Efficiency Adjustments Break the Logarithmic Rank Barrier

Gabriel Ziegler; Geng Zhao; Josue Ortega

Efficiency Adjustments Break the Logarithmic Rank Barrier

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that EADA's expected average rank is at most $4\log\log n+O(1)$ in i.i.d. one-to-one markets, breaking DA's logarithmic rank barrier.

desk verdict This paper delivers the first asymptotic average-rank bounds for EADA and for every Pareto-efficient mechanism weakly dominating DA; the core proof is sound and the caveats are honestly stated. read the letter →

arxiv 2608.09984 v1 pith:SQF6N2KI submitted 2026-08-05 cs.GT econ.TH

Josue Ortega , Geng Zhao , Gabriel Ziegler This is my paper · ORCID

classification cs.GTecon.TH MSC 91B6860C05

keywords efficiency-adjusteddeferredacceptanceexpectedaveragerankrandommatchingmarketsParetoefficiencylognschoolchoiceone-to-one

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Deferred Acceptance (DA) is known to give students an expected average rank of order $\log n$, while the rank-minimizing assignment keeps expected average rank bounded. This paper establishes the first asymptotic guarantee for the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism: in i.i.d. one-to-one markets with $n$ students and $n$ schools, the expected average rank under EADA is at most $4\log\log n+O(1)$. The paper further proves that every Pareto-efficient matching that weakly Pareto-dominates the DA matching has expected average rank $O((\log\log n)^2)$, so the logarithmic barrier falls for an entire class of mechanisms, not just EADA. A sympathetic reader would care because this shows that respecting priorities and Pareto efficiency is compatible with an order-of-magnitude improvement over DA.

What carries the argument

The argument rests on a sequential implementation of EADA in which under-demanded schools are settled one at a time, rerunning DA after each settlement; the order of settlement is shown to be irrelevant. This turns a student's final rank into the number of applications she makes when she is settled, so the total rank decomposes into a cut term bounded by DA's worst rank and a residual term bounded by the number of applications in the residual DA run. The residual market reached by EADA is student-closed: every survivor prefers her residual assignment to every school already removed. For fixed survivor sets, closure combined with the uniformity of preferences yields the key estimate that the probability of a residual run with at least $p$ applications is at most $(m/n)^p$, because conditional on relative rankings the surviving schools occupy a uniform subset of each student's full list. A union bound over all possible survivor sets then controls the endogenously selected residual market. The general theorem replaces closure by acyclicity of the top-$k$ envy graph, which Pareto efficiency forces and which random preferences make unlikely at large size.

What would settle it

Compute exactly, for $n=6$, the probability in Lemma 8 for a fixed survivor pair with $m=3$ and $p=2$: if $\Pr(\text{closed} \cap \{P\ge 2\}) > (3/6)^2 = 1/4$ under i.i.d. uniform preferences, the key estimate fails; a broader check is exhaustive enumeration of all profiles at small $n$ to see whether EADA's expected average rank exceeds $4\log\log n + C$ for every fixed $C$.

Watch

Extended reading notes

Core claim

The central claim is that EADA breaks DA's logarithmic rank barrier: Theorem 1 states $\mathbb{E} R(\mu_{\mathrm{EADA}})\le 4\log\log n+O(1)$ under i.i.d. uniform preferences, where $R$ is the average rank of assigned schools. Because $\log\log n=o(\log n)$, this improves the asymptotic order of students' assignments. Theorem 2 shows the same qualitative conclusion for every Pareto-efficient matching $\nu$ that weakly Pareto-dominates the student-proposing DA matching, with the weaker rate $\mathbb{E} R(\nu)=O((\log\log n)^2)$. The paper also extends both conclusions to balanced many-to-one markets with bounded quotas and to bounded Plackett-Luce common popularity, and it gives a tiered example showing that some restriction on preference correlation is necessary.

Load-bearing premise

The load-bearing assumption is that, after conditioning on the relative order of the surviving schools on a student's list, the positions those schools occupy in her full list form a uniformly random subset, independently across students; this holds under i.i.d. uniform preferences, and the paper's tiered example shows that without some such restriction average rank becomes linear.

