Exact Maximin Share Fairness via Adjusted Supply

Aditi Sethia; Satyanand Rammohan; Siddharth Barman

REVIEW 3 major objections 5 minor 24 references

Exact Maximin Share Fairness via Adjusted Supply

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper proves that exact maximin share fairness is always achievable when goods may be duplicated or chores discarded, with only logarithmic per-item duplication under completely general monotone preferences.

desk verdict The upper-bound half is real and clean; the lower-bound proof for goods has an unquantified Poisson approximation and needs repair before 'essentially tight' is justified. read the letter →

arxiv 2502.03789 v2 pith:DWO2GZAN submitted 2025-02-06 cs.GT

Siddharth Barman , Satyanand Rammohan , Aditi Sethia This is my paper · ORCID

classification cs.GT MSC 91B32

keywords maximinsharefairdivisionmulti-allocationmonotonevaluationschoresprobabilisticmethodcapacityadjustment

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

Maximin share (MMS) fairness asks that each agent receive a bundle worth at least the best guaranteed share it could secure by partitioning all goods into n bundles and taking the worst. For indivisible goods, exact MMS is known to be impossible even with additive preferences. This paper shows the obstruction disappears if the supply of items can be adjusted after the fact: there is always a multi-allocation giving every agent its MMS in which each good is copied at most $3\log m$ times and the total number of assigned goods, copies included, is still at most $m$, even for completely monotone valuations. The analogous statement for chores is that at most $m/e$ chores ever need to be discarded. The consequences are concrete: exact fairness is a supply-adjustment problem, not merely an allocation problem.

What carries the argument

The central object is the characteristic vector $\chi_A$ of a multi-allocation, whose $\ell_1$ norm counts the total number of assigned copies, whose $\ell_\infty$ norm is the maximum number of copies of any single item, and whose zero-coordinate count $\|\chi_A\|_z$ measures unassigned chores in the chores setting. The proof mechanism is the probabilistic method: independently, each agent samples one bundle uniformly from its MMS-inducing $n$-partition. Because each good lies in exactly one bundle of every such partition, it lands in any given agent's sampled bundle with probability $1/n$, so the expected copy count of every good is $1$ and the expected total is $m$; Chernoff, Hoeffding, and Chebyshev bounds then show the $\ell_\infty$ and $\ell_1$ events co-occur with positive probability. For identically ordered instances, the sampled allocation is post-processed by a copy-redistribution algorithm that replaces higher-indexed, lower-marginal-value goods with lower-indexed, higher-marginal-value ones, preserving MMS while compressing multiplicities; for chores, the same sampling gives an expected unassigned count of $m(1-1/n)^n \approx m/e$, and a covering-plus-redistribution step converts concentration on chore intervals into a bound on unassigned chores.

What would settle it

Recompute the lower-bound argument of Section 3.4 with the exact Binomial(n,1/n) law for each good's copy count instead of Poisson(1), then redo the union bound over all $n^n$ candidate multi-allocations; if the resulting probability that every allocation has some good with at least $\log m/\log\log m$ copies is not strictly positive for large $m$, the claimed asymptotic tightness of Theorem 3.1's duplication bound is not established. For the upper bound itself, a counterexample search over small monotone instances would settle Theorem 3.1 for those sizes.

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Extended reading notes

Core claim

Every fair division instance with goods and monotone valuations admits an MMS multi-allocation $A=(A_1,\dots,A_n)$ whose characteristic vector $\chi_A$ satisfies $\|\chi_A\|_\infty \le 3\log m$ and $\|\chi_A\|_1 \le m$: no individual good needs to be handed to more than logarithmically many agents, and the total number of goods handed out, counting copies, is at most the original number of goods. The same random-sampling idea shows that for $m$ chores with monotone costs an MMS multi-allocation can leave at most $m/e$ chores unassigned. Under identically ordered valuations or costs the bounds improve to $O(\sqrt{\log m})$ multiplicity and $m + O(m\sqrt{\log m}/\sqrt{n})$ total assigned goods; under additive valuations, two copies of any good suffice. Matching lower bounds show that the monotone results are essentially tight, so these guarantees cannot be substantially improved in the most general model.

