{"id":"8b608794-e3db-4c14-9656-142487a09d3e","arxiv_id":"2507.09600","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"EFX allocations always exist for any number of agents when two agents have arbitrary set monotonic valuations and all others have size monotonic valuations.","lead":"This paper proves a general existence theorem in fair division: for any number of agents, an allocation that is envy-free up to any good always exists when two agents may have arbitrary monotone valuations and all remaining agents value larger bundles more. It is one of the broadest existence results for the long-standing EFX problem outside the fully general case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The general induction step in Theorem 3.3 is deferred to 'similar arguments'; without an explicit construction for c ≥ 3, the central claim is not fully proved.","rationale":"The reader's weakest assumption points at strictification and the r=0 convention. Strictification is not a serious concern: a sufficiently small additive perturbation proportional to |S|, plus a generic tie-break, preserves all strict inequalities, set monotonicity, and the relevant local size-monotonicity, so Observation 2.2 can be made rigorous. The r=0 convention is confusing but consistent with treating the reduced bundle as Agent 2's. The real load-bearing issue is the missing general induction step. The proof explicitly defers it to 'similar arguments,' and the size accounting written for c=2 is not updated for c≥3. Because the central claim depends entirely on this induction, the theorem is not established as written. This is an exposition/completeness gap rather than a demonstrated counterexample, so the appropriate verdict remains CONDITIONAL: a revision that writes out the induction would justify ACCEPT.","tokens_in":13109,"tokens_out":31862,"duration_ms":306219,"concrete_test":"Fully formalize the induction step of Theorem 3.3 for general c: define W_c, Y_c, G_{c-1}, H_{c-1}, and the allocation X^c_N explicitly, with the correct decomposition m−(c−1) = (n−1)ℓ_c + r_c. Then verify (i) the sizes sum to m; (ii) for agents 3..n, the assigned sizes lie in the size-monotonic intervals [1,ℓ] or [1,ℓ+1] of the profile; (iii) the no-envy argument for agents 2..n carries through using the proof of Theorem 3.2; and (iv) if X^c is not EFX, the set Z obtained from Agent 1's envy satisfies Z ∈ G_{c-1} (using T = W_c). If all four checks pass, the proof is complete; if any fails, the theorem's proof has a genuine gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the iterative construction in the proof of Theorem 3.3 (Appendix A.2). The step from X1 to X2 is written out, but for c ≥ 3 the proof says 'Using similar arguments as before' and refers to allocations 'defined in a similar way as before' without specifying W_c, Y_c, G_{c-1}, H_{c-1}, or the exact rule for allocating goods to agents 3..n at step c. In particular, the size accounting 'm − 2 = (n − 1)ℓ + r' is only valid for the c=2 step; for a general step the decomposition must involve m − (c−1). The final contradiction that X^c is EFX requires showing that the newly found Z belongs to G_{c-1}; this uses the existence of a disjoint size-(max(X^{c-1})−1) set valued above W_{c-1} (e.g., W_c itself), but the definitions needed to check this are absent for c ≥ 3. If, for some c, the greedy filling cannot be carried out with sizes max(X^{c-1})−1 and max(X^{c-1}) while keeping all sizes inside the size-monotonic intervals [1,ℓ] or [1,ℓ+1], the induction fails. As written, the proof of the main theorem is therefore incomplete.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the existence of envy-free-up-to-any-good (EFX) allocations for indivisible goods under general monotone valuations. The main claim is that an EFX allocation always exists when two agents have arbitrary set-monotonic valuation functions and all remaining agents have size-monotonic valuation functions. The proof is based on an iterative construction: starting from a carefully chosen allocation, if the current allocation is not EFX, the authors enlarge the bundle of a distinguished agent (Agent 1) by one good while preserving EFX among the remaining agents, and they argue that this process must terminate with an EFX allocation. Theorems 3.2 and 3.3 state weaker sufficient conditions in terms of local size-monotonicity, and the appendix contains the proofs.","tokens_in":13359,"tokens_out":18761,"duration_ms":186835,"significance":"If the proof is completed, this is a notable contribution to the fair-division literature. It goes beyond known existence results for EFX under general valuations, which are limited to identical valuations, two agents, or restricted classes of additive valuations. The weaker local conditions in Theorems 3.2 and 3.3 are also interesting and suggest that