Editorial extensions

If this is right

EADA's expected average rank is $o(\log n)$, so the efficiency adjustment changes the asymptotic order of what students receive, not merely the constant factor.
Any Pareto-efficient mechanism that weakly Pareto-dominates DA, even one selected after the preference profile is observed, has expected average rank $O((\log\log n)^2)$, ruling out logarithmic average rank throughout the class.
The bounds carry over to many-to-one markets with fixed quotas: under a common quota $q$, EADA attains at most $2(1+1/q)\log\log m+O_q(1)$.
Under bounded Plackett-Luce common popularity with weight ratio $\kappa$, EADA attains at most $4\kappa\log\log n+O_\kappa(1)$, and every Pareto-efficient improvement of DA remains sublogarithmic.
The coefficient $4$ is not claimed sharp; tightening the closure argument is the identified route toward possibly lower rates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the proof's union bound charges every possible survivor set, the actual distribution of markets reached by EADA is plausibly much more concentrated; a sharper analysis of that joint distribution could yield a matching lower bound or a rate below $\log\log n$.
Theorem 2's uniformity over mechanisms chosen after seeing preferences suggests the sublogarithmic property is a consequence of Pareto efficiency plus weak dominance of DA; a testable extension would ask whether weakening 'weakly Pareto-dominates DA' to 'improves a positive fraction of students' already forces the same escape from the logarithmic barrier.
The tiered counterexample indicates the boundary is quantitative rather than structural: between uniform preferences and complete tier agreement, the expected average rank may interpolate from doubly logarithmic to linear, and the bounded-popularity model is one natural way to trace that transition.
Simulations in the paper cannot distinguish bounded from doubly logarithmic growth, so an exact computation of EADA's rank distribution for small $n$, or a matching lower bound, would be the next decisive step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real result, not a rounding error. Theorem 1 (EADA expected average rank at most 4 log log n + O(1)) and Theorem 2 (every Pareto-efficient matching that weakly dominates DA achieves O((log log n)^2)) are the first asymptotic guarantees for those objects in iid uniform markets. The paper deserves a serious referee.

The new machinery is the combination of a sequential implementation of EADA, deterministic rank accounting via application counts, and a union bound over closed residual markets. I checked the main lemmas—serial implementation, the rank identity in Lemma 5, application nesting in Lemma 6, the closure bound in Lemma 8, the tail bound in Proposition 3, the acyclicity argument in Lemma 9, and the uniform event in Lemma 10—and the stress-test pass agrees: no load-bearing flaw. Lemma 8 is exactly right under iid uniform preferences, and the paper is explicit that some distributional restriction is necessary by giving a tiered-agreement example where average rank is linear. The extensions to bounded quotas and bounded Plackett-Luce popularity are substantial and appear to follow the same architecture.

What the paper does especially well is honesty about slack. It states that the constants are not sharp, that no lower bound is known, and that simulations at these n cannot distinguish a bounded average rank from log-log divergence. That is the right way to present a first asymptotic result.

Soft spots, in proportion. The model is stylized: complete lists, balanced market, iid uniform preferences. That is not a flaw given the question, but it limits the practical takeaway. The lack of code is not a concern for a theory paper; the proofs are detailed enough to referee. One minor citation check: the Pittel bound is quoted as (2+a)(log n)^2; if Pittel's theorem states (2+a) log n, the conclusion still follows by monotonicity, but the reference should be made precise. Also, the general theorem's rate O((log log n)^2) has no matching lower bound; the paper says this openly.

Verdict: accept for peer review. I would cite Theorem 1 in future work on average-rank performance of matching mechanisms, and I would bring the paper to reading group. A serious referee will find a few things to polish, but the core contribution is solid and clearly beyond what was known.

Referee Report

0 major / 5 minor

Summary. The paper studies the expected average rank (equivalently, the average position of students' assignments in their reported preferences) achieved by the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism in i.i.d. one-to-one matching markets. It proves Theorem 1, which bounds E[R(\mu_{EADA})] by 4 log log n + O(1), and Theorem 2, which shows that every Pareto-efficient matching weakly Pareto-dominating the student-proposing DA matching has expected average rank O((log log n)^2). The proof of Theorem 1 proceeds through a sequential implementation of EADA (Proposition 1), a deterministic rank accounting (Proposition 2), a closure probability bound for residual markets (Lemma 8), a union-bound and tail-sum estimate (Proposition 3), and a worst-rank bound for DA from Pittel (Proposition 4). Theorem 2 is proved via an acyclicity property of Pareto-efficient matchings (Lemma 9) and a uniform probabilistic bound on large acyclic sets in random preference graphs (Lemma 10). Appendices extend both results to balanced many-to-one markets with bounded quotas and to bounded Plackett-Luce common popularity, and a tiered example in Section B.5 shows that some correlation restriction is necessary.

