Proportionality in Practice: Quantifying Proportionality in Ordinal Elections

David McCune; Jannik Peters; Markus Brill; Tuva Bardal

REVIEW 2 major objections 3 minor 48 references

Proportionality in Practice: Quantifying Proportionality in Ordinal Elections

T0 review · 2 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2505.00520 v1 pith:RNW4X27J submitted 2025-05-01 cs.GT

Tuva Bardal , Markus Brill , David McCune , Jannik Peters This is my paper · ORCID

classification cs.GT MSC 91B1291B14

keywords proportionalrepresentationsingletransferablevotesolidcoalitionsproportionalityaxiomsordinalelectionsmultiwinnervotingballottruncationScottishlocal

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

Now 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.

Other 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.

Verdict: 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[§4.2 (Definition 7, Theorem 1)] 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.
[§6 and Appendix D.1] 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.

minor comments (3)

[§4.1] 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.
[§2.2, Appendix D.3, Appendix D.6, References] 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.
[Table 4 and Table 9] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circularity by construction; only a minor, non-load-bearing self-citation of Brill and Peters (2023).

full rationale

The quantitative measures (α-PSC, α-LS, α-EJR+, α-priceability) are defined directly from the proportionality axioms in Section 4 and are then evaluated on external Scottish ballot data. The optimal α-values are computed by explicit threshold enumeration and ILP from those same definitions, not fitted to the rule rankings. The voting rules are independent algorithms with fixed behaviors, and the claims that PSC rarely binds and that SNTV often achieves optimal α-values are outputs of the computation rather than built into the measures. The synthetic ballot completion model in Appendix D.1 uses observed conditional distributions to extend truncated ballots and does not tune the completion procedure to any target conclusion; its limitations are about external validity, not circularity. The only self-citation that appears—the Brill and Peters (2023) axioms and the background claim that STV fails stronger axioms—is not load-bearing: the definitions are restated in the paper, and the empirical derivation does not invoke any unverified theorem from that citation to force its conclusion. I additionally note that Section 4.2 writes the threshold set T as containing α^1,...,α^{|C'|} without explicitly capping ℓ at k, which conflicts with Definition 7's ℓ∈[k] bound; this is a computational/correctness inconsistency, not a circular reduction. No equation is equivalent to its inputs by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The α-measures are computed, not fitted; no numerical constants are tuned to the conclusions. The free parameters appear in the synthetic ballot completion procedure. The main assumptions are representativeness of the Scottish dataset, correctness of the implemented rules, and the normative reasonableness of scaling thresholds by α.

free parameters (2)

Ballot extension cutoff = 0.10 (10%)
In the synthetic ballot completion method (Appendix D.1), ballots are extended only while the number of longer ballots is at least 10% of the current ballot count. This threshold is chosen by the authors and affects the completed-preference dataset.
Random completion distribution = Uniform over unranked candidates
Ballots that cannot be extended by observed longer ballots are completed by choosing uniformly at random among unranked candidates. This choice affects the completed dataset and the conclusions about truncation.

assumptions (5)

standard math 3-Hitting Set is NP-complete
Used in the proof of Theorem 2 to establish NP-completeness of deciding whether an α-PSC-satisfying committee exists.
domain assumption The Scottish local election ballot data from McCune and Graham-Squire accurately reflects the ballots actually cast
All empirical results in Sections 3 to 6 depend on this dataset being correct and representative.
domain assumption The described implementations of Scottish STV and Meek-STV correspond to the rules used in the data
Section 2.2 describes the STV variants; if these differ from actual practice, the rule comparisons could be affected.
ad hoc to paper Proportionality can be meaningfully quantified by a single multiplicative factor α applied to the size threshold of cohesive groups
Section 4 proposes this parameterization as the core quantitative measure; it is a normative modeling choice rather than a derived fact.
ad hoc to paper The synthetic completion model approximates the preferences voters would have expressed under full rankings
Section 6 and Appendix D.1 assume that completing truncated ballots by conditional distributions of longer ballots gives a realistic counterfactual full-preference profile.

