{"id":"dd479271-390a-4020-9315-217f6883db93","arxiv_id":"2508.00130","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A 3.65-approximately stable committee always exists in approval-based elections and can be computed using a Lindahl equilibrium and a strongly Rayleigh distribution.","lead":"Approval-based committees can be chosen so that no alternative committee is supported by too large a fraction of voters. The paper proves that a 3.65-approximately stable committee always exists and can be found by an algorithm based on a Lindahl equilibrium and a strongly Rayleigh distribution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 3.65 bound depends entirely on an unverified probabilistic premise: the Lindahl equilibrium distribution is strongly Rayleigh; the supplied full text is garbled, so this step cannot be checked.","rationale":"I read the paper in good faith. The paper promises an existential and algorithmic theorem with a specific constant; for that theorem to be true, the Lindahl equilibrium must produce a joint distribution with sufficiently strong negative correlation, and the concentration argument must cover all deviations simultaneously. This is exactly the weakest assumption the reader identified. Because the supplied full text is unreadable, I cannot audit those steps; however I found no detectable internal inconsistency from the abstract and visible fragments. The correct outcome remains UNVERDICTED with low confidence. I am not claiming fraud or error; the failing is verifiability. A reparsed source and independent re-derivation would settle whether the concern actually lands.","tokens_in":7006,"tokens_out":3499,"duration_ms":37069,"concrete_test":"Obtain the uncorrupted TeX/PDF source and isolate the lemma that says the Lindahl equilibrium distribution is strongly Rayleigh. Independently re-derive it from the definition of the Lindahl equilibrium; then re-derive the subsequent bound showing that for every committee T the probability that T beats S by the lambda|T|/k threshold is negligible for lambda = 3.65. If either derivation cannot be reconstructed, or a counterexample profile violates the Rayleigh premise, the 3.65 claim is unproven. As a secondary computational check, exhaustively enumerate all approval profiles with up to n=4 voters and m=5 candidates and k=2 target committee sizes and test whether any committee violates 3.65-stability.","verdict_should_be":"UNCHANGED","load_bearing_attack":"From the abstract, the central claim is that for every approval-based election there is an algorithmically computable committee S with lambda <= 3.65 approximate stability. The proof architecture is Lindahl equilibrium plus sampling from an associated strongly Rayleigh distribution. The load-bearing step is that the candidate distribution generated by the Lindahl equilibrium has negative dependence strong enough that no deviating committee T is preferred by lambda|T|/k times |V| voters with high probability after sampling. This premise is asserted in the abstract, but the supplied full text is character-corrupted; no lemma, equation, or derivation for the Rayleigh property or for the uniform deviation bound can be inspected. Without a readable proof, the existence theorem is unsupported. This is not a claimed contradiction in the mathematics; it is an epistemic gap that prevents a correctness determination. The manuscript itself contains no legible limitation statement that mitigates this gap, and no machine-checked proof or reproducible code is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript (arXiv:2508.00130) studies approximately stable committees in approval-based elections. It defines λ-approximate stability by requiring that no alternative committee T is preferred by at least (λ|T|/k)|V| voters, and it claims that for every instance a 3.65-approximately stable committee exists and can be computed algorithmically. The proposed method is to find a Lindahl equilibrium and then sample from a strongly Rayleigh distribution associated with it.","tokens_in":7168,"tokens_out":3207,"duration_ms":32871,"significance":"If the proof were correct, the result would be a substantial existence-and-computation guarantee for a natural stability notion in committee selection, and the Lindahl-equilibrium/strongly-Rayleigh approach appears to be a novel and promising technique. A strength of the claim is that the approximation constant is explicit and algorithmic; however, no machine-checked proof or reproducible code is supplied, and the proof text cannot be inspected because of corruption. The significance can therefore not be fully assessed from the submitted version.","major_comments":[{"comment":"The supplied full text is almost entirely corrupted after the title page, so no lemma, theorem statement, or proof step is readable. The central existence theorem for λ=3.65 and the algorithmic claim are therefore unsupported in the version under review. The load-bearing step—that the Lindahl equilibrium induces a strongly Rayleigh distribution whose sampled committees satisfy the uniform deviation bound—cannot be verified.","section":"Full text (after title page)"},{"comment":"The abstract asserts that sampling from the strongly Rayleigh distribution yields the 3.65 factor, but no derivation is visible; in particular, there is no legible statement of the negative-correlation/strongly Rayleigh condition for the Lindahl equilibrium distribution, nor a proof that the probability of a large-coalition deviation