Smart Lotteries in School Choice: Ex-ante Pareto-Improvement with Ex-post Stability

2026-02-11

AI Summary

This work addresses the inefficiency of the standard Deferred Acceptance (DA) algorithm in school choice settings where coarse priorities—such as distance or sibling attendance—induce ties that are typically broken uniformly at random, yielding ex ante inefficient probabilistic assignments. The authors propose a novel “smart lottery” mechanism that, while strictly preserving ex post stability (i.e., every realized matching is stable under the original priority structure), achieves the first ex ante stochastic dominance Pareto improvement over the standard DA outcome. By integrating column generation with integer programming, the method efficiently solves the NP-hard optimization problem subject to stability constraints. Empirical evaluations on both synthetic and real-world data demonstrate significant welfare gains, consistently outperforming benchmark approaches that separate lottery-based tie-breaking from subsequent optimization.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• school choice
• ex-ante efficiency
• ex-post stability
• probabilistic assignment
• Pareto improvement

Innovation

• smart lottery
• ex-ante Pareto improvement
• ex-post stability
• school choice
• integer programming with column generation