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Cite this review

Pith. "Pith review of Proportionality in Practice: Quantifying Proportionality in Ordinal Elections." pith.science (2026). https://pith.science/paper/RNW4X27J

@misc{pith2026250500520,
  author       = {Pith},
  title        = {Pith review of: Proportionality in Practice: Quantifying Proportionality in Ordinal Elections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RNW4X27J}},
  note         = {Machine review of arXiv:2505.00520}
}

read the original abstract

Proportional representation plays a crucial role in electoral systems. In ordinal elections, where voters rank candidates based on their preferences, the Single Transferable Vote (STV) is the most widely used proportional voting method. STV is considered proportional because it satisfies an axiom requiring that large enough solid coalitions of voters are adequately represented. Using real-world data from local Scottish elections, we observe that solid coalitions of the required size rarely occur in practice. This observation challenges the importance of proportionality axioms and raises the question of how the proportionality of voting methods can be assessed beyond their axiomatic performance. We address these concerns by developing quantitative measures of proportionality. We apply these measures to evaluate the proportionality of voting rules on real-world election data. Besides STV, we consider SNTV, the Expanding Approvals Rule, and Sequential Ranked-Choice Voting. We also study the effects of ballot truncation by artificially completing truncated ballots and comparing the proportionality of outcomes under complete and truncated ballots.

Figures

Figures reproduced from arXiv: 2505.00520 by the authors.

**Figure 2.** The PSC values achieved by voting rules, together with optimal PSC values [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Left: PSC values achieved by Scottish STV and seq-RCV for the 485 elections where the rules disagree. Right: PSC values achieved by Scottish STV and EAR over the 262 elections where the rules disagree. Elections are ordered by increasing optimal PSC value. < 25% < 50% < 100% PSC 49 (4.6%) 305 (28.5%) 837 (78.2%) EJR+ 81 (7.5%) 430 (40.2%) 1030 (96.3%) LS 240 (22.4%) 749 (70.0%) 1066 (99.6%) Priceability 115 (10.7%) … view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Histograms of LS and priceability values achievable in our elections, rounded [PITH_FULL_IMAGE:figures/full_fig_p026_4.png]

**Figure 5.** Figure 5: The EJR+ values achieved by voting rules [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

**Figure 6.** Figure 6: The LS values achieved by voting rules 28 [PITH_FULL_IMAGE:figures/full_fig_p028_6.png]

**Figure 7.** Figure 7: The priceability values achieved by voting rules [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

**Figure 8.** Figure 8: Histograms of optimal values, rounded to one decimal place (completed pref [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]

**Figure 9.** Figure 9: The PSC values achieved by voting rules (completed preferences). [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]

**Figure 10.** Figure 10: The EJR+ values achieved by voting rules (completed preferences). [PITH_FULL_IMAGE:figures/full_fig_p034_10.png]

**Figure 11.** Figure 11: The priceability values achieved by voting rules (completed preferences). [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]

**Figure 12.** Figure 12: The LS values achieved by voting rules (completed preferences). [PITH_FULL_IMAGE:figures/full_fig_p035_12.png]

**Figure 13.** Figure 13: Differences in PSC values between Scottish STV and Sequential RCV when [PITH_FULL_IMAGE:figures/full_fig_p036_13.png]

**Figure 14.** Figure 14: Differences in PSC values between Scottish STV and EAR when elections [PITH_FULL_IMAGE:figures/full_fig_p037_14.png]

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Works this paper leans on

48 extracted references · 40 canonical work pages

[1]

D. J. Amy. Behind the Ballot Box: A Citizen's Guide to Voting Systems. Bloomsbury Academic, 2000

work page 2000
[2]

Aziz and B

H. Aziz and B. E. Lee. The expanding approvals rule: improving proportional representation and monotonicity. Social Choice and Welfare, 54: 0 1--45, 2020

2020
[3]

Aziz and B

H. Aziz and B. E. Lee. Proportionally representative participatory budgeting with ordinal preferences. In Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI), pages 5110--5118. AAAI Press, 2021

work page 2021
[4]

