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

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.

💡 Research Summary

The paper addresses the classic school‑choice problem where students have strict preferences over schools and schools possess coarse priority structures (e.g., distance, siblings). In practice, the Deferred Acceptance (DA) algorithm is run after breaking ties randomly, producing a probability distribution over stable matchings. While each realized matching is stable with respect to the coarse priorities, the induced random matching is generally not ex‑ante efficient: there exist other random matchings that stochastically dominate it (sd‑dominance), meaning every student’s probability of receiving a school at least as good as a given one is weakly higher and strictly higher for some pair.

The authors propose a “smart lottery” mechanism that improves the ex‑ante welfare of the random matching while preserving the possibility of implementing the outcome through a lottery over stable matchings. They formalize this as the class of Parity‑Improving Random Matchings under Ex‑post Stability (PIRMES) mechanisms and focus on a concrete instance called DA‑PIRMES, which starts from the standard DA‑with‑uniform tie‑breaking outcome.

The mechanism works in three steps:

1. STEP 0 – Compute the baseline random matching (p) (e.g., DA with uniform tie‑breaking).
2. STEP 1 – Search for a random matching (q) that (i) sd‑dominates (p), (ii) maximizes the improvement measured by average rank reduction, and (iii) is ex‑post stable, i.e., it can be expressed as a convex combination of matchings that are stable with respect to the original coarse priorities. If no such (q) exists, the algorithm returns (p).
3. STEP 2 – Decompose (q) into its constituent stable matchings and run a lottery (the “smart lottery”) that selects one of them according to the decomposition weights.

The central computational challenge is that finding an ex‑post stable, sd‑efficient random matching is NP‑hard. The authors prove several hardness results: (a) checking whether a given random matching can be improved under the above constraints is NP‑hard even with unit capacities and short preference lists; (b) finding a weakly stable matching that Pareto‑dominates another weakly stable matching by a specified rank improvement is NP‑hard; (c) deciding whether a deterministic weakly stable matching can be sd‑dominated by any ex‑post stable random matching is also NP‑hard. Moreover, they show that no mechanism can simultaneously be strategy‑proof, ex‑post stable, and constrained‑sd‑efficient.

To overcome these difficulties, the paper introduces an integer programming model with column generation. The master problem minimizes the average rank of the random matching subject to the sd‑dominance constraints, while the pricing subproblem searches for a new weakly stable matching that can improve the objective. Starting from a small set of stable matchings, the algorithm iteratively adds columns until optimality is certified. This approach leverages the structure of the stable‑matching polytope and recent advances in decomposition techniques.

Empirical evaluation is conducted on both synthetic instances (varying numbers of students, schools, capacities, and tie structures) and a real‑world dataset from an Estonian kindergarten allocation system. The results demonstrate that DA‑PIRMES consistently yields random matchings with substantially lower average ranks than the baseline DA‑with‑lottery and the deterministic improvement method of Erdil and Ergin (EE). In many settings, the fraction of students who strictly improve and the magnitude of their rank gains are several times larger than under EE. Notably, in some preference‑priority configurations, even the Efficiency‑Adjusted Deferred Acceptance (EADA) mechanism— which improves ex‑post welfare but does not guarantee stability—fails to sd‑dominate the baseline, whereas DA‑PIRMES succeeds while preserving stability.

The paper’s contributions are threefold:

1. Conceptual – It introduces the notion of ex‑ante Pareto improvement under the strict requirement of ex‑post stability, thereby reconciling efficiency and stability in a randomized setting.
2. Theoretical – It establishes a suite of complexity results that delineate the computational limits of achieving constrained sd‑efficiency.
3. Algorithmic – It provides a practical column‑generation based integer programming framework that solves realistic instances to optimality or near‑optimality, enabling the deployment of smart lotteries in actual school‑choice programs.

Overall, the work advances market‑design theory for many‑to‑one matching markets by showing that, despite inherent computational hardness, sophisticated operations‑research tools can produce mechanisms that are both welfare‑enhancing and legally defensible (because the final outcomes remain stable with respect to the original coarse priorities). This opens the door for policy makers to replace naïve tie‑breaking lotteries with smarter, data‑driven lotteries that deliver measurable gains for students without compromising fairness guarantees.