{"id":"79622753-a777-45b8-8940-089571fbba04","arxiv_id":"2505.00520","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors define α-parameterized versions of four proportionality axioms and show on 1,070 real Scottish elections that solid coalitions are rarely large enough for classic PSC to bind, with SNTV matching STV under the new measures.","lead":"This paper develops quantitative versions of proportionality axioms for ranked-choice elections by scaling the required size of voter coalitions. Applying them to 1,070 Scottish council elections, it finds that classic proportionality axioms rarely bind and that simple plurality-based SNTV often performs as well as STV.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.2's threshold set T includes ℓ>k constraints that Definition 7 does not; if taken literally, the full candidate set forces αPSC≥k/(k+1) for every committee, contradicting Figs. 1 and 8 unless the implementation silently caps ℓ at k.","rationale":"The reader's weakest assumption is the synthetic ballot-completion model in Appendix D.1. That model affects only Section 6, where the paper argues that truncation is not the primary cause of PSC's weak discriminatory power; the core observation that PSC rarely binds on the actual truncated data, and the SNTV comparison in Tables 3 and 4, do not depend on it. My concern is more load-bearing: Section 4.2's construction of T is internally inconsistent with Definition 7 for solid coalitions with |C′|>k, and since the full candidate set is always a solid coalition, a literal reading of the algorithm would force every committee to have αPSC≥k/(k+1). The reported distributions contradict that floor, so the implementation almost certainly deviates from the written algorithm, but without code the numerical results cannot be checked. This strengthens the reader's CONDITIONAL verdict rather than changing it: the condition should be to correct the algorithm statement, provide the code, and verify the diagnostic. I disagree with the reader's choice of the single weakest assumption because the completion model is a secondary robustness check, whereas the Section 4.2 ambiguity threatens every computed PSC value in the paper.","tokens_in":20180,"tokens_out":29969,"duration_ms":326392,"concrete_test":"Run a diagnostic instance with n voters, k=3, m=5, and every voter submitting the full ranking 1≻2≻3≻4≻5. Under Definition 7 every size-3 committee satisfies α-PSC for all α>0, so αPSC(W)=0 for all W. Under the literal T construction of Section 4.2, the grand coalition (N′=all voters, C′=C) has α4=n·3/(n·4)=0.75, so every W has αPSC(W)=0.75. Recompute the optimal PSC value and the Table 4 counts for this instance with the provided implementation. If the result is 0.75 (or any nonzero value), the implementation follows the uncapped T and all reported PSC values are suspect; if 0, the implementation caps ℓ at k and only the paper's description, not the experiments, needs correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4.2 defines the threshold set T as the union over maximal solid coalitions (N′,C′) of {α1,...,α|C′|}, and Theorem 1's algorithm checks |W∩C′|≥ℓ for each such threshold. But Definition 7 quantifies α-PSC only over ℓ∈[k]. For any solid coalition with |C′|>k, the thresholds ℓ=k+1,...,|C′| are not part of α-PSC, yet the algorithm as written would impose them. The worst case is C′=C, the full candidate set, which is a solid coalition of size n in every election: for ℓ=k+1 the threshold is k/(k+1), so every committee of size k would have αPSC(W)≥k/(k+1) (0.75 for k=3, 0.8 for k=4). The reported histograms (Fig. 1, Fig. 8) contain optimal PSC values far below that floor, so either the implementation silently caps ℓ at min(|C′|,k) or the published numbers do not follow the stated algorithm. Because no code is provided, this gap is load-bearing: all αPSC values, and hence Tables 4 and 9 and the SNTV comparison, rest on an ambiguity in the core computation. The paper should state the ℓ≤k cap explicitly and include a diagnostic check.