Aziz and B

H. Aziz and B. E. Lee. A characterization of proportionally representative committees. Games and Economic Behavior, 133: 0 248--255, 2022

work page 2022
[5]

H. Aziz, M. Brill, V. Conitzer, E. Elkind, R. Freeman, and T. Walsh. Justified representation in approval-based committee voting. Social Choice and Welfare, 48 0 (2): 0 461--485, 2017 a

work page 2017
[6]

H. Aziz, E. Elkind, P. Faliszewski, M. Lackner, and P. Skowron. The C ondorcet principle for multiwinner elections: from shortlisting to proportionality. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), pages 84 -- 90, 2017 b

work page 2017
[7]

H. Aziz, B. E. Lee, S. Morota Chu , and J. Vollen. Proportionally representative clustering. In Proceedings of the 20th International Workshop on Internet and Network Economics (WINE), 2024. Forthcoming

work page 2024
[8]

M. L. Balinski and H. P. Young. The webster method of apportionment. Proceedings of the National Academy of Sciences (PNAS), 77 0 (1): 0 1--4, 1980

work page 1980

Show all 48 references

[9]

M. L. Balinski and H. P. Young. Fair Representation: M eeting the Ideal of One Man, One Vote . Yale University Press, 1982

1982
[10]

Boehmer, P

N. Boehmer, P. Faliszewski, . Janeczko, and A. Kaczmarczyk. Robustness of participatory budgeting outcomes: Complexity and experiments. In Proceedings of the 16th International Symposium on Algorithmic Game Theory (SAGT), pages 161--178, 2023

2023
[11]

Boehmer, M

N. Boehmer, M. Brill, A. Cevallos, J. Gehrlein, L. S \'a nchez-Fern \'a ndez, and U. Schmidt-Kraepelin. Approval-based committee voting in practice: A case study of (over-)representation in the P olkadot blockchain. In Proceedings of the 38th AAAI Conference on Artificial Inte...

2024
[12]

Boehmer, P

N. Boehmer, P. Faliszewski, . Janeczko, A. Kaczmarczyk, G. Lisowski, G. Pierczy \'n ski, S. Rey, D. Stolicki, S. Szufa, and T. W a s. Guide to numerical experiments on elections in computational social choice. In Proceedings of the 33rd International Joint Conference on Artifi...

2024
[13]

Boehmer, P

N. Boehmer, P. Faliszewski, . Janeczko, D. Peters, G. Pierczy \'n ski, S . Schierreich, P. Skowron, and S. Szufa. Evaluation of project performance in participatory budgeting. In Proceedings of the 33rd International Joint Conference on Artificial Intelligence (IJCAI), pages 2...

2024
[14]

Brill and J

M. Brill and J. Peters. Robust and verifiable proportionality axioms for multiwinner voting. In Proceedings of the 24th ACM Conference on Economics and Computation (ACM-EC), page 301. ACM Press, 2023. Full version arXiv:2302.01989 [cs.GT]

2023 arXiv
[15]

Brill, J.-F

M. Brill, J.-F. Laslier, and P. Skowron. Multiwinner approval rules as apportionment methods. Journal of Theoretical Politics, 30 0 (3): 0 358--382, 2018

2018
[16]

Brill, S

M. Brill, S. Forster, M. Lackner, J. Maly, and J. Peters. Proportionality in approval-based participatory budgeting. In Proceedings of the 37th AAAI Conference on Artificial Intelligence (AAAI), pages 5524--5531. AAAI Press, 2023

2023
[17]

Burnett and V

C. Burnett and V. Kogan. Ballot (and voter) ``exhaustion''' under instant runoff voting: An examination of four ranked-choice elections. Electoral Studies, 37: 0 41--49, 2015

2015
[18]

Caragiannis, E

I. Caragiannis, E. Micha, and N. Shah. Proportional fairness in non-centroid clustering. In Proceedings of the 38th Conference on Neural Information Processing Systems (NeurIPS), 2024. Forthcoming

2024
[19]

