{"id":"921dfd51-58f1-4508-a8bc-8d4c4341fa0f","arxiv_id":"2508.16177","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New proportional rank aggregation rules PSB and FB guarantee that every input ranking, and even every group of rankings, is represented in the output ranking in proportion to its weight.","lead":"Voting rules that merge ranked lists now have a fairness guarantee: each input list gets a say in the final ranking proportional to its importance, measured by how many pairwise orderings it shares with the result. The new rules, Proportional Sequential Borda and Flow-adjusting Borda, solve an open social choice problem and give practical benchmarks for aggregator rankings and AI model evaluation.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No internal gap found in PSB rank-priceability proof; the real caveat is the explicitly admitted pairwise-agreement utility model, not a correctness flaw.","rationale":"The reader's weakest assumption is the pairwise-agreement utility model, and the paper itself flags this in Section 6. I agree that this is the main limitation for applications, but it is a modeling assumption rather than a flaw in the theorem. I independently walked through Theorem 4.5 and found the rank-priceability proof internally consistent: the per-round payment cap is justified by the Borda-winner inequality, the budget accounting sums correctly, and the leftover-budget bound of 3/4 is established by a valid induction with separate handling of the last three rounds. The central claim that PSB satisfies uPJR therefore holds under the paper's stated definitions. The reader's conditional verdict mentions also the unshipped computer-assisted check in Proposition 5.1; that is a reproducibility gap, but it does not bear on the uPJR theorem, so it does not change the formal central claim. The appropriate verdict remains conditional on the utility model being the intended objective, exactly as the reader concluded.","tokens_in":49657,"tokens_out":17406,"duration_ms":176653,"concrete_test":"Simulate PSB on a battery of adversarial profiles for m = 5, 6, 7 (including opposed rankings with extreme weights) and compute the payment scheme from Theorem 4.5, checking Conditions (1)–(4) exactly; in particular, verify that the total leftover budget never exceeds 0.75. If a counterexample with leftover > 0.75 appears, the rank-priceability proof has a hidden flaw; otherwise the central mechanism is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the proof of Theorem 4.5 carefully. Each priceability condition is derived from the Borda-maximization inequality U(b_i,x_i,X_i) ≥ (m−i)b_i(≻), the min-based payment update, and the leftover-budget bound of at most 3/4. The induction for rounds i ≤ m−3 uses the correct total-Borda-score lower bound and verifies that the min is not binding; the final three rounds are handled by separate case arguments. I find no algebraic gap in Conditions (1)–(4). Thus the central formal claim — PSB is rank-priceable and therefore satisfies uPJR — is supported as a theorem under the paper's definitions. What remains load-bearing outside the formal claim is the modeling choice in Section 2.1: utility is defined as the number of agreeing pairwise comparisons, with all pairs weighted equally, and this decomposition is used in every proof. Section 6 explicitly acknowledges this as the biggest restriction. If an application values top positions more heavily, the uPJR guarantee does not transfer. This is a scoping caveat about external applicability, not an internal inconsistency in the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies proportional representation in rank aggregation. It defines utility as the number of pairwise comparisons on which an input ranking and the output ranking agree, and formulates three axioms: uJR, uPJR, and sPJR. The main results are: (i) the Squared Kemeny rule fails even uJR (Proposition 3.1); (ii) a Chamberlin-Courant-style SWF satisfies uJR (Proposition 3.2); (iii) the Proportional Sequential Borda rule (PSB) is rank-priceable and hence satisfies uPJR (Theorem 4.5 and Proposition 4.2); (iv) PSB guarantees a linear-in-weight average utility bound for every subprofile (Theorem 4.6); (v) PSB fails sPJR (Proposition 5.1, via a computer-assisted argument); and (vi) the Flow-adjusting Borda rule (FB) is pair-priceable and hence satisfies sPJR, with the same average-utility bound (Theorems 5.5 and 5.6). Detailed proofs are provided in the appendices.","tokens_in":49932,"tokens_out":8245,"duration_ms":93048,"significance":"If the results are correct, this is the first exact resolution of the open problem of designing fully proportional social welfare functions in the pairwise-agreement model of Lederer et al. [2024]. The paper makes a substantive methodological contribution by transferring priceability ideas from participatory budgeting to rank aggregation. The main theorems are backed by detailed, structured proofs, and the implications rank-priceability implies uPJR and pair-priceability implies sPJR are clean and short. The paper also gives explicit quantitative guarantees rather than only existential ones. The main reservation concerns the reproducibility of the computer-assisted counterexample in