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EFX for Additive Chores: Nonexistence, Pareto Incompatibility, and Bi-Valued Existence

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Pith's one-line read Additive chores need not admit EFX allocations once there are four or more agents, even when costs take only three values.

desk verdict Clean non-existence of EFX for additive chores (even tri-valued, n≥4), first positive-cost EFX-PO separation, and a solid four-agent bi-valued existence proof. read the letter →

arxiv 2606.08872 v2 pith:FQDA5B6N submitted 2026-06-07 cs.GT econ.TH

Wentao He , Biaoshuai Tao This is my paper · ORCID
classification cs.GTecon.TH
keywords EFXindivisiblechoresadditivecoststri-valuedbi-valuedPareto-optimalityfairdivisionexistence
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 settles a long-open existence question for fair division of indivisible chores: envy-freeness up to any chore (EFX) is not guaranteed for additive cost functions. For every number of agents at least four the authors exhibit a concrete tri-valued instance that admits no EFX allocation at all; the construction uses only three chore types and two agent types, both of which are tight. They further show that, already for strictly positive bi-valued costs, every EFX allocation can fail to be Pareto-optimal when there are four or more agents. At the same time they prove that an EFX allocation does exist for every four-agent bi-valued instance. Together the results draw a sharp line: EFX can fail for additive chores, and even when it exists it can be incompatible with efficiency.

What carries the argument

A pair of carefully ratio-tuned constructions (tri-valued costs 1, q = s+2, r = s(q+1)/2 and bi-valued costs {1,r} with r > ⌈n/2⌉+1) that force every candidate allocation to leave some agent strongly envious after the removal of any single chore, together with a multi-case constructive algorithm that always finds an EFX allocation when n=4 and costs are bi-valued.

What would settle it

Either exhibit an EFX allocation for the explicit four-agent, thirteen-chore tri-valued instance of Table 1 (or its general-n extension in Appendix A), or produce a positive bi-valued instance with four or more agents in which some EFX allocation is also Pareto-optimal.

Watch

Extended reading notes

Core claim

For every n ≥ 4 there exist additive tri-valued chore instances with no EFX allocation, and there exist strictly positive bi-valued instances in which every EFX allocation is Pareto-dominated; yet every four-agent bi-valued instance still admits an EFX allocation.

Load-bearing premise

The non-existence proofs rely on specific numerical ratios among the three (or two) cost values; if those exact ratios were forbidden by modelling constraints the counter-examples would no longer work.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies envy-freeness up to any item (EFX) for indivisible chores under additive costs. Theorem 1 shows that for every n≥4 there exist tri-valued additive instances (three positive cost levels, three chore types, two agent types) with no EFX allocation; the construction is tight with respect to the known positive results for two chore types or identical agents. Theorem 2 exhibits, for every n≥4 and sufficiently large r, strictly positive bi-valued instances in which every EFX allocation fails to be Pareto-optimal—the first such incompatibility that avoids zero costs—and notes that the agent threshold is tight by prior three-agent results. Theorem 3 proves that every four-agent bi-valued instance nevertheless admits an EFX allocation, via a modular argument that first inserts M34 items, then concatenates carefully chosen canonical M01 prefixes with multigraph orientations of M2 items (including gap-filling and residual exceptional configurations).

Significance. The non-existence result resolves a long-standing open question for additive chores and cleanly separates the chore setting from additive goods, where EFX existence remains open. The positive-cost EFX–PO incompatibility for bi-valued instances is new and tight in the number of agents. The four-agent existence theorem, while technical, supplies the first general positive result beyond the previously settled cases of two agents, identical orderings, binary costs, or two chore types. All constructions are fully explicit and finite; the counting and modular-arithmetic arguments are self-contained and do not rely on external data or hidden modelling constraints. These contributions are of clear interest to the fair-division community.

minor comments (4)
  1. Section 5 (especially 5.5) is extremely dense. A short high-level roadmap or a table summarising the gap-filling choices for each b∈{0,1,2,3} would help readers navigate the case analysis without changing any technical content.
  2. In the n=4 tri-valued example (Table 1 and Proposition 1) the total cost 100 and the bound 25 are clear; a one-line remark that the same modular arithmetic reappears with the general parameters of Appendix A would make the generalisation more transparent.
  3. Definition 2 introduces the threshold τi; later proofs sometimes switch between the “remove any item” language and the τi formulation. Consistent use of one of the two would improve readability.
  4. A few minor typos appear (e.g., “disutility must be at most 25 up to removing to one item” in the proof of Proposition 1; “the numbers of types … are both tight” in the abstract). A light copy-edit pass would suffice.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all three theorems rest on explicit finite constructions and self-contained case analyses, with parameters chosen once to satisfy modular/counting inequalities rather than fitted to data or smuggled via self-citation.