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies whether the proportionality axiom PSC has empirical force in Scottish local elections, and proposes quantitative relaxations of PSC, local stability, EJR+, and priceability by scaling coalition-size thresholds by a factor α. It defines α-PSC (Definition 7), gives a polynomial-time algorithm for the α-value of a fixed committee, proves NP-completeness for finding a committee satisfying α-PSC with α<1 (Theorem 2), and computes optimal α-values via an ILP over threshold constraints. On 1070 real-world Scottish elections, the authors find that PSC rarely rules out any committee, that optimal α-values are usually around 0.4–0.6, that SNTV is the rule most often achieving optimal α under the PSC and local-stability measures, and that STV variants are most aligned with EJR+ and priceability. Experiments with synthetically completed ballots yield qualitatively similar conclusions, suggesting ballot truncation is not the main reason for PSC's weak discriminatory power.","tokens_in":20510,"tokens_out":10954,"duration_ms":109157,"significance":"If correct, the empirical findings are significant for computational social choice and for debates about electoral reform: the paper provides a large-scale real-world quantification of proportionality axioms for ordinal elections and demonstrates that STV's flagship axiom is nearly vacuous on actual ballot data, while a simple plurality-style rule matches or beats STV on the proposed quantitative measures. The technical contributions—the α-parameterization, the polynomial-time fixed-committee computation, and the NP-hardness reduction—are coherent and reusable. The paper uses a substantial public dataset and reports detailed tables and histograms, which is a strength. The main reservations are the absence of released code, an ambiguity in the core threshold description in Section 4.2, and the reliance on a single synthetic ballot-completion model for the truncation analysis; these limit independent verification but appear addressable in revision.","major_comments":[{"comment":"The threshold set T contains {α^1_{N',C'}, ..., α^{|C'|}_{N',C'}} for every maximal solid coalition, but Definition 7 quantifies α-PSC only over ℓ∈[k]. For any coalition with |C'|>k—in particular C'=C, the full candidate set, which is a solid coalition of size n in every election—the constraints for ℓ=k+1,...,|C'| are not part of α-PSC. If the algorithm as written imposed them, the constraint |W∩C|≥k+1 would be unsatisfiable for every committee of size k, forcing αPSC(W)≥k/(k+1) (0.75 for k=3, 0.8 for k=4) for all W. The optimal PSC values reported in Figures 1 and 8, e.g., values near 0.08 and the bulk below 0.7, are incompatible with such a floor. The implementation must therefore be silently capping ℓ at min(|C'|,k). Since no code is provided, this is a load-bearing ambiguity: every reported αPSC value, and hence Tables 4 and 9 and the comparison of SNTV with the other rules, depends on this detail. Please state the cap explicitly in Definition 7 and Theorem 1, and add a diagnostic check that no computed value falls below the k/(k+1) floor.","section":"§4.2 (Definition 7, Theorem 1)"},{"comment":"The conclusion that ballot truncation is not the main cause of PSC's low discriminatory power rests entirely on the synthetic completion method. That method extends a truncated ballot using the conditional distribution of observed longer ballots with the same prefix, but stops extending when fewer than 10% of ballots of the next length remain and then completes uniformly at random. This is a reasonable baseline, but it is an untested modeling assumption, and the 10% cutoff and uniform completion are arbitrary. The paper reports no sensitivity analysis, so it is unclear whether the near-invariance of the conclusions under completion would survive alternative, equally plausible completion models (e.g., party-based completion or models with stronger or weaker continuation rates). I ask the authors to provide such robustness checks or to soften the Section 6 and abstract claims accordingly.","section":"§6 and Appendix D.1"}],"minor_comments":[{"comment":"The sentence 'Observe that W satisfies PSC if and only if αPSC(W)<1' is not correct at the boundary: since the set of α satisfying α-PSC is open, a committee violating 1-PSC only through a coalition whose threshold equals 1 has αPSC(W)=1. The paper should either define αPSC with a convention for this case or state the one-sided implications; this also affects the counting of 'seq-RCV fails 1-PSC' in Section 5.","section":"§4.1"},{"comment":"There are several small typos and formatting issues: 'saitsfies' in §2.2, 'prcieability' in Appendix D.3, 't> 100 more instances' in Appendix D.6, and 'V ollen' in the Aziz et al. reference.","section":"§2.2, Appendix D.3, Appendix D.6, References"},{"comment":"The tables report counts and average distances without any measure of dispersion or significance. Since the differences between S-STV and SNTV on the PSC measure (856 vs. 901) are modest, a paired comparison or confidence interval would strengthen the claim that SNTV is 'most aligned' with PSC and LS.","section":"Table 4 and Table 9"}],"recommendation":"major_revision","confidential_remarks":"The Section 4.2 cap issue is likely an implementation detail rather than a conceptual flaw, but because the empirical part has no code release, the authors should be asked to confirm the cap and provide at least pseudocode for the computation. The synthetic completion model is also fixable with sensitivity analysis. I would not want to see the 'SNTV matches STV' claim over-interpreted without uncertainty quantification. The paper is otherwise a good fit for a computational social choice audience."