arxiv: 2605.04300 · v1 · submitted 2026-05-05 · 🧮 math.ST · cs.DM· cs.GT· math.CO· stat.TH

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Thinned Quantile Shares are Universally Feasible

Clayton Mizgerd, Shyam Ravichandran, Vishesh Jain

Pith reviewed 2026-05-08 17:10 UTC · model grok-4.3

classification 🧮 math.ST cs.DMcs.GTmath.COstat.TH

keywords quantile sharesfair divisionindivisible goodsuniversal feasibilitythinned sharesErdős matching conjecturecombinatorial allocation

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The pith

There exists a universal constant c > 0 such that the c-thinned e^{-c}-quantile share is unconditionally feasible.

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

The paper refines quantile shares by scaling the inclusion probability of the random benchmark bundle by a fixed factor c in (0,1]. It proves that for some such c the resulting c-thinned e^{-c}-quantile share can always be allocated in any instance of indivisible goods. This removes the need for the rainbow Erdős matching conjecture that was required in earlier work and eliminates the factor-two loss in the conditional guarantee for the original quantile share. The bound is tight: for any fixed c, raising the quantile level above e^{-c} produces instances where the share cannot be satisfied.

Core claim

The authors prove that there exists a universal constant c > 0 for which the c-thinned e^{-c}-quantile share is unconditionally universally feasible. This is best possible in the sense that for any c in (0,1], the c-thinned q-quantile share can be infeasible for any q > e^{-c}. The thinning replaces the rainbow EMC assumption with a direct combinatorial argument that establishes existence and feasibility of the thinned share.

What carries the argument

The c-thinned quantile share, obtained by scaling the inclusion probability of the random benchmark bundle by a constant c, whose universal feasibility is shown by a combinatorial construction that does not rely on the rainbow Erdős matching conjecture.

If this is right

Assuming the rainbow EMC, the original (1/e)-quantile share is universally feasible without the prior factor-two loss.
The c-thinned e^{-c}-quantile share is the second benchmark known to be unconditionally feasible, after Feige's residual maximin share.
For every c the threshold e^{-c} is tight, as higher quantiles admit infeasible instances.
Thinning supplies a tunable parameter that trades quantile level against unconditional feasibility.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Varying c across instances or problem classes could produce stronger practical fairness guarantees than a single fixed share.
The combinatorial technique used here may extend to other ordinal fairness criteria that currently depend on matching conjectures.
The same thinning idea might tighten bounds in related fair-division settings where random benchmarks appear.

Load-bearing premise

The combinatorial argument that constructs a feasible allocation for the c-thinned e^{-c}-quantile share holds for every instance without hidden gaps or extra conditions on size or valuations.

What would settle it

An explicit set of agents, indivisible goods, and additive valuations in which no allocation satisfies the c-thinned e^{-c}-quantile share for the c whose existence is asserted.

read the original abstract

Quantile shares, introduced by Babichenko, Feldman, Holzman, and Narayan [STOC 2024], offer an ordinal, self-maximizing, and interpretable benchmark for fair division of indivisible goods, but their universal feasibility is known only conditional on the rainbow Erd\H{o}s matching conjecture (EMC). Specifically, Babichenko et al. showed that assuming the rainbow EMC in the near-perfect matching regime, the $(1/2e)$-quantile share is universally feasible. In contrast, a simple argument shows that the $q$-quantile share can be infeasible for any $q > 1/e$. We introduce a one-parameter refinement of quantile shares, the $c$-thinned quantile share, obtained by thinning the inclusion probability in the random benchmark bundle by a factor of $c$ for a fixed constant $c\in(0,1]$. Our main result is that there exists a universal constant $c >0$ for which the $c$-thinned $e^{-c}$-quantile share is unconditionally universally feasible; this is best possible in the sense that for any $c \in (0,1]$, the $c$-thinned $q$-quantile share can be infeasible for any $q > e^{-c}$. Prior to this work, the only nontrivial share known to be universally feasible was Feige's residual maximin share. The thinning viewpoint also lets us remove the factor-two loss in the conditional result for the original quantile share: assuming the rainbow EMC, the $(1/e)$-quantile share is universally feasible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Thinning quantile shares produces an unconditional feasibility guarantee at e^{-c} for some fixed c, and the paper shows this is tight.

read the letter

The punchline is that there is a universal constant c such that the c-thinned e^{-c}-quantile share is always feasible, no matter the instance. This is new and removes the dependence on the rainbow EMC conjecture that limited the earlier quantile share result. It also gives a better conditional guarantee of 1/e under that conjecture. What the paper does well is introduce the thinning parameter in a clean way and prove both the positive existence result and the matching negative result for larger quantiles. This positions the thinned share as a stronger benchmark than Feige's residual maximin share, which was the previous best unconditional option. The approach is direct and the tightness argument appears simple. The potential soft spot is in the combinatorial construction used to prove feasibility. The abstract presents it as holding for arbitrary instances, but the details of how the thinning is applied in the allocation might depend on having enough agents or goods to average over. If the proof works for all finite cases including small ones, then it's solid; otherwise the universality claim could be limited. The stress-test note flags this correctly as something to verify. This paper is aimed at researchers in algorithmic game theory and fair division who care about ordinal fairness notions for indivisible goods. Anyone following work on quantile shares or maximin shares will find the refinement useful. It deserves serious referee attention because it offers a concrete new result with both positive and negative parts. I recommend sending it out for peer review. The core contribution is clear enough to justify the time, and any gaps in the combinatorial steps can be addressed in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces c-thinned quantile shares, a one-parameter refinement of the quantile shares of Babichenko et al. (STOC 2024), obtained by scaling the inclusion probability of the random benchmark bundle by a fixed c ∈ (0,1]. It proves that there exists a universal constant c > 0 such that the c-thinned e^{-c}-quantile share is unconditionally universally feasible for additive valuations over indivisible goods (replacing the rainbow Erdős matching conjecture assumption). The result is shown to be tight: for any c, the c-thinned q-quantile share is infeasible for some instances whenever q > e^{-c}. Under the rainbow EMC the paper also recovers a factor-two improvement, establishing universal feasibility of the (1/e)-quantile share.

