Constrained School Choice

performance of the mechanisms, does not alter too much the relative hierarchy of SOSM,
TTC, and BOS (in this order) when one is concerned with diverse issues such as stability,
efficiency, truthtelling, or even social mobility. In 2003 the New York City Department
of Eduction (NYCDOE) adopted a centralized mechanism based on SOSM (Abdulka-
diroglu, Pathak, and Roth, 2005). Although choice in this mechanism is constrained,
Abdulkadiroglu, Pathak, and Roth (2008) provide evidence that over the years partici-
pants learned how to make sound choices. Also, the school district of Boston removed the
constraint on the length of submittable preference lists for the school year 2007-2008 (see
see Abdulkadiroglu, Pathak, Roth and Sonmez (2006)). This suggests that a (reasonable)
constraint would be desirable when the authorities change their mechanisms and adopt
either SOSM or TTC, and then after a few years the constraint could be dropped.

From a theoretical perspective, one possible extension of our model is the incorpora-
tion of incomplete information. Ehlers and Masso (2008) study a many-to-one matching
market with incomplete information. They show that at least for stable mechanisms (i.e.,
in particular SOSM) there is a strong link between the ordinal Bayesian Nash equilib-
ria under incomplete information and the Nash equilibria under complete information.²⁵
More precisely, Ehlers and Masso’s results show that a characterization of the equilib-
ria under complete information immediately leads to a characterization of the equilibria
under incomplete information.

A Appendix: Proofs for SOSM

Let Q ∈ Q^I. We denote DA(Q) for the application of the DA algorithm (with students
proposing) to Q. We will make use of the following two results to prove Theorem 5.3.

Lemma A.1 (Roth, 1982b, Lemma 1; cf. Roth and Sotomayor 1990, Lemma 4.8)

Let Q ∈ QI and i ∈ I. Let Qi ∈ Q be a preference list whose first choice is γ(Q)(i) if
γ(Q)(i) = i, and the empty list otherwise. Then, γ(Qi,Q_-i)(i) = γ(Q)(i).

Lemma A.2 For any school choice problem P and quota k, O^γ(P, k) ⊆ IR(P)∩NW(P).

Proof Let Q ∈ E^γ(P, k). It is immediate that γ(Q) ∈ IR(P). Suppose γ(Q) ∈
NW(P). Then, there are i ∈ I and s ∈ S with sP_iγ(Q)(i) and ∣γ(Q)(s)∣ < q_s∙ Let Q_i be

²⁵A strategy profile is an ordinal Bayesian Nash equilibrium is if it is a a Bayesian Nash equilibrium
for every von Neumann-Morgenstern utility representation of individuals’ true preferences.

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