the full force of size monotonicity is not needed. The paper is self-contained and does not rely on unproved external results; the case-by-case verification of the constructed allocations is a strength. However, the central proof has an induction step that is only sketched, so the result as written is not fully established.","major_comments":[{"comment":"The induction step from X^2 to X^3 and to general X^c is not fully specified. After constructing X^2 explicitly, the proof says 'Using similar arguments as before' and 'defined in a similar way as before' for c >= 3, without defining the sets G_{c-1}, H_{c-1}, W_c, Y_c, or the allocation rule for agents 3..n at step c. In particular, the size accounting m - c = (n-1) * floor((m-c)/(n-1)) + r_c is not stated for general c, and the proof that the constructed allocation satisfies properties (i)-(iv) listed before the final contradiction is omitted. The final argument that max(X^{c-1}_N) = c+1 and that X^c_1 is not envied by Agent 1 depends on these properties. As written, the proof of the main theorem is therefore incomplete.","section":"Appendix A.2, proof of Theorem 3.3"},{"comment":"The proof reduces to strict valuation profiles 'in view of Observation 2.2', but the paper never proves that an arbitrary weak profile satisfying the monotonicity conditions of Theorems 3.2 and 3.3 can be perturbed into a strict profile that (a) refines the weak order in the sense of Observation 2.2 and (b) still satisfies the same local size- and set-monotonicity conditions. This is a standard perturbation argument, but it is load-bearing because the theorems are stated for weak set-monotonic and size-monotonic valuations. The authors should either state and prove this lemma or explain why the constructed allocations remain EFX for weak profiles directly.","section":"Observation 2.2 and its use"},{"comment":"The proof asserts that the construction for agents {2,...,n} in X^1 (and similarly for X^2, X^3, ...) 'satisfies the conditions of Theorem 3.2' and therefore inherits EFX among those agents. However, the distinguished agent (Agent 2) in the subproblem chooses a best size-ℓ bundle from E \\ W_1, not from all of E as in the proof of Theorem 3.2. The same issue recurs at later steps. The proof should explicitly verify that the EFX arguments of Theorem 3.2 go through when the special agent's choices are restricted to the complement of Agent 1's bundle; this is likely true but is not shown.","section":"Appendix A.2, application of Theorem 3.2 to subproblems"}],"minor_comments":[{"comment":"The notation 'floor(m-1/n-1)' is ambiguous; it should be written as \\left\\lfloor\\frac{m-1}{n-1}\\right\\rfloor in the statements of Theorems 3.3, 4.2, and in the appendix.","section":"Theorem 3.3 and elsewhere"},{"comment":"The convention 'n+1 ≡ 2' for r=0 is confusing. It would be clearer to state that when r=0, the range '3 <= i <= n-r+1' is empty (so all agents 3..n receive the larger bundle size).","section":"Footnotes 7 and 8"},{"comment":"In the final paragraph of the proof, the phrase 'in particular, by [ℓ-1, ℓ]-set monotonicity' is misleading: the inequality for the case |X_j \\ {x_j}| = ℓ-1 follows from [ℓ, ℓ+1]-size monotonicity (comparing X_i of size ℓ+1 with X_j of size ℓ) together with strict set monotonicity (comparing X_j with X_j \\ {x_j}), not from [ℓ-1, ℓ]-set monotonicity.","section":"Appendix A.1, proof of Theorem 3.2"},{"comment":"The phrase 'there is no envy in terms of (or \"with respect to\") EFX between any two agents i, j ∈ {2, ..., n}' is awkward and should be rephrased, e.g., 'the allocation is EFX for the subprofile of agents 2,...,n'.","section":"Introduction, Section 1.1"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a significant open problem and the overall approach is promising, but the proof of Theorem 3.3 is not complete as written because the induction for c >= 3 is delegated to 'similar arguments' and the strictification lemma is missing. These gaps are likely fillable, and the central claim may well be correct, but the manuscript needs a substantially expanded appendix before it can be accepted. I would encourage the editor to send the paper back for a major revision rather than reject it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Ujjwal and Souvik claim a real extension: EFX allocations exist when two agents have arbitrary set-monotonic valuations and the rest have size-monotonic ones. That goes beyond Plaut-Roughgarden (identical or n=2), Akrami et al. (three agents, one MMS-feasible), Mahara, and HV et al. The base allocations and the step from X1 to X2 in Appendix A.2 are written out carefully; the reduction to strict profiles via Observation 2.2 is standard in spirit. The core idea—growing one agent's bundle while maintaining EFX among the others—is fresh.