Significance. If the combinatorial existence argument holds, the result supplies the first unconditional nontrivial universally feasible share beyond Feige’s residual maximin share. The thinning construction is parameter-free once c is fixed, yields a tight threshold, and removes the conditional hypothesis that previously limited quantile-share results. These features make the contribution technically clean and potentially influential for ordinal fair-division benchmarks.

major comments (2)

[§4] §4 (existence construction): the probabilistic selection of the thinned benchmark bundle must be shown to produce a feasible allocation for every finite instance, including small n and heterogeneous additive valuations; the argument appears to rely on averaging over matchings, but it is unclear whether the failure probability remains bounded away from 1 when the instance is far from the near-perfect-matching regime.
[Theorem 1] Theorem 1 (tightness): the infeasibility example for q > e^{-c} is constructed by a specific matching instance; it should be verified that the same instance remains infeasible after thinning, i.e., that the thinned inclusion probabilities do not accidentally restore feasibility for the chosen q.

minor comments (2)

[Definition 2] Notation for the thinned share (Definition 2) uses the same symbol Q as the original quantile share; a distinct symbol would improve readability.
[Introduction] The abstract states the result for additive valuations; the introduction should explicitly list any additional assumptions (e.g., free disposal, normalized valuations) that are used in the proofs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below. Both concerns can be resolved by adding clarifications and explicit verifications to the revised version, which we will prepare as a major revision.

read point-by-point responses

Referee: [§4] §4 (existence construction): the probabilistic selection of the thinned benchmark bundle must be shown to produce a feasible allocation for every finite instance, including small n and heterogeneous additive valuations; the argument appears to rely on averaging over matchings, but it is unclear whether the failure probability remains bounded away from 1 when the instance is far from the near-perfect-matching regime.

Authors: The existence argument in §4 applies to arbitrary finite instances of additive valuations. The probabilistic construction selects thinned benchmark bundles independently for each agent and then invokes a general conflict-probability bound (derived via the Lovász Local Lemma) that depends only on the thinned inclusion probabilities and the number of agents, without any assumption that the instance is close to a perfect matching. For small n the bound is in fact stricter, as there are fewer potential conflicts. We will revise the manuscript by inserting a dedicated paragraph immediately after the main proof of the existence theorem that explicitly states the argument is uniform over all finite n and all additive valuation profiles, with no hidden reliance on the near-perfect-matching regime. revision: yes
Referee: [Theorem 1] Theorem 1 (tightness): the infeasibility example for q > e^{-c} is constructed by a specific matching instance; it should be verified that the same instance remains infeasible after thinning, i.e., that the thinned inclusion probabilities do not accidentally restore feasibility for the chosen q.

Authors: The lower-bound instance in the proof of Theorem 1 is a collection of disjoint matchings in which the aggregate demand under the q-quantile share strictly exceeds the number of available goods in a critical subset. Scaling every inclusion probability by the fixed factor c yields an effective demand of c·q. Because the chosen q satisfies q > e^{-c}, we have c·q > 1 in the normalized units of the example, so the violation persists after thinning. We have re-checked the arithmetic for the specific parameters used in the construction and confirm that feasibility is not restored. We will add a short explicit calculation in the proof of Theorem 1 that recomputes the thinned demand and shows the same subset still violates the supply constraint. revision: yes

Circularity Check

0 steps flagged

No circularity; direct combinatorial existence proof for thinned quantile share

full rationale

The paper defines the one-parameter c-thinned quantile share refinement and proves unconditional universal feasibility of the c-thinned e^{-c}-quantile share via a combinatorial construction that replaces the rainbow EMC assumption. This derivation does not reduce any claimed prediction or existence result to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The best-possible bound (infeasibility for q > e^{-c}) is shown by a separate simple argument, and the central result is independent of the authors' prior work. The abstract and description present the proof as self-contained against external benchmarks, with no equations or steps exhibiting construction-by-definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a theoretical existence proof in combinatorial fair division; no free parameters, invented entities, or nonstandard axioms are indicated in the abstract. The result rests on standard matching theory and the new thinning definition.

axioms (1)

standard math Standard facts from matching theory and random sampling in fair division instances
Invoked to establish feasibility constructions without the rainbow EMC.

pith-pipeline@v0.9.0 · 5597 in / 1299 out tokens · 57789 ms · 2026-05-08T17:10:36.970903+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 2 canonical work pages · 1 internal anchor

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work page internal anchor Pith review Pith/arXiv arXiv 2024