Ergin-acyclic
Kesten-acyclic
X -acyclic
strongly X -acyclic
Figure 1: Venn diagram of the acyclicity conditions
We have made use of four different acyclicity conditions. Below we indicate the absence
or existence of each of the 12 possible logical implications between pairs of acyclicity
conditions.
• [Kesten-acyclicity ⇒ Ergin-acyclicity]: Kesten (2006, Lemma 1);
• - [Kesten-acyclicity ⇒ Ergin-acyclicity]: Example 2 in Kesten (2006) which is given
by fs1 = i1, i2, i3 and fs2 = i3, i1, i2, qs1 = 1 and qs2 = 2;
• [strong X -acyclicity ⇒ X -acyclicity]: immediate from definition;
• - [X -acyclicity ⇒ strong X -acyclicity]: Example 2 in Kesten (2006) which is given
by fs1 = i1, i2, i3 and fs2 = i3, i1, i2, qs1 = 1 and qs2 = 2;
• [strong X -acyclicity ⇒ Ergin-acyclicity]:
if ((s, s′), (i, j, l)) constitutes an Ergin-cycle, then ((s, s′), (i, l)) constitutes a weak
X -cycle;
• - [strong X -acyclicity ⇒ Kesten-acyclicity]:
fs1 = i1, i2 , i3 and fs2 = i3 , i1 , i2 , qs1 = 1 and qs2 = 3;
• - [X -acyclicity ⇒ Kesten-acyclicity]: follows from [strong X -acyclicity ⇒ X -acyclicity]
and - [strong X -acyclicity ⇒ Kesten-acyclicity];
19A proof that [Kesten-acyclicity and X -acyclicity ⇒ strong X -acyclicity] is as follows. Suppose that
the priority structure is Kesten-acyclic, X -acyclic, but not strongly X -acyclic. By Theorem 7.5(ii)⇒(i),
there is a school choice problem P with |S(P)| ≥ 2. By Theorem 6.8, S(P) = Oτ (P, k). By Theorem
7.2, Oτ (P, k) ⊆ PE(P). Hence, S(P) ⊆ PE(P) and |S(P)| ≥ 2, which contradicts the optimality of the
Student-Optimal Stable matching.
22