Charikar, A

M. Charikar, A. Lassota, P. Ramakrishnan, A. Vetta, and K. Wang. Six candidates suffice to win a voter majority. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing (STOC), 2025. Forthcoming

2025
[20]

X. Chen, B. Fain, L. Lyu, and K. Munagala. Proportionally fair clustering. In Proceedings of the 36th International Conference on Machine Learning (ICML), pages 1032--1041, 2019

2019
[21]

Delemazure and D

T. Delemazure and D. Peters. Generalizing instant runoff voting to allow indifferences. In Proceedings of the 25th ACM Conference on Economics and Computation (ACM-EC), page 50. ACM Press, 2024

2024
[22]

M. Dummett. Voting Procedures. Oxford University Press, 1984

1984
[23]

Ebadian and E

S. Ebadian and E. Micha. Boosting sortition via proportional representation. In Proceedings of the 24th International Conference on Autonomous Agents and Multiagent Systems (AAMAS), 2025. Forthcoming

2025
[24]

Ebadian, A

S. Ebadian, A. Kahng, D. Peters, and N. Shah. Optimized distortion and proportional fairness in voting. ACM Transactions on Economics and Computation, 12 0 (1): 0 1---39, 2024

2024
[25]

Elkind, J

E. Elkind, J. Lang, and A. Saffidine. Condorcet winning sets. Social Choice and Welfare, 44 0 (3): 0 493--517, 2015

2015
[26]

Elkind, P

E. Elkind, P. Faliszewski, J.-F. Laslier, P. Skowron, A. Slinko, and N. Talmon. What do multiwinner voting rules do? A n experiment over the two-dimensional E uclidean domain. In Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI), pages 494--501. AAAI Pr...

2017
[27]

Faliszewski, J

P. Faliszewski, J. Fils, D. Peters, G. Pierczy \'n ski, P. Skowron, D. Stolicki, S. Szufa, and N. Talmon. Participatory budgeting: Data, tools, and analysis participatory budgeting: Data, tools, and analysis. In Proceedings of the 32nd International Joint Conference on Artific...

2023
[28]

o lz, D. C. Parkes, A. D. Procaccia, G. Rusak, I. Shapira, and M. W \

S. Fish, P. G \"o lz, D. C. Parkes, A. D. Procaccia, G. Rusak, I. Shapira, and M. W \"u thrich. Generative social choice. In Proceedings of the 25th ACM Conference on Economics and Computation (ACM-EC), page 985. ACM Press, 2024

2024
[29]

Graham-Squire and D

A. Graham-Squire and D. McCune. An examination of ranked-choice voting in the united states, 2004–2022. Representation, pages 1--19, 2023

2004
[30]

I. D. Hill, B. A. Wichmann, and D. R. Woodall. Algorithm 123: Single transferable vote by meek's method. The Computer Journal, 30 0 (3), 1987

1987
[31]

Hoffman, J

C. Hoffman, J. Kauba, J. Reidy, and T. Weighill. Proportionality in multi-winner RCV elections: A simulation study with ballot truncation. Communications in Statistics - Simulation and Computation, 2024. Forthcoming

2024
[32]

S. Janson. Thresholds quantifying proportionality criteria for election methods. Technical report, arXiv:1810.06377 [cs.GT], 2018

2018 arXiv
[33]

Jiang, K

Z. Jiang, K. Munagala, and K. Wang. Approximately stable committee selection. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 463--472. ACM, 2020

2020
[34]

Y. H. Kalayci, D. Kempe, and V. Kher. Proportional representation in metric spaces and low-distortion committee selection. In Proceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI), pages 9815--9823. AAAI Press, 2024

2024
[35]

Kellerhals and J

L. Kellerhals and J. Peters. Proportional fairness in clustering: A social choice perspective. In Proceedings of the 38th Conference on Neural Information Processing Systems (NeurIPS), 2024

2024
[36]

Kilgour, J

D. Kilgour, J. Gr \'e goire, and A. Foley. The prevalence and consequences of ballot truncation in ranked-choice elections. Public Choice, 184: 0 197--218, 2020