Load-bearing premise

The tightness lower bound assumes the number of copies of each good in the random construction follows a Poisson(1) distribution, when it actually follows Binomial(n,1/n), and does not quantify how small the error from that approximation is before union-bounding over all allocations.

Editorial extensions

If this is right

Exact MMS is feasible for every monotone valuation, which is impossible without duplication: the barrier is item supply, not preference structure.
In any instance with $m$ goods, one can find a fair multi-allocation that hands out no more than $m$ goods in total, so duplication never increases the overall volume of allocated resources.
For additive valuations, an MMS assignment exists in which no good is used more than twice and the total number of assigned goods is at most $2m$, a mild adjustment in realistic settings.
For chores, MMS fairness is always achievable by discarding at most $m/e$ chores; under additive costs the discarded count drops to $2m/11 + n$.
The lower bounds show the general guarantees are essentially optimal: some instances require $\Omega(\log m/\log\log m)$ copies of a good, and every MMS assignment leaves $(1-o(1))m/e$ chores unassigned.

Reading between the lines

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

If the paper is right, fair division can be treated as capacity planning: duplicating a few scarce goods and leaving unwanted goods unassigned lets an algorithm meet an exact fairness target without changing preferences, so practical systems could adjust class sizes, inventory, or hospital capacities rather than rationing by approximation.
The same random-sampling template likely extends to other share notions and to envy-freeness up to any good (EFX) with duplication or charity, a direction the paper itself hints at; a testable next step is to run the sampling argument with EFX-inducing partitions and see what multiplicity bounds emerge.
The gap between the $3\log m$ upper bound and the $\log m/\log\log m$ lower bound for monotone goods suggests the true worst-case multiplicity may be $\Theta(\log m/\log\log m)$, but that is an editorial guess, not a claim of the paper.
For identically ordered valuations, the dyadic-prefix redistribution suggests a broader principle: whenever items can be sorted by marginal value, copy counts can be equalized across a prefix structure, which may carry over to online or dynamic allocation.

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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

Let me give you the short version: the main existence result is true and the proof is as clean as advertised; the lower bound that is supposed to make it tight is not yet proven. The paper deserves a serious referee, but the 'essentially tight' claim needs work.

The new model—exact MMS with limited duplication/disposal—is a good idea and the paper executes it well for the upper bounds. Theorem 3.1 is a genuinely neat observation: sample each agent's bundle uniformly from its MMS partition, then Chernoff plus Markov gives ℓ∞ ≤ 3 log m and ℓ1 ≤ m with positive probability. The identically ordered results (Theorems 3.4 and 4.4) are more involved—dyadic prefixes plus a copy-redistribution argument—and they appear sound to me. The chores upper bound m/e is a simple and correct application of the same sampling idea. The additive results depend on imported theorems ([HH22], [HL21]); that is fine, but the dependence should be flagged clearly.

The soft spot is Theorem 3.10. The proof replaces Bin(n,1/n) with Poisson(1) and then treats equation (14) as an exact identity inside a union bound over n^n allocations. That step is not justified as written. The approximation error is not quantified, and the parameters have to satisfy two competing constraints—the binomial tail needs n large, while the union bound needs n small relative to m^{o(1)}. The lower bound may well be true, and the argument is probably repairable with Le Cam's inequality or a direct binomial tail, but as it stands the matching lower bound is unproven. Relatedly, the statement should say something about the regime of n: for n < log m/log log m, a lower bound of log m/log log m on the max multiplicity is trivially impossible. Also, Theorem 3.1's constant 3 only works with base-2 logs in the Chernoff bound as written; with natural logs the constant needs to grow. These are minor fixes, but they are real.

Bottom line: the upper-bound half is a solid, citable contribution. The lower-bound half needs a rigorous repair before the 'essentially tight' claims can be trusted. I would send it to review, and I would expect the authors to fix the Poisson step. Worth a reading-group slot once the revised version is out.