Significance. If the results are correct, they resolve a natural open question about the rank efficiency of EADA and provide the first sublogarithmic asymptotic guarantee for any Pareto-efficient improvement of DA. The main argument is self-contained given the cited external lemmas, and the leading-order bound is derived rather than fitted to simulations. The paper is careful to identify the distributional limit of its main probabilistic lemma and to exhibit a counterexample showing that unrestricted correlation destroys the result. The authors also honestly state that no matching lower bound is established and that the simulations cannot distinguish bounded from slowly diverging behavior; these limitations do not undermine the stated theorems. The results are likely to be of broad interest to the random matching markets and school choice communities.

minor comments (5)

[Section 2.5, Proposition 4] The statement of Pittel's Theorem 6.1(b) should be checked against the original source; if the original threshold is of the form C log n rather than C (log n)^2, equation (16) still follows by monotonicity and E[M_DA]=O((log n)^2) is unchanged, so the issue is not load-bearing, but the citation and displayed formula should be accurate.
[Section 2.5, after equation (16)] The phrase 'we may therefore take c=3, 4' appears garbled; it should presumably read 'c = 3.4' or otherwise be stated consistently with the value c(20) = (sqrt(89)-3)/2.
[Section 3, Proposition 5] The notation log^2(1+M) is used for the square of the logarithm; this should be defined explicitly in the notation paragraph of Section 2, which currently defines only o and O notation.
[Appendix A.4, Proposition 9, display (44)] The last term in the displayed inequality is ambiguous as typeset; it should be written as (1+m/K)/(qm) or equivalently 1/(qm)+1/(qK) to avoid confusion.
[Section 4, Figure 1] The text refers to Figure 1, but the figure itself is not visible in the manuscript text; the simulation values and standard errors are given, but the figure should be included or the reference adjusted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived from independent external lemmas (Tang–Yu, Pittel) and from deterministic accounting plus fresh probabilistic bounds, not from fitted inputs or self-referential definitions.

full rationale

The paper's central claim, Theorem 1, is obtained by combining a deterministic rank-accounting bound (Proposition 2), a union-bound tail estimate for residual DA application counts (Lemma 8 and Proposition 3), and Pittel's external worst-rank estimate (Proposition 4). None of these inputs is defined in terms of EADA's expected average rank, and no parameter is fitted to the quantity being predicted. Lemma 8 is a genuine distributional calculation under i.i.d. uniform preferences: conditional on relative rankings of the surviving schools, the positions of those schools in each full list form a uniform subset, so the closure probability factors as (m/n)^P; this is not an assumption that encodes the target bound. The 4 log log n coefficient emerges from optimizing the cut q = n/(log n)^2 against the two independent estimates, rather than from matching data. Theorem 2 similarly relies on acyclicity of envy graphs (Lemma 9), a probabilistic uniform bound over all matchings (Lemma 10), and Pittel's bound; it does not presuppose sublogarithmic rank. The self-citations in the introduction and positioning section (Ortega et al. 2026a,b) are used only to motivate the question and to compare with prior qualitative welfare-improvement results; they are not load-bearing in the proofs of Theorems 1 or 2. The only caveat, that Lemma 8 requires the i.i.d. assumption and fails under unrestricted correlation, is explicitly acknowledged by the tiered counterexample in Section B.5 and is correctly framed as a scope condition, not a circular step. The appendices extend the same architecture to many-to-one and bounded Plackett–Luce models with self-contained analogues of the external lemmas. Therefore no step in the derivation chain reduces, by construction or by self-citation, to the paper's own conclusions.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on the standard i.i.d. uniform random market model, the full-consent version of EADA, the external Tang-Yu equivalence for under-demanded schools, and Pittel's worst-rank bound for DA. No parameters are fitted to data; constants in proofs are chosen for convenience and do not affect asymptotics.

