Longer Lists Yield Better Matchings

Aapeli Vuorinen; Yuri Faenza

REVIEW 3 major objections 4 minor 31 references

Longer Lists Yield Better Matchings

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In a large Serial Dictatorship market with one seat per school, longer uniformly random preference lists weakly increase the match probability of every student among the first n positions, for all sufficiently large markets.

desk verdict A rigorous continuous-limit comparison with a genuinely useful welfare message, but the proof of the headline discrete theorem skips a uniformity step that the pointwise limit cannot supply. read the letter →

arxiv 2506.06217 v1 pith:VR7ORBWO submitted 2025-06-06 cs.GT cs.DSecon.TH

Yuri Faenza , Aapeli Vuorinen This is my paper

classification cs.GTcs.DSecon.TH MSC 91B6860F1734A34

keywords serialdictatorshiprandompreferencelistscentralizedmatchingschoolchoicetruncatedcontinuumlimitfunctionallawoflargenumbersbalancedmarkets

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

The paper asks whether limiting how many schools a student may list hurts the student in a large centralized matching market. In its model, $n$ schools each have one seat, students arrive in a fixed order, and each student's acceptable-school list of length $d$ is drawn uniformly at random; a student takes the favorite listed school that is still free, or goes unmatched. The central claim (Theorem 2.1) is that for any $d\le\ell$, once $n$ is large enough, every student in one of the first $n$ positions is matched to some listed school with probability at least as high under lists of length $\ell$ as under lists of length $d$. Since a companion lemma shows each school is also matched with higher probability under longer lists, the paper concludes that in balanced markets—no more students than seats—longer lists favor every agent, so a planner should make lists as long as practical.

What carries the argument

The central object is the continuous-market initial value problem $x_d(0)=0,\; x_d'(t)=1-x_d(t)^d$, the unique deterministic trajectory that the discrete occupied-school process $T_{\lfloor tn\rfloor}^{n,d}/n$ approaches uniformly in probability. The proof never solves the ODE in closed form (it is a Chini-type equation); instead it uses the integral equation $t=\int_0^{x_d(t)}(1-u^d)^{-1}\,du$, implicit differentiation with respect to $d$, and an auxiliary integral inequality $\int_0^{x_d(t)}(1+\log u)/(1-u^d)\,du\le0$ for $x_d(t)$ with $t\le1$ to establish $\partial^2 x/\partial d\partial t\ge0$, i.e. the match rate $x_d'$ increases in $d$ before time $1$. Lemma 2.7 then converts the continuous comparison into the discrete probability comparison.

What would settle it

Compute exact or high-precision simulated match probabilities in the discrete model for, say, $n=2000$, $d=1$, $\ell=2$, and every student $1\le i\le2000$; if any such student has $P(M_i^{n,1}=1)>P(M_i^{n,2}=1)$, Theorem 2.1 fails for that pair. The same check can be run in the continuous model by evaluating $x'_2(t)$ and $x'_1(t)$ on $[0,1]$ to any desired precision.

Watch

Extended reading notes

Core claim

Under the Serial Dictatorship mechanism with $n$ one-seat schools and uniformly random preference lists, the paper proves a monotonicity theorem: longer lists weakly increase the match probability of every student among the first $n$, for all sufficiently large $n$. The proof identifies the large-market limit of the discrete random market with the deterministic initial value problem $x_d(0)=0,\; x_d'(t)=1-x_d(t)^d$, where $x_d(t)$ is the proportion of schools already taken by time $t$. It then shows $x'_\ell(t)\ge x'_d(t)$ for all $t\in[0,1]$, and transfers this comparison back to finite markets via a functional law of large numbers. The paper also proves that the student in position $n$ gets matched with probability approaching $1/2$ as $d$ grows, and that for late students the monotonicity reverses: for instance, near position $1.22n$, lists of length $1$ beat lists of length $2$.

Load-bearing premise

The load-bearing premise is that the finite-market inequality follows from its large-market limit: the proof shows $P(M_{\lfloor tn\rfloor}^{n,d}=1)\to x_d'(t)$ and $x'_\ell(t)\ge x'_d(t)$ for $t\le1$, but it does not supply a uniformity estimate that would carry the comparison to every fixed student $i\le n$ in a single finite market, and students with $i=o(n)$ are exactly where the continuous gap is smallest.

Editorial extensions

If this is right

In markets with at most as many students as seats, the policy prescription is unambiguous: extending preference lists to any longer length weakly raises every student's and every school's chance of being matched, once the market is large.
The student at the end of the balanced segment, position $n$, is matched with probability near $1/2$ when lists are long, which gives a concrete benchmark for the match probability of the last student.
For unbalanced markets the monotonicity breaks: early students still favor longer lists, but some late students beyond position $n$ may strictly prefer shorter lists, so list-length policy involves a trade-off between early and late cohorts.
Under Random Serial Dictatorship, where student order is random, a randomly selected student's match probability also increases with list length (Corollary 2.9).
The bound in Lemma 2.5 quantifies how much a move from $d$ to $d+1$ can change a student's chance of landing in their top $k$ choices, giving a control valid for all sufficiently large markets.