Referee Report

3 major / 5 minor

Summary. The paper studies exact maximin share (MMS) fairness when the supply of indivisible items can be adjusted after the fact: goods may be duplicated, or chores may be discarded. The proposed solution concept is an MMS multi-allocation in which each agent receives a subset of value at least its MMS, while the characteristic vector of the multi-allocation is bounded in ℓ1 and ℓ∞ for goods, and in the number of zero entries for chores. For monotone goods valuations, Theorem 3.1 gives an MMS multi-allocation with ℓ∞ ≤ 3 log m and ℓ1 ≤ m; for identically ordered valuations, Theorem 3.4 improves the multiplicity to O(√log m) at the cost of an additive m·Õ(1/√n) in ℓ1; and for additive valuations, Theorem 3.9 gives ℓ∞ ≤ 2 and ℓ1 ≤ 2m. On the chores side, Theorem 4.2 shows that at most m/e chores need be left unassigned under monotone costs, Theorem 4.4 gives an Õ(m/n^{1/4}) bound under identically ordered costs, and Theorem 4.9 gives a 2m/11 + n bound for additive costs. The paper also provides lower bounds claiming that the monotone-valuation goods bound and the monotone-cost chores bound are essentially tight, plus an entitlement generalization in Appendix A and an NP-hardness result in Appendix B.

Significance. If the results hold, the paper makes a clean conceptual contribution: the well-known impossibility of exact MMS under general monotone preferences can be bypassed by allowing limited duplication or disposal, and the quantitative bounds are quite strong. The upper-bound proofs are mostly self-contained and use standard probabilistic tools: random sampling from MMS-inducing partitions, Chernoff/Hoeffding bounds, and Chebyshev's inequality. There are no fitted parameters and no circularity: the existence arguments follow directly from the definition of MMS. The paper is honest about its external dependencies, notably the use of the cardinality-constrained MMS result of Hummel and Hetland in Theorem 3.9. The main caveat is the lower-bound proof for goods in Theorem 3.10, which currently uses an unquantified Poisson approximation; that gap is localized but load-bearing for the claimed tightness.

major comments (3)

[§3.4, Eq. (14)–(16)] The proof of Theorem 3.10 replaces the exact distribution χ_g ∼ Bin(n,1/n) by χ_g ∼ Poi(1) and treats this as equality in Eq. (14), then uses the Poisson CDF in the union bound in Eq. (16). No bound on the approximation error is supplied. The total-variation error of the Poisson approximation is of order 1/n, which is not automatically negligible compared with the n^{-2n} target probability when n may be as small as 2 under the stated condition m ≥ 2 n log n e(ℓ+1)!. As written, the ``essentially tight'' claim for Theorem 3.1 is not established. Please either prove a direct binomial tail bound or quantify the approximation (e.g., via Le Cam's inequality), and state explicitly which ranges of n and m are used.
[§3.1, Lemma 3.2] The Chernoff step uses t = 3 log m and the bound Pr{χ_g ≥ t} ≤ 2^{-t}. With natural logarithms this gives only m^{-3 ln 2}, not m^{-3}, so the union-bound estimate Pr{G_1^c} ≤ 1/m^2 does not follow; with base-2 logarithms the constant 3 is correct, but the base is never stated. The same issue appears in Lemma A.4 for the entitlement result. Please state the logarithmic base explicitly, or adjust the constants and the admissible range of m if natural logarithms are intended.
[§3.3, Theorem 3.9] The proof of Theorem 3.9 is essentially a reduction to the external result of Hummel and Hetland [HH22] on maximin shares under cardinality constraints, but the cited theorem is not stated in the paper. The reader cannot verify that the required guarantee (exact or 1/2-approximate) and the feasibility condition match exactly what is needed in the auxiliary instance ~I. Since the additive-goods result is one of the headline contributions, please state the invoked result precisely or give a proof of the needed consequence.

minor comments (5)