full rationale

The paper is a pure existence/non-existence theory paper in algorithmic fair division. Theorem 1 constructs an explicit tri-valued instance (three item classes A/B/C, two agent types, concrete costs p=1, q=s+2, r=s(q+1)/2) and derives a contradiction from the EFX definition via bundle-size counting and modular arithmetic (Propositions 1–2 and Appendix A); the same pattern holds for the bi-valued PO-incompatibility construction of Theorem 2 (r > ⌈n/2⌉+1) and the exhaustive case analysis of Theorem 3 (M01/M2/M34 partition, canonical prefixes, multigraph orientation, gap-filling). None of these steps define a quantity in terms of the claimed conclusion, fit a free parameter to a target quantity and then “predict” it, or rest on a uniqueness theorem or ansatz imported solely from overlapping-author prior work. Citations (Aziz et al. 2023 for two chore types, Garg et al. 2023 for n≤3 bi-valued EFX+fPO, Tao et al. 2025 for binary EFX+PO) are used only for tightness statements and related-work context; they are not load-bearing premises inside the new proofs. Consequently the derivation chain is independent of its own outputs and contains no circular reduction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The paper works entirely inside the standard additive-chore model. The only non-standard ingredients are the concrete numerical cost triples chosen for the counter-examples; those numbers are free parameters of the constructions, not fitted quantities. No new physical or mathematical entities are postulated.

free parameters (2)
  • tri-valued cost triple (p,q,r) = 1, 2t2+3, (2t2+1)(t2+2)
    Chosen as p=1, q=s+2, r=s(q+1)/2 so that modular arithmetic and counting inequalities become strict; any other triple that satisfies the same inequalities would work, but the paper fixes one concrete triple.
  • bi-valued threshold r = > ceil(n/2)+1
    Any r>⌈n/2⌉+1 makes the Pareto-domination argument go through; the paper treats r as a free parameter above that bound.
assumptions (3)
  • domain assumption Standard definition of additive cost functions and of EFX for chores (envy disappears after removal of any single chore from the envious agent's bundle).
    Taken as given from the fair-division literature; restated in Section 2.
  • domain assumption Pareto optimality is defined with respect to complete allocations of indivisible items (no fractional or incomplete allocations).
    Standard; used throughout Sections 3-4.
  • standard math Existence of EFX for two chore types (Aziz et al. 2023) and for three bi-valued agents (Garg et al. 2023).
    Cited only for tightness; not used inside the new proofs.

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Cite this review

Pith. "Pith review of EFX for Additive Chores: Nonexistence, Pareto Incompatibility, and Bi-Valued Existence." pith.science (2026). https://pith.science/paper/FQDA5B6N

@misc{pith2026260608872,
  author       = {Pith},
  title        = {Pith review of: EFX for Additive Chores: Nonexistence, Pareto Incompatibility, and Bi-Valued Existence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FQDA5B6N}},
  note         = {Machine review of arXiv:2606.08872}
}
abstract

We consider the fair division problem of indivisible chores and resolve the long-standing open problem for the existence of EFX (envy-free up to any item) allocations with additive cost functions. We show that, even for tri-valued additive cost functions, for every $n\geq 4$, there exists an instance with $n$ agents where no EFX allocation exists. Our counterexample only uses three types of chores and two types of agents. The numbers of types for chores and agents are both tight: an EFX allocation is known to exist for one type of agents (i.e., with identical cost functions) or two types of chores. We then consider bi-valued instances. We show that, for every $n\geq 4$, there exists an instance with $n$ agents where every EFX allocation is not Pareto-optimal. This is also the first example showing the incompatibility of EFX and Pareto-optimality when the costs of items are positive: existing examples showing the incompatibility of EFX and Pareto-optimal exploit items with $0$ costs. Our result shows such an example exists even for bi-valued instances. The number of agents $n$ is also tight: for $n\leq 3$, it is known that EFX is compatible with Pareto-optimality. Finally, we also show that an EFX allocation is guaranteed to exist for $n=4$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Non-Existence of EFX Chore Allocations for Monotone Cost Functions with Binary Marginals

    cs.GT 2026-08 conditional novelty 8.0 of 10

    EFX allocations need not exist for chores when agents have binary XOS or binary supermodular costs, as shown by 18-agent, 53-chore counterexamples.

Reference graph

Works this paper leans on

15 extracted references · 2 linked inside Pith · cited by 1 Pith paper

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