Constrained School Choice

idea would be to establish that at equilibrium the number of unassigned students equals
the number of unassigned students in stable matchings. The following two examples show
that in general this is not true. In other words, the number of unassigned students at equi-
librium is not inherited from that of the set of stable matchings. Given Proposition 6.2,
this in particular implies for the Student-Optimal Stable mechanism that the number of
unassigned students can vary from one equilibrium outcome to another.

Example 8.3 For both γ and τ: Less Assigned Students in an Equilibrium than
in Stable Matchings

Let I = {i₁, i₂, i₃} be the set of students, S = {s₁, s₂, s₃} be the set of schools, and
q = (1, 1, 1) be the capacity vector. The students’ preferences P and the priority structure
f are given in the table below. One easily verifies that strategy profile Q given below is
a Nash equilibrium in Γ^γ(P, 2) and Γ^τ (P, 2).

Since γ(Q) = τ(Q) = {{i1, s1}, {i3, s3}, {i2}, {s2}} andγ(P) = {{i1, s1}, {i2, s3}, {i3, s2}},
there are less assigned students at γ(Q) = τ(Q) than in any stable matching.          ^

Example 8.4 For both γ and τ: More Assigned Students in an Equilibrium
than in Stable Matchings

Let I = {i₁, i₂, i₃} be the set of students, S = {s₁, s₂, s₃} be the set of schools, and
q = (1, 1, 1) be the capacity vector. The students’ preferences P and the priority structure
f are given in the table below. One easily verifies that strategy profile Q given below is
a Nash equilibrium in Γ^γ(P, 2) and Γ^τ (P, 2).

Since γ(Q) = τ(Q) = {{i1, s2}, {i2, s3}, {i3, s1}} andγ(P) = {{i2, s3}, {i3, s2}, {i1}, {s1}},
there are more assigned students at γ(Q) = τ(Q) than in any stable matching.        ^

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