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is one of the few papers to push proportionality axioms into a quantitative regime and test them on a large real-world set of ranked ballots. Second, the main computational description in Section 4.2 has a gap that, taken literally, would invalidate every α-PSC number in the paper; the fix is probably trivial, but it has to be stated.\n\nWhat is genuinely good. The α-parameterization (Definition 7 and the parallel definitions for LS/EJR+/priceability) is clean and useful. The move from ‘does the axiom hold?’ to ‘what is the smallest α?’ is exactly what applied social choice needs. The empirical core is solid in spirit: 1070 Scottish STV elections, the observation that PSC rarely excludes any committee, and the results in Table 4 showing SNTV matching or beating STV on the PSC and LS measures. The NP-completeness reduction in Theorem 2 looks correct. The paper does not oversell; the conclusion is appropriately cautious about reconciling theory and practice.\n\nNow the problem. Definition 7 defines α-PSC only for ℓ∈[k]. In Section 4.2, however, the threshold set T includes αℓ for ℓ=1,...,|C′|, with no cap at k. Since the full candidate set C is a solid coalition for the whole electorate, the stated algorithm would impose a constraint |W∩C|≥k+1, forcing αPSC(W)≥k/(k+1) for every committee of size k. That contradicts Example 2 (α=0.392 with k=3) and the histograms in Figures 1 and 8. So the implementation must be silently capping ℓ at min(|C′|,k). The paper needs to say that explicitly, and ideally include code. Without code, the reader cannot check whether other silent choices were made. This is addressable, not fatal, but it sits in the definition of the paper’s central quantity.\n\nOther soft spots, in decreasing order. The ballot completion model in Appendix D.1 is reasonable but unvalidated; the conclusion that truncation isn’t the main cause of PSC’s weakness depends on it. No error bars or variance estimates anywhere. The claim that SNTV’s alignment stems from singleton first-place constraints is plausible but not tested. On the citation side, Janson (2018) and Skowron (2021) are acknowledged; I don’t see a self-citation problem.\n\nVerdict: this deserves a serious referee. Send it to review, with requests to fix the ℓ≤k cap, release code, and add error bars. I would bring it to a reading group and likely cite it.","headline":"A useful quantitative take on proportionality in ordinal elections with a real-world empirical payload; the core PSC computation has an unstated but likely applied cap on ℓ that should be made explicit.","tokens_in":21000,"tokens_out":3418,"would_cite":true,"duration_ms":33943,"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":"In real Scottish council elections, the standard proportionality axiom behind STV rarely restricts the winning committee, and the simple plurality rule SNTV matches STV on the paper's quantitative proportionality measures.","keywords":["proportional representation","single transferable vote","solid coalitions","proportionality axioms","ordinal elections","multiwinner voting","ballot truncation","Scottish local elections"],"falsifier":"Collect genuine full rankings for the same 2007–2022 Scottish wards, for example from a survey eliciting complete rankings or from a jurisdiction that requires full rankings, and recompute the fraction of PSC-unrestricted elections and the optimal $\\alpha$-values. If full-preference data show that large solid coalitions are common and the share of elections where every committee satisfies PSC falls far below the 27.5% reported for the truncated data and the 21.8% reported for the completed data, the paper's account would be refuted.","tokens_in":19987,"feed_emoji":"🗳️","tokens_out":7439,"duration_ms":69107,"temperature":0.7,"pith_summary":"This paper argues that the standard axiomatic justification for the Single Transferable Vote's proportionality, the property known as proportionality for solid coalitions (PSC), has almost no force in real elections. Looking at 1,070 Scottish local council elections, the authors find that in 27.5% of them every possible committee satisfies