Reading between the lines

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

The continuous comparison $x'_\ell(t)\ge x'_d(t)$ is proved for all real $d,\ell\in[1,\infty)$, so the monotonicity is not tied to integer list lengths; if the discrete-to-continuous transfer could be made uniform, the theorem would extend to comparisons of non-integer effective list lengths in finite markets.
Numerics for schools with multiple seats support the same monotonicity, but Conjecture 2.10 is left open; a proof would likely require a multivariate analogue of the scalar ODE comparison and exactly the uniformity argument the one-seat proof lacks.
The one non-uniform preference distribution where simulations fail is the degenerate one that effectively halves the number of schools, suggesting the balanced-market guarantee should be stated in terms of effective market size rather than raw counts $n$.
A natural next question, which the paper does not address, is whether Deferred Acceptance with bounded lists obeys a similar monotonicity; the Serial Dictatorship analysis gives no direct handle on that mechanism.

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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

You asked about arXiv:2506.06217. The continuous-limit half is a real contribution; the discrete headline is not proved by the argument in the paper.

The ODE comparison theorem (Thm 2.8) is the solid part. The functional LLN (Thm 2.6) is rigorous, and the appendix on the Chini equation and the integral inequalities shows real work. The paper is honest about the model's scope, and the numerics for non-uniform preferences and multi-seat schools are a useful sanity check.

The problem is Theorem 2.1. Its proof is one line: 'This follows directly from Theorem 2.8 and Lemma 2.7.' Lemma 2.7 is pointwise in t. It says that for each fixed t, the discrete match probability at student ⌊tn⌋ converges to x'_d(t). That gives you, for each fixed student ratio, an inequality for n large enough depending on t. It does not give 'for every n large enough and every i≤n' simultaneously at a fixed n. Early students i=o(n) are the trouble: as t→0 the continuous gap x'_ℓ(t)−x'_d(t) vanishes linearly (d=1 vs ℓ=2 gives a gap ~ i/n), while the LLN approximation error is O_p(1/√n). So the limit cannot certify the sign. The theorem might be true, but the paper needs a separate argument for early students, or uniform convergence with a positive gap on [ε,1]. Neither is present. This is a load-bearing proof gap, not a stylistic quibble.

I mildly disagree with the reader on Lemma 2.2: the stochastic dominance step is fine. Same number of taken schools means longer lists help conditionally; induction gives stochastic dominance of the matched count. Nothing wrong there.

Bottom line: this paper deserves a serious referee. The continuous comparison is a genuine technical contribution, and the welfare message matters for school-choice design. But as it stands, the main discrete theorem is unproven. Ask for the uniformity fix and a small-i argument; that is very likely repairable, and the paper would then be solid.

Referee Report

3 major / 4 minor

Summary. This paper studies a random centralized market with n one-seat schools and infinitely many students, each submitting a uniformly random preference list of length d, matched by Serial Dictatorship. The main result (Theorem 2.1) claims that for any d≤ℓ and every sufficiently large n, every student i≤n is matched to some acceptable school with probability at least as high under list length ℓ as under d. The paper develops a continuous ODE model x'_d(t)=1-x_d(t)^d, proves via a functional law of large numbers that the discrete process converges to this ODE (Theorem 2.6, Lemma 2.7), proves the continuous comparison x'_ℓ(t)≥x'_d(t) for t∈[0,1] (Theorem 2.8), and then asserts the discrete theorem by 'follows directly.' It also states school-side monotonicity (Lemma 2.2), a limit of 1/2 for the nth student's match probability (Theorem 2.4), a bound on top-k match probabilities (Lemma 2.5), and numerical experiments for non-uniform preferences and multi-seat schools.

