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Under binary valuations, EF1, strategyproofness, neutrality, minimal completeness and IDU uniquely force the maximum-Nash-welfare rule for any number of agents.

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2026-07-14 00:39 UTC pith:3BQKFQ5Z

load-bearing objection First full-space uniqueness theorems for MNW under binary valuations; clean proofs, necessary axioms, and a usable two-agent alternative that avoids the new IDU axiom.

arxiv 2607.10064 v1 pith:3BQKFQ5Z submitted 2026-07-11 econ.TH cs.GT

Fair Division with Binary Valuations: Characterizations

Florian Brandl , Warut Suksompong , Nicholas Teh This is my paper
classification econ.TH cs.GT MSC 91B3291B14
keywords fair divisionbinary valuationsmaximum Nash welfareEF1strategyproofnessresource-monotonicityaxiomatic characterization
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

When goods are indivisible and every agent simply approves or disapproves each good, three classic welfare rules become identical: maximum Nash welfare, leximin, and every additive strictly-concave welfarist rule. The paper proves that this single rule is the only allocation rule that simultaneously satisfies envy-freeness up to one good, strategyproofness, neutrality, minimal completeness and a new invariance property called IDU. For two agents the same uniqueness holds after IDU is replaced by non-redundancy and resource-monotonicity. Both characterizations are tight: drop any listed axiom and a non-MNW rule appears that still satisfies the rest. The result therefore supplies the first axiomatic foundation that singles out this common rule among all conceivable allocation procedures in the binary domain.

Core claim

Any rule that is EF1, strategyproof, neutral, minimally complete and IDU must return a maximum-Nash-welfare allocation for every binary-valuation instance with any number of agents; for two agents the same conclusion holds when IDU is replaced by non-redundancy plus resource-monotonicity, and every listed axiom is indispensable.

What carries the argument

Critical-path characterization of MNW (Lemma 3.1): an allocation maximises Nash welfare if and only if it is Pareto optimal and contains no critical path of agents along which one agent’s bundle is larger by more than one than a later agent’s bundle; the uniqueness proofs reduce every counter-example to a shortest critical path and then use the remaining axioms to produce a shorter one.

Load-bearing premise

The new invariance axiom IDU—that an agent’s decision to stop approving an unassigned good never changes the chosen allocation—is indispensable for the multi-agent uniqueness claim.

What would settle it

Exhibit any allocation rule, other than an MNW rule, that is EF1, strategyproof, neutral, minimally complete and IDU on binary instances with three or more agents; or show that the five axioms force only a proper subclass of MNW allocations once n>2.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper studies fair division of indivisible goods under binary additive valuations, where MNW, leximin, and strictly concave additive welfarist rules coincide. It proves that, for any fixed n≥2, this common rule is the unique allocation rule satisfying EF1, strategyproofness, neutrality, minimal completeness, and the new axiom IDU (invariance under disapproving unassigned goods). For n=2 an alternative characterization replaces IDU by non-redundancy plus resource-monotonicity. Both uniqueness results are tight: each axiom is shown necessary by explicit counter-examples (Propositions 3.4 and 4.5). Additional theorems refine the consistent tie-breaking behaviour that any such rule must exhibit on instances with the same number of (valued) goods.

Significance. These appear to be the first characterizations of MNW among the class of all (not merely welfarist) allocation rules in any valuation domain. Because MNW is already known to be strategyproof, resource-monotone and polynomial-time computable under binary valuations, the axiomatic uniqueness results supply strong normative justification for its use in this domain. The proofs are fully written, rest on the critical-path characterization of MNW (Halpern et al., 2020), employ careful shortest-path and inductive arguments, and include complete independence examples. The two-agent alternative characterization further shows that the newly introduced IDU axiom is not indispensable for the n=2 case.

minor comments (5)
  1. Section 2 (Definitions 2.3–2.4 and Lemma 2.8): the logical relationships among PO, non-redundancy and minimal completeness are stated clearly, but a short remark that any two of them imply the third would help readers who skip the lemma.
  2. Definition 2.7 (IDU) and the surrounding paragraph: the analogy to IIA / non-bossiness is helpful; a one-sentence example of a natural rule (e.g., a consistent picking sequence) that satisfies IDU would make the axiom more immediately intuitive.
  3. Proof of Theorem 3.2, Case 1: the reductions that prune valuations via IDU while preserving the critical path are correct, yet the notation for the successive profiles P, P', P'', P''' becomes dense; a short table or diagram summarizing the successive changes would improve readability.
  4. Throughout the manuscript (especially Appendices A–B): several author names contain residual LaTeX accent artefacts (e.g., "Lemaˆ ıtre", "Herv´ e"); these should be cleaned for the camera-ready version.
  5. Section 5: the open question of extending the resource-monotonicity characterization beyond n=2 is well-posed; a brief remark on the combinatorial obstacle (the 2^n-1 numbers needed to describe a profile) already present in the text could be moved into the main body for emphasis.

