Constrained School Choice

We do obtain a positive result for γ if we restrict ourselves to equilibria in trunca-
tions. More precisely, the following proposition says that if a profile of truncations is a
Nash equilibrium in the game Γ^γ(P, k) then the set of assigned students at the equilib-
rium coincides with the set of assigned students at any stable matching. In fact, each
Nash equilibrium in truncations in the game Γ^γ (P, k) yields a matching that is either
the student-optimal stable matching γ(P) or Pareto dominates γ(P). For a matching μ,
denote M(μ) for the set of assigned students, i.e., M(μ) := {i ∈ I : μ(i) = i}.

Proposition 8.5 Let P be a school choice problem. Let 1 ≤ k ≤ m. If P^k ∈ E^γ(P, k),
then M(γ(P^k)) = M(γ(P)). In fact, γ(P^k)Riγ(P) for all i ∈ I.

For τ we cannot obtain a similar result as the following proposition shows.

Proposition 8.6 Let P be a school choice problem. Let 1 ≤ k ≤ m. If P^k ∈ E^τ (P, k),
then possibly |M(τ(P^k))| < |M(γ(P))| or |M(τ(P^k))| > |M(γ(P))|.

9 Discussion

We studied in this paper the stability and efficiency of Nash equilibrium outcomes in a
school choice problem when either BOS, SOSM, or TTC is used. At first sight, the most
robust mechanism is BOS, for Nash equilibrium outcomes are always stable. In all other
cases we need to impose a condition on the priority structure to guarantee stability or
efficiency.²³ The problem is that these conditions are very restrictive, and hence not likely
to be met in practice.²⁴ Also, it is interesting to note that for SOSM, the implementability
of efficient matchings implies the implementability of stable matchings (see Figure 1). This
is not the case for TTC.

Presumably then, constraining students’ choices is a very costly policy. It de facto
forces them to strategize, which in turns may slash the designer or the policy maker’s in-
terest for using either SOSM or TTC. The results we obtained should be contrasted with
experimental real-life data, however. From the experimental side, Calsamiglia, Haeringer,
and Klijn (2008) show that constraining choices, although having a clear impact on the

²³Other recent papers on implementation in various settings of two-sided matching include Pais (2008),
Shinotsuka and Takamiya (2003), Sotomayor (2003), and Suh (2003).

²⁴Another negative feature of SOSM and TTC is that there are equilibria that match (unmatch) stu-
dents that are unassigned (assigned) at the stable matchings. In particular, the number of matched stu-
dents may vary within the set of equilibrium outcomes — see the examples in Haeringer and Klijn (2008).

27

<<   24   25   26   27   28   29   30   >>

More intriguing information