Constrained School Choice

• - [Ergin-acyclicity ⇒ X -acyclicity]:
f_s1 = i1, i2 and f_s2 = i2, i1 , q_s1 = q_s2 = 1;

• - [Ergin-acyclicity ⇒ strong X -acyclicity]: follows from [strong X -acyclicity ⇒
X -acyclicity] and - [Ergin-acyclicity ⇒ X -acyclicity];

• - [Kesten-acyclicity ⇒ X -acyclicity]:
f_s1 = i1, i2 and f_s2 = i2, i1 , q_s1 = q_s2 = 1;

• - [Kesten-acyclicity ⇒ strong X -acyclicity]: follows from [strong X -acyclicity ⇒
X -acyclicity] and - [Kesten-acyclicity ⇒ X -acyclicity];

• - [X -acyclicity ⇒ Ergin-acyclicity]:

f_s1 = i1, i4, i5 , i2 , i3 and f_s2 = i2 , i4, i5, i3 , i1 , q_s1 = q_s2 = 2.

Finally, we give an example of a priority structure that satisfies the requirements that are
associated with each of the 7 nodes in Figure 1. Note that all examples can be extended to
incorporate additional students or schools by (1) giving additional students lower priority
and (2) introducing multiple copies of the priority ordering of an existing school.

1. f_s1 = i1, i2 and f_s2 = i2, i1 , q_s1 = q_s2 = 1;

2. fs₁ = i1, i2, i3, i4 and fs₂ = i3, i1, i4, i2, qs₁ = 1 and qs₂ = 2;

3. any f where schools have identical priority over students;

4. fs₁ = i1, i2, i3 and fs₂ = i3, i1, i2, qs₁ = 1 and qs₂ = 3;

5. Example 2 in Kesten (2006) which is given by fs₁ = i1, i2, i3 and fs₂ = i3, i1, i2,
qs₁ = 1 and qs₂ = 2;

6. f_s1 = i1, i2 , i3 and f_s2 = i3 , i2 , i1 , q_s1 = 1 and q_s2 = 1;

7. fs₁ = i1, i4, i5, i2, i3 and fs₂ = i2, i4, i5, i3, i1, qs₁ = qs₂ = 2.

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