is controlled by that condition. Please provide the complete proof with numbered lemmas and all constants traced.","section":"Abstract"},{"comment":"The claimed algorithmic computation is not substantiated: no running time bound, representation of the Lindahl equilibrium, or sampling procedure is legible. Even if the existence proof were sound, a clear statement of the algorithmic steps would be needed to support the assertion that the committee 'can be computed algorithmically.'","section":"Algorithmic claim"}],"minor_comments":[{"comment":"Please resubmit a clean, non-corrupted PDF; as submitted, the paper cannot be read after the first page.","section":"Submission quality"},{"comment":"The abstract would benefit from a comparison with the best previous approximation factor; the reference list is not readable in this version.","section":"Related work"},{"comment":"There are equation fragments in the corrupted text that appear to be part of the proof; ensure all displayed equations are typeset correctly in the final version.","section":"Typesetting"}],"recommendation":"uncertain","confidential_remarks":"The full-text corruption appears to be an extraction artifact rather than a scientific flaw. If the authors can supply a clean PDF, the paper should be re-reviewed; as it stands, no referee can certify soundness. I would recommend requesting a clean version before any substantive decision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the paper claims the best known approximation factor for stable committees in approval-based elections, 3.65 down from 4, and gives an algorithmic proof. Second, we cannot actually read the proof in the supplied file; the text is corrupted, so my judgment rests on the abstract and the authors' track record. That is frustrating, because the claim is either a small but genuine advance or a non-event depending on the details.\n\nWhat looks genuinely new is the use of strongly Rayleigh distributions as the sampling device attached to a Lindahl equilibrium. The abstract states the theorem cleanly: for any approval profile and committee size k, there exists a committee S such that no other committee T of size |T| is preferred by more than (3.65|T|/k)|V| voters. That is the right statement, and the improvement over the previous 4 is meaningful. Vondrák and collaborators usually do careful work, and the technique, if it works, is likely to be reusable.\n\nThe soft spot is the load-bearing probabilistic premise. The proof must show that the Lindahl equilibrium induces a strongly Rayleigh distribution over candidates, and that sampling from it makes all deviations unlikely in a uniform way. Neither step is visible in the corrupted text. The stress-test note is right to flag this. It is not a contradiction inside the paper, but an epistemic gap: a referee with expertise in negative dependence has to check that step. If the strongly Rayleigh property holds and the concentration argument goes through, the constant follows. If not, the method may still yield a worse constant, or may fail entirely.\n\nNo circularity is apparent in the abstract; 3.65 is presented as an outcome, not an input. The comparison to prior work is thin in the abstract, but that is probably an artifact of us only seeing the abstract.\n\nBottom line: this deserves a serious referee. Do not desk reject it. Send it to someone who knows both Lindahl equilibria and strongly Rayleigh distributions. If the key lemma survives, it will be a solid, potentially transferable result. I would not cite it yet and would not bring the corrupted file to our reading group, but I would want to see the referee reports.","headline":"Improved constant for approximate stability in approval committees, via Lindahl equilibrium and strong Rayleigh sampling; proof not inspectable in our copy, but the claim is coherent and deserves refereeing.","tokens_in":7659,"tokens_out":2224,"would_cite":false,"duration_ms":25166,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B12","91B14"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every approval-based election admits a 3.65-approximately stable committee, and such a committee can be computed algorithmically.","keywords":["approval-based committee selection","approximate stability","Lindahl equilibrium","strongly Rayleigh distribution","proportional representation","social choice theory","algorithmic game theory","randomized algorithms"],"falsifier":"For a fixed small instance, compute the Lindahl-equilibrium distribution and exhaustively test every committee in its support: if any support committee $S$ has a rival $T$ preferred by at least $3.65|T|/k$ of the voters, the claimed guarantee fails. Because the instance is finite, this search is a concrete computation that would settle the theorem.","tokens_in":6825,"feed_emoji":"🗳️","tokens_out":7077,"duration_ms":69970,"temperature":0.7,"pith_summary":"This paper asks whether a choice of $k$ candidates can be made so that no rival committee can attract a coalition much larger than its proportional share of voters. It proves that in every approval-based election a $3.65$-approximately stable committee exists and that such a committee can be computed algorithmically. The result matters because approximate stability is a direct measure of justified dissatisfaction: if a large voting bloc would rather replace the committee, the chosen committee fails to represent them. The proof gives an economic meaning to stable representation by deriving the committee from