Circularity Check

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No significant circularity: pure axiomatic uniqueness proofs with independent definitions and fully explicit case analysis

full rationale

The paper defines MNW independently (Def. 2.9) via the product of positive utilities (equivalently leximin or strictly concave additive welfarism) and states the axioms (EF1, strategyproofness, neutrality, minimal completeness, IDU/non-redundancy/resource-monotonicity) without reference to that product. Theorems 3.2 and 4.2 prove uniqueness by assuming a rule satisfies the axioms, invoking the external critical-path characterization of MNW (Lemma 3.1 of Halpern et al. 2020, no author overlap), and deriving contradictions via explicit instance modifications that preserve the allocation under IDU (or non-redundancy + resource-monotonicity) while producing shorter critical paths or EF1 violations. All reductions are written out in Cases 1–2 of Thm. 3.2 and the induction of Lemma 4.3; necessity is shown by concrete counter-example rules (Props. 3.4, 4.5). The sole self-citations (e.g., Suksompong–Teh 2022 that MNWtie satisfies resource-monotonicity) merely confirm that the target rule meets the axioms; they are not used inside the uniqueness arguments and do not force the result by construction. No fitted parameters, self-definitional loops, or load-bearing uniqueness theorems imported from the authors appear. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 1 invented entities

No free parameters. The result rests on five (respectively six) standard or newly introduced axioms of fair division plus the domain restriction to binary additive valuations. IDU is the sole ad-hoc axiom introduced by the paper; the rest are classical. No new physical or mathematical entities are postulated.

axioms (7)
  • domain assumption Binary additive valuations (each agent values each good 0 or 1)
    Stated in Section 2 and used throughout; the coincidence of MNW/leximin/concave welfarists and the strategyproofness of MNW hold only in this domain.
  • standard math EF1 (envy-freeness up to one good)
    Definition 2.1; classical relaxation of envy-freeness (Budish 2011).
  • standard math Strategyproofness
    Definition 2.5; standard non-manipulability.
  • standard math Neutrality (permutation of good labels does not change utilities)
    Definition 2.6.
  • standard math Minimal completeness (allocate all valued goods, none of the unvalued)
    Definition 2.4; previously used by Halpern et al. 2020.
  • ad hoc to paper IDU (invariance under disapproving unassigned goods)
    Definition 2.7; newly introduced; load-bearing for the n-agent uniqueness proof.
  • standard math Non-redundancy + resource-monotonicity (two-agent alternative)
    Definitions 2.3 and 4.1; classical efficiency and solidarity axioms.
invented entities (1)
  • IDU (invariance under disapproving unassigned goods) no independent evidence
    purpose: Closes the uniqueness proof for arbitrary n by allowing controlled pruning of valuations while preserving the returned allocation.
    The axiom is new to this paper; while motivated by independence-of-irrelevant-alternatives and non-bossiness, it is not previously named or used as a primitive in the fair-division literature.

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read the original abstract

We consider the fair allocation of indivisible goods with binary valuations. In this setting, the maximum Nash welfare rule, the leximin rule, and all additive welfarist rules with a strictly concave function coincide. We show that for any number of agents, this rule is the only rule that satisfies envy-freeness up to one good, strategyproofness, neutrality, minimal completeness, and invariance under disapproving unassigned goods (IDU). Moreover, we present an alternative characterization for two agents, where we replace IDU with non-redundancy and resource-monotonicity. In both characterizations, all axioms are necessary.

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Works this paper leans on

38 extracted references · 1 linked inside Pith

  1. [1]

    Online fair division: Analysing a food bank problem

    Martin Aleksandrov, Haris Aziz, Serge Gaspers, and Toby Walsh. Online fair division: Analysing a food bank problem. In Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI), pages 2540--2546, 2015

    2015

  2. [2]

    Voudouris

    Georgios Amanatidis, Georgios Birmpas, Aris Filos-Ratsikas, Alexandros Hollender, and Alexandros A. Voudouris. Maximum N ash welfare and other stories about EFX . Theoretical Computer Science, 863: 0 69--85, 2021

    2021

  3. [3]

    Voudouris, and Xiaowei Wu

    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: 0 103965, 2023

    2023

  4. [4]