assumptions (7)

domain assumption Students' preferences and schools' priorities are mutually independent and uniformly distributed over all permutations (Section 2, Model).
Used in Theorems 1 and 2; the closure bound (Lemma 8) and acyclicity probability (Lemma 10) both require independence and uniformity of preferences.
domain assumption The market is balanced (n students, n schools) and all preference lists are complete.
Ensures DA terminates with every school receiving an application and defines average rank over n students.
domain assumption All students consent to waive their priorities in EADA (footnote 3).
Full consent is needed for EADA to reach Pareto efficiency and for the Tang-Yu implementation to apply.
standard math Tang-Yu Lemma 1: a school is under-demanded in a DA run iff it rejects no student and receives exactly one application; every DA run has at least one such school.
Used to define EADA and to prove the sequential implementation (Proposition 1).
standard math Pittel's Theorem 6.1(b) bounds the worst proposer rank in the proposer-optimal stable matching, yielding E M_DA = O((log n)^2) under uniform independent priorities.
The only input requiring uniform school priorities; used in Propositions 4, 6, and 10.
domain assumption In Appendix A, school quotas are bounded by a fixed \bar q independent of market size.
The many-to-one extension results are proved under fixed bounded quotas.
domain assumption In Appendix B, Plackett-Luce popularity weights lie in [1, \kappa] for fixed \kappa.
The correlated-preference bounds require bounded popularity ratio; the tiered example shows a restriction is necessary.

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Pith. "Pith review of Efficiency Adjustments Break the Logarithmic Rank Barrier." pith.science (2026). https://pith.science/paper/SQF6N2KI

@misc{pith2026260809984,
  author       = {Pith},
  title        = {Pith review of: Efficiency Adjustments Break the Logarithmic Rank Barrier},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQF6N2KI}},
  note         = {Machine review of arXiv:2608.09984}
}

abstract

We study the expected average rank achieved by the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism in i.i.d.\ matching markets. While student-proposing Deferred Acceptance gives students an expected average rank of logarithmic order, we prove that EADA's expected average rank is at most $4\log\log n+O(1)$. Therefore, EADA improves the asymptotic order of students' assignments. At the cost of a weaker bound, $O((\log\log n)^2)$, we extend this conclusion to a much larger class of mechanisms. Namely, every Pareto-efficient mechanism that weakly Pareto-dominates DA breaks DA's logarithmic barrier. These are the first asymptotic guarantees for the expected average rank of EADA and of the broader class of Pareto-efficient improvements of DA. The conclusions extend to many-to-one markets with bounded quotas and random markets with correlated preferences.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Each entry averages500independent markets; Monte Carlo standard errors are in parentheses

Average school rank under DA and EADA in balanced many-to-one markets. Each entry averages500independent markets; Monte Carlo standard errors are in parentheses. The separation is clear. With quota2, DA’s average rank rises from3.530to5 .539, while EADA’s rises only from1.982to2 .044. With quota5, DA rises from2 .138to 3.160, while EADA moves from1.462to1...

work page 2023

[2] [2]

https://doi.org/10

Unbalanced random matching markets: The stark effect of competition.Journal of Political Economy125, 1 (2017), 69–98. https://doi.org/10. 1086/689869 4 Claudia Cerrone, Yoan Hermstrüwer, and Onur Kesten

work page 2017

[3] [6]

arXiv preprint arXiv:2401.03598(2024)

Incontestable Assignments. arXiv preprint arXiv:2401.03598(2024). 5 Umut Dur, A Arda Gitmez, and Özgür Yılmaz

arXiv 2024

[4] [7]

5 Lars Ehlers and Thayer Morrill

School choice under partial fairness.Theoretical Economics14, 4 (2019), 1309–1346. 5 Lars Ehlers and Thayer Morrill

work page 2019

[5] [16]

SSRN preprint(2016)

Ordinal efficiency, fairness, and incentives in large markets. SSRN preprint(2016). 4 D. McVitie and L. Wilson

work page 2016

[6] [21]

4 Marek Pycia

Asymptotic welfare performance of Boston assignment algorithms.Stochastic Systems13, 2 (2023), 247–270. 4 Marek Pycia

work page 2023

[7] [22]

4 Philip J Reny

Evaluating with statistics: Which outcome measures differentiate among matching mechanisms?Unpublished paper, University of Zurich(2019). 4 Philip J Reny

work page 2019

[8] [23]

4 Ryo Shirakawa

Efficient matching in the school choice problem.American Economic Review 112, 6 (2022), 2025–43. 4 Ryo Shirakawa

work page 2022