a Lindahl equilibrium, turning a market-equilibrium concept into a sampling algorithm.","feed_headline":"Approval elections always admit a 3.65-stable committee","feed_subtitle":"A Lindahl-equilibrium lottery prevents any rival slate from winning over 3.65× its fair share of voters.","key_machinery":"The load-bearing object is the Lindahl equilibrium of the committee-selection market. In such an equilibrium each voter is assigned a personalized price for every candidate; facing these prices, the voter chooses a most-preferred committee of size $k$, and the choices clear the market in the sense that total demand equals supply. Aggregating these equilibrium choices yields a probability distribution over committees. The proof then shows this distribution is strongly Rayleigh—a negative-correlation condition saying that the selection of one candidate makes the selection of another candidate less likely. That property supplies the large-deviation bound used to argue that no deviating committee $T$ is preferred by more than $\\lambda |T|/k$ of the voters with probability larger than the union-bound threshold. The constant $3.65$ emerges from optimizing the parameters in this bound.","core_discovery":"The paper establishes that in any approval-based committee election—voters with approval sets over candidates, fixed committee size $k$—there exists a committee $S$ such that for every alternative committee $T$ of size $t$, the number of voters who prefer $T$ to $S$ is less than $\\lambda t/k$ times $|\\mathcal{V}|$, with $\\lambda = 3.65$. Equivalently, no rival slate can assemble a coalition larger than $3.65$ times its proportional share of the electorate. The proof reaches this through a Lindahl equilibrium of a voting market: voters face personalized prices for candidates, and the equilibrium demands over candidate sets define a probability distribution on committees. The paper proves that this distribution is strongly Rayleigh, and that a single sample from it is approximately stable with positive probability, so repeated sampling yields an algorithm for finding such a committee.","pith_inferences":["The paper does not claim $3.65$ is optimal; because the constant comes from a parameter optimization, a tighter large-deviation argument could plausibly lower it, and finding the true optimal constant is a natural next problem.","A testable engineering extension is to draw several candidates from the equilibrium distribution, check each by exhaustive or heuristic search for challengers, and return the best; the theorem guarantees that good committees are present in the distribution, giving such a procedure an anytime certificate.","The market-equilibrium route implies that stable committees could be produced by generic equilibrium-finding algorithms, which connects proportional representation to computational economics in a way the paper only begins to exploit."],"forward_implications":["Every approval profile has a committee that no rival of the same size can replace by attracting even $3.65$ times its proportional share of voters.","The guarantee is algorithmic: sampling from the equilibrium distribution finds such a committee, so approximate stability is not merely an existence result.","The constant $3.65$ is universal—it does not grow with the number of voters, candidates, or the committee size.","A polynomial number of samples from the equilibrium distribution suffices, because a sample is approximately stable with positive probability and each failure can be checked against all challengers.","For a challenger of size $t$, the allowed coalition threshold scales as $\\lambda t/k$, so small challengers are automatically easy to resist and the hardest case is a rival committee comparable in size to $k$."],"supporting_citations":[],"fun_headline_variants":["Approval voting always has a 3.65-stable committee","Lindahl equilibrium delivers 3.65-stable committees","3.65-stable committee exists in all approval elections","Strongly Rayleigh sampling finds stable committees"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's load-bearing premise is that the committee distribution coming out of the Lindahl equilibrium is negatively correlated strongly enough to keep the probability of any rival committee rallying a large coalition below the needed threshold; if that probabilistic control gives way, the $3.65$ guarantee collapses.","fun_headline_variants_meta":{"raw":{"variants":["Approval voting always has a 3.65-stable committee","Lindahl equilibrium delivers 3.65-stable committees","3.65-stable committee exists in all approval elections","Strongly Rayleigh sampling finds stable committees"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000263,"raw_usage":{"total_tokens":1563,"prompt_tokens":871,"completion_tokens":692,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":487,"completion_tokens_details":{"reasoning_tokens":627}},"tokens_in":487,"tokens_out":692,"duration_ms":6298,"temperature":1.0,"reasoning_tokens":627,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:20:04.418261+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed small instance, compute the Lindahl-equilibrium distribution and exhaustively test every committee in its support: if any support committee $S$ has a rival $T$ preferred by at least $3.65|T|/k$ of the voters, the claimed guarantee fails. Because the instance is finite, this search is a concrete computation that would settle the theorem.","supporting_citations":[],"review_version":1}