Constrained School Choice

3.4 An Illustrative Example

We illustrate the impact of the quota on the length of submittable preference lists through
the following example.

Let I = {i1, i2, i3, i4} be the set of students, S = {s1, s2, s3} be the set of schools,
and q = (1, 2, 1) be the capacity vector. The students’ preferences P and the priority
structure f are given in the table below. So, for instance, Pi₁ = s2, s1, s3 and fs₁ (i1) <
fs₁ (i2) < fs₁ (i3) < fs₁ (i4).

One easily verifies that if there is no quota on the length of submittable preference lists
and if the students truthfully report their preference lists, then the mechanisms yield the
following three matchings:

β(P)   =   {{s1, i2}, {s2, i1, i4}, {s3, i3}}

γ(P)   =   {{s1, i1}, {s2, i3, i4}, {s3, i2}}

τ(P)   =   {{s1, i3}, {s2, i1, i4}, {s3, i2}}.

Note that if in a direct revelation game under γ or τ students could only submit a
list of 2 schools, student i₂ would remain unassigned (and the other students unaffected),
provided that each student submits the truncated list with his two most preferred schools.
Therefore, if students can only submit short preference lists, then (at least) student i2
ought to strategize (i.e., list school s₃) to ensure a seat at some (acceptable) school. In
particular, the profile of truncated preferences does not constitute a Nash equilibrium.
Under both mechanisms in the constrained setting, truncating one’s true preferences is in
general not a (weakly) dominant strategy.

4 Constrained Preference Revelation

Fix the priority ordering f and the capacities q. We consider the following school
choice procedure. Students are asked to submit (simultaneously) preference lists Q =

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