• - [Ergin-acyclicity ⇒ X -acyclicity]:
fs1 = i1, i2 and fs2 = i2, i1 , qs1 = qs2 = 1;
• - [Ergin-acyclicity ⇒ strong X -acyclicity]: follows from [strong X -acyclicity ⇒
X -acyclicity] and - [Ergin-acyclicity ⇒ X -acyclicity];
• - [Kesten-acyclicity ⇒ X -acyclicity]:
fs1 = i1, i2 and fs2 = i2, i1 , qs1 = qs2 = 1;
• - [Kesten-acyclicity ⇒ strong X -acyclicity]: follows from [strong X -acyclicity ⇒
X -acyclicity] and - [Kesten-acyclicity ⇒ X -acyclicity];
• - [X -acyclicity ⇒ Ergin-acyclicity]:
fs1 = i1, i4, i5 , i2 , i3 and fs2 = i2 , i4, i5, i3 , i1 , qs1 = qs2 = 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. fs1 = i1, i2 and fs2 = i2, i1 , qs1 = qs2 = 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. fs1 = i1, i2 , i3 and fs2 = i3 , i2 , i1 , qs1 = 1 and qs2 = 1;
7. fs1 = i1, i4, i5, i2, i3 and fs2 = i2, i4, i5, i3, i1, qs1 = qs2 = 2.
23
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