[§3.2, Corollary 3.7] Corollary 3.7 says ``every fair division instance with additive ordered valuations'' while the surrounding section is about identically ordered valuations; the proof and Lemmas 3.5–3.6 concern identically ordered valuations. Please correct the terminology and make clear whether the corollary is intended for all identically ordered valuations or only additive ordered ones.
[§3.1, proof of Theorem 3.1] The union-bound calculation in the proof of Theorem 3.1 writes Pr{G_1^c} + Pr{G_1^c} where the second term should be Pr{G_2^c}; the numerical conclusion is unaffected.
[§4.1, Theorem 4.2] The proof says the expected number of unassigned chores ``equals'' m/e, but the exact expectation is m(1−1/n)^n, which is strictly less than m/e. The inequality E[‖χ^R‖_z] ≤ m/e is what is needed and is enough for the stated bound.
[§3.2, Algorithm/Lemma 3.8] There is a garbled cross-reference in the paragraph introducing the dyadic prefixes: ``the property laid out in /question_question'' appears to be a broken reference to the definition of identically ordered valuations. Please repair it.
[§2 and §4, notation] The quantity ‖χ^A‖_z is written as a set in the text, e.g., ‖χ^A‖_z := {j ∈ [m] | χ^A_j = 0}, even though it is used as the cardinality of that set. This is a minor notational inconsistency worth fixing.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the MMS upper bounds are derived self-containedly from the definition of maximin shares; self-citations are peripheral, and the lower-bound Poisson/Binomial gap is a correctness issue rather than a circular reduction.

full rationale

The central derivations are self-contained probabilistic-method arguments built directly on Definition 2.1, with no fitted parameters and no input redeployed as output. Theorem 3.1 samples each R_i uniformly from agent i's MMS-inducing partition M^i; the guarantee v_i(R_i) ≥ μ_i holds for every realization by definition of M^i, and the ℓ∞ and ℓ1 bounds follow from E[χ_g] = 1 (each good lies in exactly one part of each partition), a Chernoff bound, and Markov's inequality — none of which assume the conclusion. Theorems 3.4, 4.2, 4.4, and A.2 follow the same pattern: concentration bounds (Hoeffding, Chebyshev, Markov) plus deterministic redistribution lemmas (Algorithms 1–3) proved by explicit potential arguments. Theorem 3.9 reduces to the external cardinality-constrained result [HH22], with the amplification ~μ_i ≥ 2μ_i proved directly by cyclically pairing parts of an MMS partition; Theorem 4.9 transparently transfers the external 11/9 approximation [HL21]. The lower bounds (Theorems 3.10 and 4.8) construct fresh random instances and union-bound over the finite family F, never invoking the upper-bound claims. Self-citations [BB18] and [BK20] appear only as pointers alongside external results and carry no load. The skeptic's concerns are rigor gaps, not circular reductions: in Theorem 3.10 (Section 3.4, equations (14)–(16)) the count χ_g ~ Bin(n,1/n) is replaced by Poi(1) with no quantified approximation error, and the asserted independence of χ_g across goods is false (negative correlation), leaving the 'essentially tight' claim unproven as written; Theorem 4.8's lower-tail Chernoff step is applied at a threshold above the true mean. These affect correctness of the matching lower bounds, not the direction of derivation, so they do not raise the circularity score.

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

The upper-bound existence proofs are self-contained given standard concentration inequalities and the definition of MMS. Two results import external theorems ([HH22], [HL21]) as black boxes. The lower-bound proofs introduce an unquantified Poisson approximation for the Binomial copy-count distribution; this is the least supported step in the paper.

assumptions (7)

domain assumption Valuations are monotone and normalized (vi(∅)=0).
Standard in fair division; used throughout Section 3.
domain assumption Cost functions are monotone and normalized (ci(∅)=0).
Standard for chores; used throughout Section 4.
standard math MMS-inducing partitions exist for all agents.
Follows from finite maximization over all n-partitions of [m]; used in Definitions 2.1 and 4.1.
standard math Standard concentration inequalities: Chernoff, Hoeffding, Chebyshev, Markov, union bound.
Used in the probabilistic proofs of Theorems 3.1, 3.4, 4.2, and 4.4.
domain assumption The [HH22] theorem: for additive valuations under cardinality constraints, there exists an exact allocation giving each agent at least half its constrained maximin share.
Imported as a black box in Theorem 3.9; no proof is given in the paper.
domain assumption The [HL21] theorem: for additive costs, there exists an allocation giving each agent cost at most 11/9 times its MMS.
Imported as a black box in Theorem 4.9.
ad hoc to paper Binomial(n,1/n) copy counts are well approximated by Poisson(1) with negligible error in the relevant asymptotic regime.
Used in the lower-bound proofs (Theorems 3.10 and 4.8) without quantifying the approximation error.

