{"id":"2835942f-fbca-40a0-b5fa-8963eae3ce53","arxiv_id":"2606.08872","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"For every n≥4 there are tri-valued additive chore instances with no EFX allocation, bi-valued instances where every EFX allocation fails Pareto optimality, and EFX is guaranteed for four bi-valued agents.","lead":"EFX allocations of indivisible chores need not exist for additive costs once there are four or more agents, even when every cost takes only three positive values. The same paper shows that for bi-valued positive costs EFX always exists for four agents yet can be incompatible with Pareto optimality.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claims are pure existence/non-existence statements in discrete fair division. The counter-examples are finite, fully specified instances whose verification reduces to elementary arithmetic and counting; the four-agent positive result is an exhaustive case analysis that never leaves the bi-valued additive setting. The numerical ratios highlighted by the reader are not assumptions that could fail under a modelling constraint; they are free choices that make the modular contradictions appear. Because those choices are legitimate, the load-bearing concern evaporates. No independent formal verification or code is supplied, yet the arguments are short enough to be checked by hand. Consequently the reader's ACCEPT verdict with high confidence stands.","tokens_in":34086,"tokens_out":408,"duration_ms":4837,"concrete_test":"Independently recompute the n=4 tri-valued instance of Table 1 (costs 20/1/7) by exhaustive enumeration of all partitions of the 13 items into 4 labelled bundles and verify that every partition violates the EFX condition for at least one ordered pair of agents; the same check for the bi-valued instance of Table 2 with any concrete r>3 confirms that every EFX allocation is Pareto-dominated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest_assumption correctly notes that the non-existence proofs use carefully tuned ratios (p=1, q=s+2, r=s(q+1)/2 and r>⌈n/2⌉+1). Those ratios are free parameters of the model, not modelling constraints; the paper is free to choose any positive additive costs. The modular-arithmetic and counting arguments in Section 3, Appendix A and Section 4 go through exactly for those values, and the constructions are fully explicit and finite. No hidden assumption, circularity or gap appears in the three main theorems. The four-agent existence proof is long and case-heavy but self-contained and exhaustive.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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).","tokens_in":34262,"tokens_out":708,"duration_ms":14749,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"accept","confidential_remarks":"The four-agent existence proof is long but, on careful reading, appears exhaustive and free of gaps. I see no reason to request a computer-assisted check; the modular structure (insertion lemma + multigraph orientation + composition) is standard for the area. The paper is a strong fit for a theory journal in algorithmic game theory / fair division."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This paper settles the main open existence question for EFX on additive chores. Theorem 1 gives, for every n≥4, an explicit tri-valued instance with only three chore types and two agent types that admits no EFX allocation. The numbers of types are tight against known positive results. Theorem 2 then shows that every bi-valued positive instance with n≥4 has EFX allocations that are all Pareto-dominated; this is the first such separation that never uses zero costs. Theorem 3 closes the remaining gap by proving that every four-agent bi-valued instance does admit an EFX allocation.\n\nThe non-existence arguments are short modular counting arguments with fully explicit numerical tables (p=1, q=s+2, r=s(q+1)/2 and the bi-valued threshold r>⌈n/2⌉+1). You can check the n=4 case by hand in a few minutes; the general-n lift in the appendix is the same idea. The four-agent existence proof is long and case-heavy (M01/M2 partition, multigraph orientations, gap-filling, residual exceptional configurations), but every case ends with an explicit verification and the composition lemmas are clean. Citations are used only for tightness, not as hidden premises.\n\nThe only soft spot is the length and case density of Section 5; a referee will want to confirm that no residual configuration was missed. That is ordinary bookkeeping, not a conceptual hole. The free parameters the reader flagged are simply the cost values the model allows; nothing is fitted or constrained away.\n\nThis is for anyone working on fair division of chores or on the EFX/PO interface. It deserves a serious referee and should be read carefully. I would bring it to reading group and expect to cite the three theorems.","headline":"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.","tokens_in":34846,"tokens_out":483,"would_cite":true,"duration_ms":5787,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Additive chores need not admit EFX allocations once there are four or more agents, even when costs take only three values.","keywords":["EFX","indivisible chores","additive costs","tri-valued","bi-valued","Pareto-optimality","fair division","existence"],"falsifier":"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.","tokens_in":35000,"feed_emoji":"⚖️","tokens_out":636,"duration_ms":5562,"temperature":0.7,"pith_summary":"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.","feed_headline":"EFX can fail for additive chores with four agents","feed_subtitle":"Even three positive cost values are enough; bi-valued cases still exist but need not be efficient","key_machinery":"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.","core_discovery":"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.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["No EFX for tri-valued additive chores when n≥4","Tri-valued additive chores can lack any EFX for n≥4","Positive bi-valued chores can make every EFX Pareto-dominated","EFX fails to exist for additive chores with only three cost values","Every 4-agent bi-valued chore instance still admits an EFX"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["No EFX for tri-valued additive chores when n≥4","Tri-valued additive chores can lack any EFX for n≥4","Positive bi-valued chores can make every EFX Pareto-dominated","EFX fails to exist for additive chores with only three cost values","Every 4-agent bi-valued chore instance still admits an EFX"]},"model":"grok-4.5","effort":"low","cost_usd":0.004316,"raw_usage":{"total_tokens":1314,"prompt_tokens":796,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":43160000,"prompt_tokens_details":{"text_tokens":796,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":437,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":796,"tokens_out":81,"duration_ms":3633,"temperature":1.0,"reasoning_tokens":437,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-12T14:39:16.652680+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":2}