ergin, Efficient resource allocation on the basis of priorities, Econometrica 70 (2002), 2489–2497. [7] H. Ergin, T. S¨onmez, Games of school choice under the Boston mechanism, J. Pub. Econ. 90 (2006), 215–237. [8] D. Gale, L. Shapley, College admissions and the stability of marriage, Amer. Math. Monthly 69 (1962), 9–15. [9] G. Haeringer, F. Klijn, Constrained school choice, J. Econ. Theory (2009), 1921–1947. [10] G. Haeringer, F. Klijn, Constrained school choice, unpublished manuscript (2008). [11] J. W. Hatfield, F. Kojima, and Y. Narita, Promoting School Competition Through School Choice: A Market Design Approach, unpublished manuscript (2011). [12] C-L. Hsu, When is Boston Mechanism Game Dominance Solvable? unpublished manuscript (2011). [13] O. Kesten, On Two Competing Mechanisms for Priority-based Allocation Problems, J. Econ. Theory 127 (2006), 155-171

21

[14] F. Kojima, Efficient Resource Allocation under Multi-unit Demand, unpublished manuscript (2011a). [15] F. Kojima, Robust stability in matching markets, Theoretical Econ. (2011b), 257–267. [16] F. Kojima, M. Manea, Axioms for Deferred Acceptance, Econometrica, (2010), 633–653. ¨ [17] F. Kojima, M. U. Unver, The ‘Boston’ School Choice Mechanism, unpublished manuscript (2010). [18] L. Shapley, H. Scarf, On Cores and Indivisibility, J. Math. Econ. (1974), 23–28. [19] A. E. Roth, The Economics of Matching: Stability and Incentives, Math. Oper. Research. (1982), 617–628. [20] A. E. Roth, M. Sotomayor, Two-Sided Matching: A Study in Game-Theoretic Modelling and Analysis, Cambridge University Press, Cambridge, England, [Econometric Society Monograph], 1990.

22