Proposition 5.1, which is load-bearing for the claim that PSB fails sPJR and for the motivation of the stronger FB rule.","major_comments":[{"comment":"The proof of Proposition 5.1 states that 'we have shown with the help of a computer that, up to reorderings of x1,...,x4, PSB uniquely chooses' a particular ranking, and that this ranking agrees with only 292 of the required 293 pairwise comparisons. No code, no computational parameters, no verification certificate, and no explicit tie-breaking specification are supplied. Since PSB's tie-breaking can affect the chosen ranking, and since this proposition is the only evidence that PSB fails sPJR, this is a load-bearing gap: without a reproducible computation or an independent analytic proof, the claimed failure of sPJR for PSB is not established. Please provide the full computer-checkable proof (ideally with code and the exact tie-breaking rule) or replace the computer-assisted step with a hand-verifiable certificate.","section":"Section 5, Proposition 5.1"},{"comment":"The paper's fairness guarantees are all defined with respect to the utility function that counts every pairwise comparison equally. This is explicitly acknowledged in Section 6 as the biggest restriction. I do not see this as an internal flaw, but it should be made more prominent in the introduction and abstract: the claims of 'proportional representation' are with respect to this specific utility model only. If an application values top positions more heavily, the uPJR/sPJR guarantees do not transfer. A short caveat in the abstract or introduction would help avoid overclaiming.","section":"Section 6 and Section 2.1"}],"minor_comments":[{"comment":"When the computation is provided, it should also specify the tie-breaking order over rankings and over candidates, since the statement 'up to reorderings of x1,...,x4' is not enough to uniquely determine the PSB output under an arbitrary tie-breaking rule.","section":"Section 5, Proposition 5.1"},{"comment":"The proof of Proposition 3.1 refers to 'Claim (2), Claim (3), and Claim (6)' but the claims are not separately labelled in the proof. Adding explicit labels would improve readability and verifiability.","section":"Appendix B"},{"comment":"The footnote about the failure of the unconstrained equal-shares variant is somewhat confusing: it describes what would happen if the utility cap were omitted, but it is not immediately clear that this variant is not the defined RMES. Clarify that the example concerns a different, non-recommended update rule.","section":"Section A, footnote 6"},{"comment":"The notation A(▷) is introduced informally as the set of pairwise comparisons induced by ▷. Since sPJR and pair-priceability rely heavily on this set, a formal definition of A(▷) as the ordered pairs (x_i,x_j) with i<j would reduce ambiguity.","section":"Definition 2.3"},{"comment":"The final 'equivalently' step in the proof sketch and in Appendix C is somewhat compressed. A one-line justification that A U/(A+1/4) ≤ U/|S| for A=|S|·C(m,2) and |S|≤1 would help the reader.","section":"Theorem 4.6"}],"recommendation":"major_revision","confidential_remarks":"The main construction and the proof of Theorem 4.5 appear sound; the central uPJR claim is supported. However, the computer-assisted proof in Proposition 5.1 is not reproducible in the current manuscript, and this is an important part of the paper's contribution. I would be happy to support publication after the authors supply a verifiable computation or an independent proof of that proposition. The utility-model caveat is honestly stated and should not be treated as an error."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is true: this paper gives the first SWFs for rank aggregation with exact proportional representation guarantees, not just approximations. PSB satisfies uPJR via the new rank-priceability axiom, and FB satisfies the stronger sPJR via pair-priceability. The structural axioms are the right transfer from committee voting/participatory budgeting, and the design of FB via max-flow is genuinely new in this domain.\n\nThe formal core holds up. I went through the proof of Theorem 4.5 carefully; the induction on the leftover budget and the case analysis for the last three rounds are sound. The implication rank-priceability => uPJR is short and clean. Theorem 5.5's proof is sketched in the main text, but the appendix gives enough detail for a reader to verify the max-flow/min-cut argument. The authors also honestly flag in Section 6 that the whole fairness notion is tied to pairwise-agreement utility with all pairs equally weighted; if an application cares more about top positions, the guarantees do not transfer. That is a scoping caveat, not an internal inconsistency.\n\nThe real soft spot is Proposition 5.1. The paper says \"we have shown with the help of a computer\" that PSB uniquely chooses a specific ranking, but no code, no parameters, and no machine-checkable certificate are provided. That is load-bearing: it is the entire justification for introducing FB. Without reproducible evidence, the claim that PSB fails sPJR is not fully verifiable. This is a concrete reproducibility gap, not a math error that I detected. It should be fixed before publication—ship the code, or replace the computation with a hand-checkable proof or a formally verified certificate.