Smart Lotteries in School Choice: Ex-ante Pareto-Improvement with Ex-post Stability
Haris Aziz, Péter Biró, Gergely Csáji, Tom Demeulemeester
TL;DR
The paper tackles inefficiencies in school-choice mechanisms when priorities are coarse by introducing PIRMES, a smart lottery that achieves ex-ante sd-improvements while preserving ex-post stability. It formalizes a column-generation approach to construct an ex-post stable random matching that stochastically dominates a baseline $p$ and minimizes the average rank, decomposing the result into stable matchings for implementation. The authors prove coNP-completeness for constrained-sd-efficiency and establish hardness results in various regimes, while showing that the proposed method yields substantial welfare gains in synthetic and real data (including Estonian kindergartens) compared to standard post-hoc improvements like EE and EADA. Practically, the approach demonstrates significant ex-ante welfare gains and scalability via column generation, suggesting wide applicability of smart lotteries in markets with ties and coarse priorities.
Abstract
In a typical school choice application, the students have strict preferences over the schools while the schools have coarse priorities over the students based on their distance and their enrolled siblings. The outcome of a centralized admission mechanism is then usually obtained by the Deferred Acceptance (DA) algorithm with random tie-breaking. Therefore, every possible outcome of this mechanism is a stable solution for the coarse priorities that will arise with certain probability. This implies a probabilistic assignment, where the admission probability for each student-school pair is specified. In this paper, we propose a new efficiency-improving stable `smart lottery' mechanism. We aim to improve the probabilistic assignment ex-ante in a stochastic dominance sense, while ensuring that the improved random matching is still ex-post stable, meaning that it can be decomposed into stable matchings regarding the original coarse priorities. Therefore, this smart lottery mechanism can provide a clear Pareto-improvement in expectation for any cardinal utilities compared to the standard DA with lottery solution, without sacrificing the stability of the final outcome. We show that although the underlying computational problem is NP-hard, we can solve the problem by using advanced optimization techniques such as integer programming with column generation. We conduct computational experiments on generated and real instances. Our results show that the welfare gains by our mechanism are substantially larger than the expected gains by standard methods that realize efficiency improvements after ties have already been broken.