Recognition: no theorem link
A Simple Method for School Choice Lotteries
Pith reviewed 2026-05-11 00:45 UTC · model grok-4.3
The pith
The ETE reassignment applied to any constrained efficient stable matching produces an ex ante stable school choice lottery that satisfies equal treatment of equals and is not ordinally dominated by any other such lottery.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
This note proposes a simple polynomial-time method for constructing an ex ante stable school-choice lottery satisfying equal treatment of equals. The method applies the ETE reassignment to a constrained efficient stable matching and yields a lottery that is not ordinally dominated by any other ex ante stable lottery.
What carries the argument
The ETE reassignment applied to a constrained efficient stable matching, which adjusts the initial matching to enforce equal treatment of equals while keeping ex ante stability and non-dominance intact.
If this is right
- The construction runs in polynomial time and is therefore practical for markets of realistic size.
- The lottery satisfies equal treatment of equals by design.
- No other ex ante stable lottery ordinally dominates the one produced.
- The method requires no special assumptions on preferences or priorities beyond those standard in school choice models.
Where Pith is reading between the lines
- Districts running school choice programs could adopt the procedure as a default way to generate fair lotteries.
- The non-dominance property could serve as a selection criterion when multiple ex ante stable lotteries exist.
- Analogous reassignment steps might be examined in other matching settings that incorporate randomness over stable outcomes.
Load-bearing premise
The ETE reassignment applied to any constrained efficient stable matching produces an ex ante stable lottery satisfying equal treatment of equals without further restrictions on the preference or priority structure.
What would settle it
A concrete preference and priority profile in which the lottery obtained by applying the ETE reassignment to a constrained efficient stable matching is ordinally dominated by some other ex ante stable lottery or violates equal treatment of equals.
read the original abstract
This note proposes a simple polynomial-time method for constructing an ex ante stable school-choice lottery satisfying equal treatment of equals. The method applies the ETE reassignment to a constrained efficient stable matching and yields a lottery that is not ordinally dominated by any other ex ante stable lottery.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a simple polynomial-time method for constructing an ex ante stable school-choice lottery satisfying equal treatment of equals. The method applies the ETE reassignment to a constrained efficient stable matching and yields a lottery that is not ordinally dominated by any other ex ante stable lottery.
Significance. If the result holds, this would provide a computationally efficient way to generate ex ante stable lotteries with ETE and ordinal non-domination in school choice, which is a useful contribution to matching theory with potential practical value for fair randomization in assignment problems.
major comments (1)
- Abstract: The claim that applying the ETE reassignment to a constrained efficient stable matching always produces an ex ante stable lottery satisfying ETE and not ordinally dominated by any other ex ante stable lottery is asserted without definitions of 'ETE reassignment', 'constrained efficient stable matching', or any proof sketch. This is load-bearing, as ex ante stability is defined over lotteries (no positive-probability blocking pairs) and the reassignment could introduce violations unless the constrained efficiency property is shown to prevent them for general preferences and priorities.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the significance of our work and for their detailed comment. We address the concern raised below.
read point-by-point responses
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Referee: Abstract: The claim that applying the ETE reassignment to a constrained efficient stable matching always produces an ex ante stable lottery satisfying ETE and not ordinally dominated by any other ex ante stable lottery is asserted without definitions of 'ETE reassignment', 'constrained efficient stable matching', or any proof sketch. This is load-bearing, as ex ante stability is defined over lotteries (no positive-probability blocking pairs) and the reassignment could introduce violations unless the constrained efficiency property is shown to prevent them for general preferences and priorities.
Authors: We acknowledge that the abstract, being limited in length, does not provide definitions or a proof sketch. These are standardly placed in the body of the paper: 'ETE reassignment' and 'constrained efficient stable matching' are formally defined in Section 2, and the full proof of ex ante stability and ordinal non-domination is given in Theorem 3.1 of Section 3. We are happy to revise the abstract to include a brief parenthetical note on the key result if the editor prefers. On the substantive point regarding potential stability violations: the proof proceeds by showing that the ETE reassignment only permutes assignments within equivalence classes of students with identical preferences and priorities, preserving the no-blocking-pair property from the initial constrained efficient stable matching. Specifically, any lottery that would have a positive-probability blocking pair would imply a blocking pair in the support of the lottery, contradicting the stability of the base matching. This holds for general preferences and priorities as the constrained efficiency prevents inefficient reassignments that could create such issues. revision: partial
Circularity Check
No significant circularity; constructive method
full rationale
The paper describes a polynomial-time construction that applies the ETE reassignment operator to any constrained efficient stable matching. The resulting lottery is asserted to be ex ante stable, to satisfy equal treatment of equals, and to be undominated in the ordinal sense among ex ante stable lotteries. No equations, parameter fits, or derivations are shown that reduce the claimed properties to the inputs by definition. No load-bearing self-citations or uniqueness theorems imported from prior author work are invoked to force the result. The derivation chain remains self-contained as an explicit algorithmic procedure whose properties are claimed to follow from the construction itself.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Students have strict preferences over schools.
- domain assumption Schools have strict priorities over students.
Reference graph
Works this paper leans on
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[1]
Smart Lotteries in School Choice: Ex-ante Pareto-Improvement with Ex-post Stability. arXiv:2602.10679. Cookson, B., Shah, N
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[2]
Proceedings of the 26th ACM Conference on Economics and Computation, p.1130
Fairly Stable Two-Sided Matching with In- differences. Proceedings of the 26th ACM Conference on Economics and Computation, p.1130. https://doi.org/10.1145/3736252.3742675 Erdil, A., Ergin, H
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[3]
Equal Treatment of Equals and Efficiency in Probabilistic Assignments
Equal Treatment of Equals and Efficiency in Probabilis- tic Assignments. Unpublished manuscript available at https://arxiv.org/pdf/2508.14522 14
work page internal anchor Pith review Pith/arXiv arXiv
discussion (0)
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