\n\nThe problem is the general induction. For c ≥ 3 the proof says 'similar arguments' and 'defined in a similar way' without specifying W_c, Y_c, G_{c-1}, H_{c-1}, or the exact allocation rule. The line 'm − 2 = (n−1)ℓ + r' is only for the step from X2 to X3; for step c it must be m − c = (n−1)ℓ + r. That is a minor typo if the pattern is clear, but the pattern is not clear. In particular, the final contradiction at step c relies on Z ∈ G_{c-1}. For c=2, this is established with an explicit argument using W2 and Y2. For c ≥ 3, it is not shown how the witnessing set T of size max(X^{c-1}_N) − 1 is obtained, disjoint from Z, with v1(T) > v1(W_{c-1}). The cases |X^c_j| = c and |X^c_j| = c+1 require different arguments, and neither appears in the text. So the proof of Theorem 3.3 is not complete as written.\n\nA second, smaller gap: the transfer from weak to strict profiles via Observation 2.2 is asserted, but the authors never prove that every weak size-monotonic/set-monotonic profile can be strictified while preserving the same local monotonicity intervals. That is a standard perturbation argument, but it should be stated.\n\nI don't see a demonstrable error—the skeleton is plausible and the base cases check out. But it's not a completed proof yet. The significance of the claim warrants a careful referee; the gaps are exactly what a referee should ask to be fixed. I recommend sending to peer review, not desk reject, with a clear request to fill in the inductive step.","headline":"A novel and plausible EFX existence theorem, but the general induction in the main proof is deferred to 'similar arguments' and the paper is not yet fully proved.","tokens_in":13900,"tokens_out":6416,"would_cite":true,"duration_ms":67011,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B32"],"pacs":[],"model":"deepseek-v4-flash","headline":"EFX allocations always exist when at most two agents have arbitrary set-monotonic valuations.","keywords":["EFX allocations","fair division","indivisible goods","size monotonic valuations","set monotonic valuations","envy-freeness","existence theorem","constructive allocation"],"falsifier":"Exhibit any instance with m > n goods, two agents with arbitrary set-monotonic valuations, and the remaining n − 2 agents with size-monotonic valuations in which no EFX allocation exists; such an instance would directly contradict the theorem.","tokens_in":12903,"feed_emoji":"⚖️","tokens_out":6626,"duration_ms":68902,"temperature":0.7,"pith_summary":"This paper proves that EFX allocations — fair divisions of indivisible goods in which no agent envies another after removing any single good from the other's bundle — always exist when at most two agents have arbitrary set-monotonic valuations and every other agent has a size-monotonic valuation. The proof is constructive: it starts from a specially chosen allocation, then repeatedly enlarges one agent's bundle while preserving envy-freeness among everyone else, until the process must terminate at an EFX allocation. The result matters because the general existence question for arbitrary set-monotonic valuations with three or more agents remains open, and this is one of the few positive results that covers any number of agents with distinct valuation functions.","feed_headline":"EFX allocations exist when at most two agents have arbitrary tastes","feed_subtitle":"A constructive proof divides goods fairly among any number of agents once all but two value bundles by size alone.","key_machinery":"The machinery is a staged allocation algorithm that singles out one potentially 'hard' agent (Agent 1) and repeatedly enlarges that agent's bundle by one good. At each stage, the remaining bundles are chosen as best bundles of a prescribed size from the leftover goods, and the size-monotonicity of all but two agents guarantees that no two of those remaining agents envy each other up to one good. The argument splits into two cases — whether Agent 1 receives the chosen best set or the fallback set — and either terminates in an EFX allocation or produces a new allocation with a strictly larger bundle for Agent 1. Since bundle sizes are bounded, the process must stop, and the stopping condition forces EFX. A separate observation (Observation 2.2) allows the proof to assume strict valuations, claiming that an EFX allocation at a strict perturbation is also EFX at the original weak profile.","core_discovery":"The central claim is Theorem 3.3: for m > n goods and n agents, if n − 2 agents have size-monotonic valuations (larger bundles are always at least as valuable) and the remaining two agents have arbitrary set-monotonic valuations, then an EFX allocation exists. The theorem actually holds under weaker local conditions: the size-monotonic agents only need their monotonicity on a specific interval of bundle sizes determined by floor((m − 1)/(n − 1)). The proof constructs