    Kenneth J. Arrow. A difficulty in the concept of social welfare. Journal of Political Economy, 58 0 (4): 0 328--346, 1950

    1950

  5. [5]

    Fair and truthful mechanisms for dichotomous valuations

    Moshe Babaioff, Tomer Ezra, and Uriel Feige. Fair and truthful mechanisms for dichotomous valuations. In Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI), pages 5119--5126, 2021

    2021

  6. [6]

    Truthful and fair mechanisms for matroid-rank valuations

    Siddharth Barman and Paritosh Verma. Truthful and fair mechanisms for matroid-rank valuations. In Proceedings of the 36th AAAI Conference on Artificial Intelligence (AAAI), pages 4801--4808, 2022

    2022

  7. [7]

    Greedy algorithms for maximizing N ash social welfare

    Siddharth Barman, Sanath Kumar Krishnamurthy, and Rohit Vaish. Greedy algorithms for maximizing N ash social welfare. In Proceedings of the 17th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), pages 7--13, 2018

    2018

  8. [8]

    Finding fair and efficient allocations for matroid rank valuations

    Nawal Benabbou, Mithun Chakraborty, Ayumi Igarashi, and Yair Zick. Finding fair and efficient allocations for matroid rank valuations. ACM Transactions on Economics and Computation, 9 0 (4): 0 21:1--21:41, 2021

    2021

  9. [9]

    Random matching under dichotomous preferences

    Anna Bogomolnaia and Herv\' e Moulin. Random matching under dichotomous preferences. Econometrica, 72 0 (1): 0 257--279, 2004

    2004

  10. [10]

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

    Sylvain Bouveret and Michel Lemaître. Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Autonomous Agents and Multi-Agent Systems, 30 0 (2): 0 259--290, 2016

    2016

  11. [11]

    Brams and Peter C

    Steven J. Brams and Peter C. Fishburn. Approval Voting. Springer, 2007

    2007

  12. [12]

    Brams and Alan D

    Steven J. Brams and Alan D. Taylor. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge University Press, 1996

    1996

  13. [13]

    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 0 (6): 0 1061--1103, 2011

    2011

  14. [14]

    Procaccia, Nisarg Shah, and Junxing Wang

    Ioannis Caragiannis, David Kurokawa, Herv\' e Moulin, Ariel D. Procaccia, Nisarg Shah, and Junxing Wang. The unreasonable fairness of maximum N ash welfare. ACM Transactions on Economics and Computation, 7 0 (3): 0 12:1--12:32, 2019

    2019

  15. [15]

    On the fairness of additive welfarist rules

    Karen Frilya Celine, Warut Suksompong, and Sheung Man Yuen. On the fairness of additive welfarist rules. ACM Transactions on Economics and Computation, 14 0 (2): 0 5:1--5:50, 2026

    2026

  16. [16]

    Picking sequences and monotonicity in weighted fair division

    Mithun Chakraborty, Ulrike Schmidt-Kraepelin, and Warut Suksompong. Picking sequences and monotonicity in weighted fair division. Artificial Intelligence, 301: 0 103578, 2021

    2021

  17. [17]

    Maximizing N ash product social welfare in allocating indivisible goods

    Andreas Darmann and Joachim Schauer. Maximizing N ash product social welfare in allocating indivisible goods. European Journal of Operational Research, 247 0 (2): 0 548--559, 2015

    2015

  18. [18]

    On the fairness of normalized p -means for allocating goods and chores

    Owen Eckart, Alexandros Psomas, and Paritosh Verma. On the fairness of normalized p -means for allocating goods and chores. In Proceedings of the 25th ACM Conference on Economics and Computation (EC), page 1267, 2024

    2024

  19. [19]

    Equitable allocations of indivisible goods

    Rupert Freeman, Sujoy Sikdar, Rohit Vaish, and Lirong Xia. Equitable allocations of indivisible goods. In Proceedings of the 28th International Joint Conference on Artificial Intelligence (IJCAI), pages 280--286, 2019

    2019

  20. [20]

    A fair and truthful mechanism with limited subsidy

    Hiromichi Goko, Ayumi Igarashi, Yasushi Kawase, Kazuhisa Makino, Hanna Sumita, Akihisa Tamura, Yu Yokoi, and Makoto Yokoo. A fair and truthful mechanism with limited subsidy. Games and Economic Behavior, 144: 0 49--70, 2024

    2024

  21. [21]

    Procaccia

    Jonathan Goldman and Ariel D. Procaccia. Spliddit: Unleashing fair division algorithms. ACM SIG ecom Exchanges , 13 0 (2): 0 41--46, 2014