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Pith. "Pith review of Exact Maximin Share Fairness via Adjusted Supply." pith.science (2026). https://pith.science/paper/DWO2GZAN

@misc{pith2026250203789,
  author       = {Pith},
  title        = {Pith review of: Exact Maximin Share Fairness via Adjusted Supply},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DWO2GZAN}},
  note         = {Machine review of arXiv:2502.03789}
}

abstract

This work addresses fair allocation of indivisible items in settings wherein it is feasible to create copies of resources or dispose of tasks. We establish that exact maximin share (MMS) fairness can be achieved via limited duplication of goods even under monotone valuations. We also show that, when allocating chores under monotone costs, MMS fairness is always feasible with limited disposal of chores. Since monotone valuations do not admit any nontrivial approximation guarantees for MMS, our results highlight that such barriers can be circumvented by post facto adjustments in the supply of the items. We prove that, for division of $m$ goods among $n$ agents with monotone valuations, there always exists an assignment of subsets of goods to the agents such that they receive at least their maximin shares and no single good is allocated to more than $3 \log m$ agents. In addition, the sum of the sizes of the assigned subsets does not exceed $m$. For identically ordered valuations, we obtain an upper bound of $O(\sqrt{\log m})$ on the maximum assignment multiplicity across goods and an $m + \widetilde{O}\left(\frac{m}{\sqrt{n}} \right)$ bound for the total number of goods assigned. Further, for additive valuations, we prove that there always exists an MMS assignment in which no single good is allocated to more than $2$ agents and the total number of goods assigned is at most $2m$. For chores, we upper bound the number of chores that need to be discarded for ensuring MMS fairness. We prove that, under monotone costs, there exists an MMS assignment in which at most $\frac{m}{e}$ remain unassigned. For identically ordered costs, we establish that MMS fairness can be achieved while keeping at most $\widetilde{O} \left(\frac{m}{n^{1/4}} \right)$ chores unassigned. We also prove that the obtained bounds for monotone valuations and monotone costs are essentially tight.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Fair division of indivisible goods: Recent progress and open questions

Georgios Amanatidis, Haris Aziz, Georgios Birmpas, Aris Filos-Ratsikas, Bo Li, Herv \'e Moulin, Alexandros A Voudouris, and Xiaowei Wu. Fair division of indivisible goods: Recent progress and open questions. Artificial Intelligence , 322:103965, 2023

work page 2023

[2] [2]

Algorithms for max-min share fair allocation of indivisible chores

Haris Aziz, Gerhard Rauchecker, Guido Schryen, and Toby Walsh. Algorithms for max-min share fair allocation of indivisible chores. Proceedings of the AAAI Conference on Artificial Intelligence , 31(1), Feb. 2017

work page 2017

[3] [3]

Fair division under cardinality constraints

Arpita Biswas and Siddharth Barman. Fair division under cardinality constraints. In IJCAI , pages 91--97, 2018

work page 2018

[4] [4]

Course match: A large-scale implementation of approximate competitive equilibrium from equal incomes for combinatorial allocation

Eric Budish, G \'e rard P Cachon, Judd B Kessler, and Abraham Othman. Course match: A large-scale implementation of approximate competitive equilibrium from equal incomes for combinatorial allocation. Operations Research , 65(2):314--336, 2017

work page 2017

[5] [5]

Share-based fairness for arbitrary entitlements

Moshe Babaioff and Uriel Feige. Share-based fairness for arbitrary entitlements. ArXiv , abs/2405.14575, 2024

arXiv 2024

[6] [6]

Approximation algorithms for maximin fair division

Siddharth Barman and Sanath Kumar Krishnamurthy. Approximation algorithms for maximin fair division. ACM Transactions on Economics and Computation (TEAC) , 8(1):1--28, 2020

work page 2020

[7] [7]

Characterizing conflicts in fair division of indivisible goods using a scale of criteria

Sylvain Bouveret and Michel Lema\^ tre. Characterizing conflicts in fair division of indivisible goods using a scale of criteria. In Proceedings of the 2014 International Conference on Autonomous Agents and Multi-Agent Systems , AAMAS '14, pages 1321--1328, Richland, SC, 2014. International Foundation for Autonomous Agents and Multiagent Systems

work page 2014

[8] [8]

The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes

Eric Budish. The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy , 119(6):1061--1103, 2011

2011