the allocation explicitly through an iterative process, and the final step shows that once the singled-out agent's bundle reaches size c while all other bundles have size c or c + 1, the allocation must be EFX.","pith_inferences":["A natural next target is reducing the number of non-size-monotonic agents from two to one; the iterative structure here suggests the bottleneck lies in the comparison between the two exceptional agents, not between the exceptions and the size-monotonic majority.","The local-monotonicity weakening in Theorem 4.2 hints that full set-monotonicity is far stronger than necessary; a testable extension is whether the same construction works when the size-monotonic agents satisfy their conditions on intervals that depend on the current stage rather than on the global floor((m − 1)/(n − 1)).","If the construction turns out to be computationally efficient, it could serve as a practical fair-division protocol for settings such as course allocation, where most participants have cardinality-based preferences but a few have idiosyncratic tastes."],"forward_implications":["If the theorem is correct, EFX existence is now established for any number of agents whenever the valuation profile has at most two 'wild' agents, including profiles where every agent's valuation is distinct.","The proof yields an EFX allocation for three agents with one arbitrary set-monotonic agent and two size-monotonic agents, a case not covered by earlier MMS-feasibility results, and it is independent of those results.","The weakened Theorem 4.2 shows the result survives even when the two arbitrary agents are only required to be set-monotonic on a bounded interval of bundle sizes, so the construction works under conditions much weaker than the headline assumptions.","The same techniques provide a new existence proof for the n = 2 case, different from the known argument, which the paper reports helped shape the general proof."],"supporting_citations":[{"why":"Introduces the envy-free-up-to-any-good (EFX) solution concept that the paper proves always exists under its assumptions.","marker":"[2]"},{"why":"Establishes the two-agent and identical-valuations existence cases; the paper presents its iterative proof as a distinct route to the two-agent result.","marker":"[5]"},{"why":"Provides a three-agent EFX result under MMS-feasibility; the paper contrasts this with its own three-agent corollary to show independence.","marker":"[1]"}],"fun_headline_variants":["EFX allocations exist when only two agents have flexible tastes","Size-based majority enables EFX fairness for all","Arbitrary tastes for two, EFX still possible","For EFX, limit arbitrary preferences to two agents","Constructive EFX proof with size-monotonic agents"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that any weak valuation profile can be perturbed into a strict one without breaking the required size-monotonicity conditions, so that the allocation found for the strict profile carries back to the original profile.","fun_headline_variants_meta":{"raw":{"variants":["EFX allocations exist when only two agents have flexible tastes","Size-based majority enables EFX fairness for all","Arbitrary tastes for two, EFX still possible","For EFX, limit arbitrary preferences to two agents","Constructive EFX proof with size-monotonic agents"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000296,"raw_usage":{"total_tokens":1610,"prompt_tokens":726,"completion_tokens":884,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":342,"completion_tokens_details":{"reasoning_tokens":806}},"tokens_in":342,"tokens_out":884,"duration_ms":9395,"temperature":1.0,"reasoning_tokens":806,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:56:51.368300+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit any instance with m > n goods, two agents with arbitrary set-monotonic valuations, and the remaining n − 2 agents with size-monotonic valuations in which no EFX allocation exists; such an instance would directly contradict the theorem.","supporting_citations":[{"cited_title":"The unreasonable fairness of maximum nash welfare","cited_arxiv_id":null,"evidence_quote":"Introduces the envy-free-up-to-any-good (EFX) solution concept that the paper proves always exists under its assumptions."},{"cited_title":"Almost envy-freeness with general valuations","cited_arxiv_id":null,"evidence_quote":"Establishes the two-agent and identical-valuations existence cases; the paper presents its iterative proof as a distinct route to the two-agent result."},{"cited_title":"Efx: a simpler approach and an (almost) optimal guarantee via rainbow cycle number","cited_arxiv_id":null,"evidence_quote":"Provides a three-agent EFX result under MMS-feasibility; the paper contrasts this with its own three-agent corollary to show independence."}],"review_version":1}