Significance. If Theorem 2.1 is valid, the paper delivers a clean and practically relevant comparative-static: in balanced markets with uniformly random lists, longer lists weakly improve every student's chance of an acceptable match. The continuous-limit machinery is a rigorous application of a functional LLN, and the ODE comparison in Theorem 2.8 is supported by a substantial analytic appendix (Theorem C.1, Lemmas 3.3, C.2-C.7). The numerical experiments (Section 4) extend the question to non-uniform sampling and multi-seat schools and are reported with 100,000 repetitions. However, the transfer of the continuous comparison to the fixed-n discrete statement for all i≤n is not proved; this is the central logical step, so the significance claim is currently not backed by a proof.

major comments (3)

[Section 3.3, Theorem 2.1] The proof of Theorem 2.1 is the single sentence "This follows directly from Theorem 2.8 and Lemma 2.7." Lemma 2.7 gives, for each fixed t≥0, the pointwise limit P(M^{n,d}_{⌊tn⌋}=1)→x'_d(t). This does not imply the theorem's assertion for all i≤n at a fixed n. For students with i=o(n) (e.g., i fixed), t=i/n→0; the continuous gap x'_ℓ(t)-x'_d(t) is O(t) for d=1, ℓ=2, while the natural approximation error of Theorem 2.6 is O_p(n^{-1/2}). Thus the limit statement cannot certify the sign of the discrete difference for early students. No uniform-in-t convergence, quantitative rate, exact finite-n computation, or monotone coupling is provided in Sections 3.1-3.4 to cover these students. This is a load-bearing gap in the proof of the headline theorem.
[Section 3.3, Lemma 2.2] The proof of Lemma 2.2 states that the conditional match probability in (3) increases in d and then "Since T^{n,d}_i = Σ M^{n,d}_j we deduce that T^{n,ℓ}_i ≥ T^{n,d}_i in the sense of stochastic dominance." This deduction assumes that earlier students are more likely matched with longer lists, which is exactly the content that the lemma (and Theorem 2.1) must establish; as written the argument is circular. Furthermore, monotonicity of the conditional probability for a fixed value of T_i does not by itself yield stochastic dominance of the sum of dependent indicators without a coupling or induction. Consequently Lemma 2.2, and the school-side monotonicity used in Corollary 2.9, are currently unproved.
[Section 3.4, Lemma 2.5] The proof of Lemma 2.5 uses the pointwise convergence of P(K^{n,d}_i≤k) to x_d(i/n)^k (via Theorem 2.6) and then bounds the continuous difference x_{d+1}(t)^k-x_d(t)^k by its value at t=1. The lemma asserts the inequality for every i≤n and every n large enough, but the convergence is only pointwise in t; for i=o(n), the continuous difference tends to 0 and no uniform control is given, so the argument does not cover early students. The same missing finite-n transfer as in Theorem 2.1 affects this lemma.

minor comments (4)

[Section 3.3, Lemma 2.2 proof] There is a duplicated article in the phrase "in the the discrete market."
[Section 3.4, Lemma 2.5 proof] The letter k is used both for the rank threshold and for the number of matched schools in the conditioning event in P(K^{n,d}_i ≤ k | T^{n,d}_i = k), which is confusing; one of the two quantities should be renamed.
[Section 2.3] The sentence "we show in Lemma 2.7 that P(M^{n,d}_i=1)→x'_d(i/n) in probability" is imprecise: the convergence is for fixed t=i/n as n→∞, and the quantity P(M^{n,d}_i=1) is a number, not a random variable, so the phrase "in probability" is unnecessary and potentially misleading.
[Abstract and Introduction] The abstract and the introduction present Theorem 2.1 as established; if the proof is revised, the exposition should reflect that the finite-n comparison for early students requires a separate argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's theorems are derived from the stated random-list model via a functional law of large numbers and ODE comparison, with no fitted parameters and no load-bearing self-citations.

full rationale

The derivation chain is self-contained. The discrete model is defined directly (Section 2.1), the continuous model is introduced as an intended limit (Section 2.2), and the connection is established by Theorem 2.6, which is proved from a standard functional law of large numbers (Theorem 3.2) and an elementary approximation lemma (Lemma 3.1). Lemma 2.7 then converts the uniform-in-probability convergence of school counts into pointwise convergence of student match probabilities. Theorem 2.8 is a purely analytic statement about the ODE solutions x_d, proved by implicit differentiation and integral estimates in Section 3.2 and Appendix C; it does not import the discrete conclusion. The main theorem (Theorem 2.1) is asserted to follow from Theorem 2.8 and Lemma 2.7; whether that inference is fully justified for all i <= n simultaneously at finite n is a mathematical correctness concern (a possible uniformity gap for early students), not a circularity: the ingredients are distinct results, not restatements of the target inequality. No parameter is fitted to data and then renamed as a prediction, and the only self-citations (e.g., Faenza et al. 2023, 2024, 2025) appear in introductory context and are not load-bearing for the main claims. The numerical experiments are explicitly supplementary and do not define any constant used in the proofs. Hence the paper's central results do not reduce to their inputs by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the random-list, serial-dictatorship model and on standard stochastic-process approximation; no free parameters or invented entities are introduced. The main analytical risk is not a new entity but the chain of integral inequalities in Appendix C.