PSC, and in most of the rest the axiom rules out few outcomes. To get beyond this toothlessness, they replace each proportionality axiom with a parameterized version and define the smallest parameter $\\alpha$ at which a committee satisfies it, yielding quantitative proportionality measures ($\\alpha$-PSC, $\\alpha$-LS, $\\alpha$-EJR+, $\\alpha$-priceability). On these measures, SNTV, a simple rule selecting the top $k$ first-place candidates, achieves the optimal $\\alpha$-value more often than STV or the Expanding Approvals Rule for the PSC and local-stability measures, while the STV variants lead on EJR+ and priceability. The paper also artificially completes the heavily truncated Scottish ballots and finds that the weak force of PSC is largely not due to truncation, because even full rankings rarely form large cohesive groups.","feed_headline":"Simple plurality matches STV on measured proportionality","feed_subtitle":"Across 1,070 Scottish council elections, the standard STV guarantee rarely binds and SNTV scores just as well.","key_machinery":"The load-bearing object is the $\\alpha$-parameterized axiom. For a group of voters that is $\\ell$-large when $|N'| \\ge \\ell n/k$, the paper declares it $\\ell_\\alpha$-large when $|N'| \\ge \\alpha \\cdot \\ell n/k$, and defines $\\alpha$-PSC, $\\alpha$-LS, $\\alpha$-EJR+, and $\\alpha$-priceability by replacing `$\\ell$-large' with `$\\ell_\\alpha$-large' (or, for priceability, scaling the price $p$ by $\\alpha$). The $\\alpha$-value of a committee is the infimum $\\alpha$ at which it satisfies the axiom, and the $\\alpha$-value of an instance is the minimum over all committees. For PSC the thresholds reduce to values $|N'| k/(n\\ell)$ computed from maximal solid coalitions, so a committee's PSC value is found in polynomial time by checking constraints at each threshold; finding the instance-optimal value is NP-complete (by reduction from 3-Hitting Set), and the experiments use an integer linear program that adds constraints in decreasing threshold order, which the appendix reinterprets as D'Hondt-style apportionment with non-disjoint parties. For LS and EJR+, the minimal $\\alpha$ is found by checking unselected candidates, and for priceability it is found by a linear program.","core_discovery":"The central discovery is that PSC, the property usually cited to justify STV's proportionality, is empirically toothless: in 294 of the 1,070 elections every committee is PSC-compatible, in 592 only one solid coalition earns a seat, and only in 17.2% do multiple solid coalitions impose any requirement. The paper's new quantitative measures, obtained by scaling group-size thresholds by a factor $\\alpha$, show that most real elections admit optimal $\\alpha$-values around 0.4 to 0.6, and that 1-LS and 1-EJR+ committees always exist in the dataset although they are not guaranteed in general. Among the rules compared, SNTV achieves the optimal PSC value in 901 of 1,070 elections and the optimal local-stability value in 935, outperforming both STV variants and EAR, while the STV variants achieve the optimal EJR+ and priceability values most often. seq-RCV performs worst on all four measures: with artificially completed ballots it violates PSC in 55 elections and reaches local-stability $\\alpha$-values up to 1.6. The paper concludes that in practice the proportionality measures behave similarly for the proportional rules, while a majoritarian method is clearly separated.","pith_inferences":["Extending the same $\\alpha$-parameterization to approval-based settings (e.g., justified-representation axioms) would let the same degree-of-proportionality comparison be run on participatory budgeting or sortition datasets, where binary axioms are also often satisfied vacuously.","The near-equivalence between SNTV and STV outcomes (agreement in roughly 70% of elections) suggests that in these small, low-dimensional ward elections first-place votes carry most of the information that a transferable-vote count uses; a testable prediction is that the gap widens in multi-member districts with stronger party systems, such as Irish STV elections.","The NP-hardness of computing the optimal $\\alpha$-PSC value means that practical use for larger assemblies needs approximation; the paper's noted connection to D'Hondt apportionment points toward a concrete heuristic: allocate representation guarantees by a divisor method over non-disjoint coalitions and check feasibility greedily.","If replicated on other STV jurisdictions (e.g., Ireland, Malta, Australia), the results would support treating proportionality as a graded empirical property rather than a binary axiom, and would