\n\nThe citation patterns look appropriate. The paper builds correctly on Lederer et al. [2024] and Aziz et al. [2025], and the self-citations are warranted since those works define the open problem and the Squared Kemeny baseline.\n\nMy recommendation: yes, send this to serious peer review. The design contribution is substantial, the central theorems appear correct, and the computational counterexample is a fixable issue rather than a fatal flaw. I would press the authors hard on reproducibility but I expect the paper to be accepted after that is resolved.","headline":"Solid, significant paper that resolves an open design problem in proportional rank aggregation; the core axioms and proofs hold up, but the one computer-assisted counterexample needs to be made reproducible before I'd trust it fully.","tokens_in":50383,"tokens_out":1401,"would_cite":true,"duration_ms":18408,"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":"Rank aggregation can be made proportional: a sequential Borda rule guarantees every weighted input ranking its agreed share of pairwise comparisons.","keywords":["rank aggregation","proportional representation","justified representation","pairwise comparisons","Borda rule","priceability","social welfare functions","sequential rules"],"falsifier":"The theorems are universal, so a single counterexample would refute them. Run PSB (or FB) with a fixed tie-breaking on any small profile, say all rankings for m = 5 or 6, and check each input ranking's pairwise-agreement count against floor(weight * C(m,2)); any count below the bound falsifies uPJR (or sPJR for FB on subprofiles). A targeted test already appears in the paper: its Proposition 5.1 constructs a 26-candidate, 8-ranking profile on which PSB returns a ranking agreeing with only 292 pairwise comparisons of a group that deserves 293, demonstrating that PSB does not satisfy the stronge","tokens_in":49566,"feed_emoji":"⚖️","tokens_out":6765,"duration_ms":75409,"temperature":0.7,"pith_summary":"This paper tries to solve a fairness problem in rank aggregation: when weighted input rankings are merged into one output ranking, every input ranking should be represented in proportion to its weight. The author measures representation by the number of pairwise comparisons on which the output agrees with the input, and demands at least floor(alpha * C(m,2)) agreements for an input ranking of weight alpha, a condition called uPJR. The central result is the Proportional Sequential Borda rule, which repeatedly picks the budget-weighted Borda winner, deletes it, and reduces each input's budget in proportion to its contribution; via a property called rank-priceability, this rule satisfies uPJR. A more involved Flow-adjusting Borda rule satisfies the stronger group-level condition sPJR: any collection of inputs with total weight alpha gets at least floor(alpha * C(m,2)) pairwise agreements in the output. If correct, these are the first fully proportional social welfare functions in this pairwise-agreement model, making aggregators for hotels, university rankings, or AI-system evaluations responsive to minority criteria rather than merely copying the majority ranking.","feed_headline":"Every weighted ranking gets its fair count of pairwise wins","feed_subtitle":"Proportional Sequential Borda and Flow-adjusting Borda rules give weighted inputs a floor on pairwise agreements.","key_machinery":"The central object is the pairwise-agreement utility function u(≻,▷), which counts the candidate pairs on which an input ranking ≻ and the output ranking ▷ agree. The load-bearing identity is the decomposition u(≻,▷) = sum_{i=1}^{m-1} u(≻, x_i, {x_i, ..., x_m}), which lets each sequential output position be treated as a priced item whose maximum value to any ranking is m-i. Rank-priceability formalizes the resulting payment scheme: each ranking can pay only for utility it receives, never more than its budget, each position costs at most m-i, and total spending must exceed C(m,2) - 1. Pair-priceability refines this to payments for individual candidate pairs. These priceability properties are","core_discovery":"The paper's central claim is that the Proportional Sequential Borda rule (PSB) satisfies uPJR: for every profile and every input ranking with weight alpha, the output ranking agrees with at least floor(alpha * C(m,2)) pairwise comparisons. PSB works by giving each input ranking a budget of alpha * C(m,2), then in each round placing the candidate with the highest budget-weighted Borda score into the next output position, charging each ranking no more than the utility it gains from that placement, and deleting the candidate. The proof goes through rank-priceability, a payment scheme showing that the collective spending on the output ranking exceeds C(m,2) - 1; the paper proves that any rank-pr","pith_inferences":["The author flags in Section 6 that pairwise-agreement utility treats all output positions equally; a natural test is to rerun PSB and FB with top-weighted utilities, where the priceability payment constraints would need rescaling and the uPJR-style floor becomes a different target.","The budget-update construction