Constrained School Choice



Suppose fs1 (i2) < fs1 (i1). Since i1 γ(Q)(s2), setting i = i2, i = i1 , s = s1, s = s2,
Is = γ(Q)(s1)ip, and Is= γ(Q)(s2)i1 yields a weak X-cycle.

Suppose now fs1 (i1) < fs1 (i2). If fs1 (i3) < fs1 (i2), then we obtain again a weak X-
cycle by setting
i = i3, i' = i2, s = sɪ, s= S3, Is = γ(Q)(sι)ip, and Is= γ(Q)(s3)i2.
So, suppose fs1 (i2) < fs1 (i3). By repeating this reasoning with students i4, i5, . . . , ip-1 we
either obtain a weak X -cycle or establish that the priority ordering of school s
ɪ has the
following form:

fsι : ∙ ∙ ∙ Y(Q) (sɪ) ∙ ∙ ∙ iɪ ∙ ∙ ∙ i2 ∙ ∙ ∙ ip-2 ∙ ∙ ∙ ip-ɪ ∙ ∙ ∙

To deal with the latter case, recall that by construction, ip γ(Q)(s1) and ip-ɪ γ(Q)(sp).
So, we obtain a weak X-cycle by setting i = i
p, i' = ip-1, s = sɪ, s= sp, Is = γ(Q)(s1)ip,
and I
s= γ(Q)(Sp)ip-b
(ii) (i): Without loss of generality, let students iɪ = i and i2 = i and schools sɪ = s
and s
2 = s be the agents involved in a weak X -cycle. Without loss of generality we may
assume that I
s1 = {i3,..., iqs1 +1} and Is2 = {iqs1 +2,..., iqs1 +qs2}. Consider the students’
preferences P given below. (Unacceptable schools are not depicted.)

Pil

P

i2

P ...

i i3

P
_____iqs- +1

P
iqs1 +2

∙∙∙ P

_________________iqs- +qs¾

^^^⅛1 +qs¾ + 1_____

P

ггп

sɪ

s2

sɪ     sɪ

sɪ

s2

s2        s2

S2

sɪ

Note that for j = qs1 + qs2 + 1, . . . , n, student ij finds all schools unacceptable. One easily
verifies that the distinct matchings
are stable. So,
|S(P)| ≥ 2.

iɪ i2 i3
μι =

sɪ s2 sɪ

and

iɪ i2 i3
μs =

s2 sɪ sɪ


iqs1 iqs1 +2

sɪ       s2


iqs1 iqs1 +2

sɪ       s2


iqs1 +qs2   iqs1 +qs2


s2      iqs1 +qs2


iqs1 +qs2   iqs1 +qs2


s2      iqs1 +qs2


in
in


in
in


(i) (iii): Let P be a school choice problem with Q Eγ(P, k) such that γ(Q) / PE(P).
Suppose first that
|S(P)| ≥ 2. Then, by (i) (ii), f admits a weak X -cycle. So, suppose
|S(P)| = 1. By Lemma A.8, there exists Q Eγ(P, 1) with γ(Q) = γ(P) =: μ1. If
μ
1 PE(P) then by Theorem 1 of Ergin (2002) f admits an Ergin-cycle, which by

32



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