reopen the normative question of whether STV's complexity is justified over SNTV."],"forward_implications":["If the paper's findings hold, the standard claim that STV is proportionally representative because it satisfies PSC has little empirical bite in the settings studied: the axiom simply does not select among committees in most real elections.","SNTV, which carries no proportionality guarantee, would be a competitive and far simpler alternative under the PSC and local-stability measures, since it hits the optimal value more often than both STV variants and EAR.","The quantitative $\\alpha$-measures separate a majoritarian rule (seq-RCV) from the proportional and semi-proportional rules, so they can serve as a diagnostic that binary axioms alone cannot provide.","Ballot truncation is not the explanation for weak PSC restrictions: after synthetically completing ballots, the share of elections where every committee satisfies PSC drops only from 27.5% to 21.8%, and the relative performance of the rules is broadly unchanged, with seq-RCV deteriorating further.","Because in most instances the optimal PSC constraints involve singleton candidate sets (57% on average), the measures reward first-place vote share, which explains why SNTV's performance is comparable to that of more complex rules."],"supporting_citations":[{"why":"Supplies the real-world dataset of 1,100 Scottish local council elections and the version of Scottish STV used in the experiments.","marker":"[McCune and Graham-Squire, 2024]"},{"why":"Introduces proportionality for solid coalitions (PSC), the axiom whose practical force the paper measures.","marker":"[Dummett, 1984]"},{"why":"Introduces the Expanding Approvals Rule and generalized PSC; EAR is one of the rules compared.","marker":"[Aziz and Lee, 2020]"},{"why":"Formulates the EJR+ and priceability axioms and the EAR priceability guarantee on which the quantitative versions are based.","marker":"[Brill and Peters, 2023]"},{"why":"Defines local stability, the multiwinner extension of Condorcet consistency used to build the $\\alpha$-LS measure.","marker":"[Aziz et al., 2017b]"},{"why":"Documents Sequential Ranked-Choice Voting as used in Utah and its non-proportional behavior, motivating its inclusion.","marker":"[McCune et al., 2024]"},{"why":"Prior empirical study of ballot truncation in six Cambridge MA elections, the baseline the paper's truncation analysis extends.","marker":"[Hoffman et al., 2024]"},{"why":"Provides the existence of 9.8217-locally-stable committees, used as a benchmark in the discussion of $\\alpha$-LS.","marker":"[Charikar et al., 2025]"}],"fun_headline_variants":["SNTV equals STV on measured proportionality in real elections","STV's proportionality guarantee rarely bites in practice","Quantitative measures show proportional rules similar in practice","Seq-RCV lags on all four measured proportionality metrics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the synthetic method used to complete truncated ballots, extending each partial ballot according to the distribution of longer ballots that share its prefix and filling in uniformly at random when too few longer ballots exist, faithfully reconstructs the full rankings voters would have given; if that completion model is wrong, the paper's conclusion that truncation is not the main reason PSC has little force collapses.","fun_headline_variants_meta":{"raw":{"variants":["SNTV equals STV on measured proportionality in real elections","STV's proportionality guarantee rarely bites in practice","Quantitative measures show proportional rules similar in practice","Seq-RCV lags on all four measured proportionality metrics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001009,"raw_usage":{"total_tokens":4272,"prompt_tokens":964,"completion_tokens":3308,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":3245}},"tokens_in":580,"tokens_out":3308,"duration_ms":25219,"temperature":1.0,"reasoning_tokens":3245,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:40:09.505520+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Collect genuine full rankings for the same 2007–2022 Scottish wards, for example from a survey eliciting complete rankings or from a jurisdiction that requires full rankings, and recompute the fraction of PSC-unrestricted elections and the optimal $\\alpha$-values. If full-preference data show that large solid coalitions are common and the share of elections where every committee satisfies PSC falls far below the 27.5% reported for the truncated data and the 21.8% reported for the completed data, the paper's account would be refuted.","supporting_citations":[],"review_version":1}