assumptions (6)

domain assumption Each student's preference list of length d is drawn independently and uniformly at random from the n schools without replacement.
This is the defining random-market model in Section 2.1; it makes match probability depend only on the count of taken schools.
domain assumption The mechanism is Serial Dictatorship with a fixed deterministic student order and unit seat capacity per school.
The model in Section 2.1; the order allows per-student analysis and the capacity makes T_i the number of taken schools.
domain assumption There are infinitely many students and exactly n schools, so the market can be studied in the limit n to infinity.
The asymptotic model in Section 2.1; it lets the authors define the continuous limit over t in [0, infinity).
standard math The functional law of large numbers for Markov chains (Theorem 3.2, cited from Kroese et al. and Ethier and Kurtz) applies to the embedded continuous-time chain.
Used in the proof of Theorem 2.6 to pass from the discrete market to the ODE x'=1-x^d.
standard math The initial value problem x'=1-x^d, x(0)=0 has a unique global solution, and the solution is smooth enough for the Leibniz rule and mixed partial derivatives.
Used throughout Section 3.2; Lipschitz continuity of 1-x^d on [0,1] and standard ODE theory justify this.
standard math The standard calculus facts used in Appendix C, including integration by parts, geometric series expansions, and elementary logarithm inequalities, are correct.
The proof of Theorem 2.8 depends on a long chain of such inequalities; they are asserted with proofs but are not independently machine-checked.

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Pith. "Pith review of Longer Lists Yield Better Matchings." pith.science (2026). https://pith.science/paper/VR7ORBWO

@misc{pith2026250606217,
  author       = {Pith},
  title        = {Pith review of: Longer Lists Yield Better Matchings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VR7ORBWO}},
  note         = {Machine review of arXiv:2506.06217}
}

abstract

Many centralized mechanisms for two-sided matching markets that enjoy strong theoretical properties assume that the planner solicits full information on the preferences of each participating agent. In particular, they expect that participants compile and communicate their complete preference lists over agents from the other side of the market. However, real-world markets are often very large and agents cannot always be expected to even produce a ranking of all options on the other side. It is therefore important to understand the impact of incomplete or truncated lists on the quality of the resultant matching. In this paper, we focus on the Serial Dictatorship mechanism in a model where each agent of the proposing side (students) has a random preference list of length $d$, sampled independently and uniformly at random from $n$ schools, each of which has one seat. Our main result shows that if the students primarily care about being matched to any school of their list (as opposed to ending up unmatched), then all students in position $i\leq n$ will prefer markets with longer lists, when $n$ is large enough. Schools on the other hand will always prefer longer lists in our model. We moreover investigate the impact of $d$ on the rank of the school that a student gets matched to. Our main result suggests that markets that are well-approximated by our hypothesis and where the demand of schools does not exceed supply should be designed with preference lists as long as reasonable, since longer lists would favor all agents.

Figures

Figures reproduced from arXiv: 2506.06217 by the authors.

**Figure 1.** 𝑥𝑑 and 𝑥 ′ 𝑑 for various values of 𝑑: note how the values for 𝑥 ′ 𝑑 cross just after 𝑡 = 1 Similarly to the discrete model (see the discussion in Section 1.1), we see that there are again two forces at play: for small 𝑑, the rate of getting matched is smaller, but simultaneously the proportion of schools taken up by earlier students is smaller, so later students may prefer small 𝑑 as it gives them some schools to po… view at source ↗

**Figure 2.** 100 simulations of the discrete model with the continuous model overlaid. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** 𝑥𝑑 and 𝑥 ′ 𝑑 for 𝑑 = 1 and 𝑑 = 2. Note that students before the highlighted cutoff at 𝑡 = 1.219 have higher probability of being assigned to any school with longer lists, and vice versa for students after the cutoff. The case of general 𝑑. In the case of general 𝑑, we cannot hope to solve the initial value problem analytically. Our main result in the continuous realm, proved in Section 3.2 is that all students for 𝑡… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Solution to the multiple-seat continuous market for [PITH_FULL_IMAGE:figures/full_fig_p028_4.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

We have each of these flows from𝑦𝑘 𝑑 to𝑦𝑘+1 𝑑 for𝑘= 0,...,𝑞− 1and one from 𝑦𝑞−1 𝑑 to𝑥𝑑

Note further that this is a positive flow into 𝑦𝑘+1 𝑑 and negative for 𝑦𝑘 𝑑 . We have each of these flows from𝑦𝑘 𝑑 to𝑦𝑘+1 𝑑 for𝑘= 0,...,𝑞− 1and one from 𝑦𝑞−1 𝑑 to𝑥𝑑. For convenience, define 𝛾𝑑(𝑡)= 1−𝑥 𝑑(𝑡)𝑑 1−𝑥 𝑑(𝑡) . Denoting derivative with respect to time with a dot for clarity to avoid multiple superscripts, this gives us the following differential eq...

work page 2013

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