suggests a general recipe: any budget-balanced, priceable rule from participatory budgeting can likely be converted into a proportional SWF by treating each output position as an item of shrinking cost, though whether the uPJR guarantee survives for non-Borda utilities is open.","The linear group-average bound (alpha/4 * C(m,2) - 3/16) is probably not tight; the paper's own Remark 2 notes an upper bound of about 1/2 * C(m,2), so sharpening the 3/4 leftover-budget bound would improve the constant.","In practical ranking portals, these rules give each criterion a guaranteed 'screen time' in pairwise comparisons; testing them on real data against Kemeny or plain Borda aggregation would show whether users perceive the output as more balanced."],"forward_implications":["Any multi-criteria aggregator using PSB can promise each criterion a concrete lower bound on how many of its pairwise judgments survive in the output ranking.","uPJR resolves the open problem left by the Squared Kemeny rule: payoff from participatory budgeting and committee-voting fairness notions can be transferred to rank aggregation.","The Flow-adjusting Borda rule extends the guarantee to coalitions: if several input rankings jointly hold weight alpha, their preferred pairwise comparisons are represented at the floor(alpha * C(m,2)) level.","Both rules give a linear group-average utility bound, so even small subprofiles receive a predictable amount of representation, not just the individual floor.","Both rules are described as concrete sequential procedures, so implementation does not require solving a global optimization over all rankings."],"supporting_citations":[{"why":"Supplies the pairwise-agreement utility model, the Squared Kemeny rule that this paper shows fails uJR, and the open problem of fully proportional SWFs that uPJR resolves.","marker":"[Lederer et al., 2024]"},{"why":"Provides the priceability concept and the Method of Equal Shares, which inspire rank-priceability and the budget design of PSB.","marker":"[Peters and Skowron, 2020]"},{"why":"Extends priceability in participatory budgeting, giving the template for the payment-scheme conditions that rank-priceability adapts.","marker":"[Brill et al., 2023]"},{"why":"Defines proportional justified representation (PJR) in committee voting, which uPJR adapts to rankings by viewing them as approval ballots over candidate pairs.","marker":"[Sánchez-Fernández et al., 2017]"},{"why":"Defines justified representation (JR), the approval-voting notion that uJR adapts to the ranking setting.","marker":"[Aziz et al., 2017]"},{"why":"The Chamberlin-Courant rule is the template for the simple SWF that the paper shows satisfies uJR.","marker":"[Chamberlin and Courant, 1983]"},{"why":"Underlies the Ranked Method of Equal Shares and the connection between PSB and the Method of Equal Shares discussed in the appendix.","marker":"[Peters et al., 2021]"},{"why":"Provides committee-voting rules with ranked preferences that satisfy approximate uPJR, serving as the comparison baseline for the exact guarantee PSB achieves.","marker":"[Aziz et al., 2025]"}],"fun_headline_variants":["Proportional Sequential Borda: every rank gets its pairwise due","Rank aggregation with guaranteed pairwise fairness","Pairwise wins allotted by weight in rank merging","Weighted rankings get proportional pairwise wins","Fair share of pairwise agreements in rank voting"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The whole framework rests on measuring 'representation' as the plain count of pairwise comparisons on which two rankings agree, treating every position in the output ranking as equally valuable; if applications care more about the top of the ranking, the fairness guarantees target the wrong quantity and the rules' proportional outcomes may not be the desired ones.","fun_headline_variants_meta":{"raw":{"variants":["Proportional Sequential Borda: every rank gets its pairwise due","Rank aggregation with guaranteed pairwise fairness","Pairwise wins allotted by weight in rank merging","Weighted rankings get proportional pairwise wins","Fair share of pairwise agreements in rank voting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000618,"raw_usage":{"total_tokens":2687,"prompt_tokens":710,"completion_tokens":1977,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":454,"completion_tokens_details":{"reasoning_tokens":1909}},"tokens_in":454,"tokens_out":1977,"duration_ms":14599,"temperature":1.0,"reasoning_tokens":1909,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T17:27:27.499717+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorems are universal, so a single counterexample would refute them. Run PSB (or FB) with a fixed tie-breaking on any small profile, say all rankings for m = 5 or 6, and check each input ranking's pairwise-agreement count against floor(weight * C(m,2)); any count below the bound falsifies uPJR (or sPJR for FB on subprofiles). A targeted test already appears in the paper: its Proposition 5.1 constructs a 26-candidate, 8-ranking profile on which PSB returns a ranking agreeing with only 292 pairwise comparisons of a group that deserves 293, demonstrating that PSB does not satisfy the stronge","supporting_citations":[],"review_version":1}