2020
[37]

Lackner and P

M. Lackner and P. Skowron. Multi-Winner Voting with Approval Preferences. Springer, 2022

2022
[38]

Marsh and C

M. Marsh and C. Plescia. Split-ticket voting in an STV system: choice in a non-strategic context. Irish Political Studies, 31 0 (2): 0 163--181, 2016

2016
[39]

S. T. McCormick. Handbook on Discrete Optimization, chapter 7. Elsevier, 2007

2007
[40]

McCune and A

D. McCune and A. Graham-Squire. Monotonicity anomalies in scottish local government elections. Social Choice and Welfare, 63 0 (1): 0 69--101, 2024

2024
[41]

McCune and L

D. McCune and L. McCune. Does the choice of preferential voting method matter? an empirical study using ranked choice elections in the united states. Representation, 60 0 (1): 0 1--16, 2024

2024
[42]

McCune, E

D. McCune, E. Martin, G. Latina, and K. Simms. A comparison of sequential ranked-choice voting and single transferable vote. Journal of Computational Social Science, 7 0 (1): 0 643--670, 2024

2024
[43]

Peters and P

D. Peters and P. Skowron. Proportionality and the limits of welfarism. In Proceedings of the 21st ACM Conference on Economics and Computation (ACM-EC), pages 793--794. ACM Press, 2020

2020
[44]

Peters, G

D. Peters, G. Pierczy \'n ski, and P. Skowron. Proportional participatory budgeting with additive utilities. In Advances in Neural Information Processing Systems (NeurIPS), volume 34, pages 12726--12737, 2021

2021
[45]

P. Skowron. Proportionality degree of multiwinner rules. In Proceedings of the 22nd ACM Conference on Economics and Computation (ACM-EC), pages 820--840. ACM Press, 2021

2021
[46]

N. Tideman. The single transferable vote. Journal of Economic Perspectives, 9 0 (1): 0 27--38, 1995

1995
[47]

N. Tideman. Collective Decisions And Voting: T he Potential for Public Choice . Ashgate, 2006

2006
[48]

Tomlinson, J

K. Tomlinson, J. Ugander, and J. Kleinberg. Ballot length in instant runoff voting. In Proceedings of the 37th AAAI Conference on Artificial Intelligence (AAAI), pages 5841--5849. AAAI Press, 2023

2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

D. J. Amy. Behind the Ballot Box: A Citizen's Guide to Voting Systems. Bloomsbury Academic, 2000

work page 2000

[2] [2]

Aziz and B

H. Aziz and B. E. Lee. The expanding approvals rule: improving proportional representation and monotonicity. Social Choice and Welfare, 54: 0 1--45, 2020

2020

[3] [3]

Aziz and B

H. Aziz and B. E. Lee. Proportionally representative participatory budgeting with ordinal preferences. In Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI), pages 5110--5118. AAAI Press, 2021

work page 2021

[4] [4]

Aziz and B

H. Aziz and B. E. Lee. A characterization of proportionally representative committees. Games and Economic Behavior, 133: 0 248--255, 2022

work page 2022

[5] [5]

H. Aziz, M. Brill, V. Conitzer, E. Elkind, R. Freeman, and T. Walsh. Justified representation in approval-based committee voting. Social Choice and Welfare, 48 0 (2): 0 461--485, 2017 a

work page 2017

[6] [6]

H. Aziz, E. Elkind, P. Faliszewski, M. Lackner, and P. Skowron. The C ondorcet principle for multiwinner elections: from shortlisting to proportionality. In Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI), pages 84 -- 90, 2017 b

work page 2017

[7] [7]

H. Aziz, B. E. Lee, S. Morota Chu , and J. Vollen. Proportionally representative clustering. In Proceedings of the 20th International Workshop on Internet and Network Economics (WINE), 2024. Forthcoming

work page 2024

[8] [8]

M. L. Balinski and H. P. Young. The webster method of apportionment. Proceedings of the National Academy of Sciences (PNAS), 77 0 (1): 0 1--4, 1980

work page 1980