    2014

  22. [22]

    Procaccia, Alexandros Psomas, and Nisarg Shah

    Daniel Halpern, Ariel D. Procaccia, Alexandros Psomas, and Nisarg Shah. Fair division with binary valuations: One rule to rule them all. In Proceedings of the 16th Conference on Web and Internet Economics (WINE), pages 370--383, 2020. Extended version available at arXiv:2007.06073v2

    Pith/arXiv arXiv 2020

  23. [23]

    Fair division through information withholding

    Hadi Hosseini, Sujoy Sikdar, Rohit Vaish, Hejun Wang, and Lirong Xia. Fair division through information withholding. In Proceedings of the 34th AAAI Conference on Artificial Intelligence (AAAI), pages 2014--2021, 2020

    2014

  24. [24]

    Marc Kilgour

    D. Marc Kilgour. Approval balloting for multi-winner elections. In Jean-Fran c ois Laslier and M. Remzi Sanver, editors, Handbook on Approval Voting, chapter 6, pages 105--124. Springer, 2010

    2010

  25. [25]

    Strategy-proofness, solidarity, and consistency for multiple assignment problems

    Bettina Klaus and Eiichi Miyagawa. Strategy-proofness, solidarity, and consistency for multiple assignment problems. International Journal of Game Theory, 30 0 (3): 0 421--435, 2002

    2002

  26. [26]

    Voudouris

    Maria Kyropoulou, Warut Suksompong, and Alexandros A. Voudouris. Almost envy-freeness in group resource allocation. Theoretical Computer Science, 841: 0 110--123, 2020

    2020

  27. [27]

    APX -hardness of maximizing N ash social welfare with indivisible items

    Euiwoong Lee. APX -hardness of maximizing N ash social welfare with indivisible items. Information Processing Letters, 122: 0 17--20, 2017

    2017

  28. [28]

    Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi

    Richard J. Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi. On approximately fair allocations of indivisible goods. In Proceedings of the 5th ACM Conference on Electronic Commerce (EC), pages 125--131, 2004

    2004

  29. [29]

    Pareto efficient combinatorial auctions: Dichotomous preferences without quasilinearity

    Komal Malik and Debasis Mishra. Pareto efficient combinatorial auctions: Dichotomous preferences without quasilinearity. Journal of Economic Theory, 191: 0 105128, 2021

    2021

  30. [30]

    Weighted envy-freeness for submodular valuations

    Luisa Montanari, Ulrike Schmidt-Kraepelin, Warut Suksompong, and Nicholas Teh. Weighted envy-freeness for submodular valuations. Social Choice and Welfare, 65 0 (4): 0 983--1016, 2025

    2025

  31. [31]

    Fair Division and Collective Welfare

    Herv\' e Moulin. Fair Division and Collective Welfare. MIT Press, 2003

    2003

  32. [32]

    Cake-Cutting Algorithms: Be Fair if You Can

    Jack Robertson and William Webb. Cake-Cutting Algorithms: Be Fair if You Can. Peters/CRC Press, 1998

    1998

  33. [33]

    A characterization of maximum N ash welfare for indivisible goods

    Warut Suksompong. A characterization of maximum N ash welfare for indivisible goods. Economics Letters, 222: 0 110956, 2023

    2023

  34. [34]

    On maximum weighted N ash welfare for binary valuations

    Warut Suksompong and Nicholas Teh. On maximum weighted N ash welfare for binary valuations. Mathematical Social Sciences, 117: 0 101--108, 2022

    2022

  35. [35]

    Weighted fair division with matroid-rank valuations: M onotonicity and strategyproofness

    Warut Suksompong and Nicholas Teh. Weighted fair division with matroid-rank valuations: M onotonicity and strategyproofness. Mathematical Social Sciences, 126: 0 48--59, 2023

    2023

  36. [36]

    Non-bossiness

    William Thomson. Non-bossiness. Social Choice and Welfare, 47 0 (3): 0 665--696, 2016

    2016

  37. [37]

    A general framework for fair allocation under matroid rank valuations

    Vignesh Viswanathan and Yair Zick. A general framework for fair allocation under matroid rank valuations. ACM Transactions on Economics and Computation, 13 0 (3): 0 14:1--14:32, 2025

    2025

  38. [38]

    Extending the characterization of maximum N ash welfare

    Sheung Man Yuen and Warut Suksompong. Extending the characterization of maximum N ash welfare. Economics Letters, 224: 0 111030, 2023

    2023