arxiv: 2605.04612 · v1 · submitted 2026-05-06 · 💻 cs.GT

Recognition: unknown

An Axiomatic Analysis of Proportionality Notions in Approval-Based Multiwinner Voting

Chris Dong, Jannik Peters

Pith reviewed 2026-05-08 17:00 UTC · model grok-4.3

classification 💻 cs.GT

keywords proportionality notionsapproval-based multiwinner votingPJR+EJR+axiomatic characterizationmonotonicitywitness-based notions

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The pith

PJR+ is the canonical minimal proportionality notion under monotonicity and three other mild axioms in approval-based multiwinner voting.

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

The paper defines axioms that proportionality notions in approval-based multiwinner voting ought to satisfy and uses them to characterize PJR+ and EJR+. It proves that any notion meeting monotonicity, independence of losers, robustness to fully satisfied voters, and lower quota must refine PJR+, establishing PJR+ as the weakest such notion. The authors further introduce witness-based proportionality notions that certify misrepresentation through a specific set of voters and show PJR+ and EJR+ are the strongest within that class under the axioms. This supplies normative reasons to prefer PJR+ over earlier concepts like PJR. A reader would care because the work clarifies how to choose among competing proportionality standards when designing multiwinner voting rules.

Core claim

The paper shows that monotonicity distinguishes PJR+ from PJR and that any proportionality notion obeying the four mild axioms refines PJR+. Within the class of witness-based proportionality notions, which certify misrepresentation by exhibiting a witness set of voters, PJR+ and EJR+ are the strongest notions satisfying the relevant axioms. Combining both directions yields exact axiomatic characterizations of PJR+ and EJR+.

What carries the argument

Witness-based proportionality notions, which certify misrepresentation of a group of voters by exhibiting a witness set of those voters whose approved candidates are insufficiently represented.

If this is right

Any voting rule whose proportionality guarantee meets the four axioms automatically satisfies PJR+.
PJR+ serves as the base that stronger notions such as EJR+ build upon.
Earlier notions like PJR are ruled out once monotonicity is required.
The witness-based framework allows systematic comparison of proportionality concepts by their strength under fixed axioms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Voting system designers can now rank candidate proportionality guarantees by checking which axioms they obey.
The same axiomatic lens could be applied to proportionality in other multiwinner settings such as ranked ballots.
Monotonicity appears to be the property that forces refinement of PJR+, suggesting it should be checked first when evaluating new notions.

Load-bearing premise

That monotonicity together with independence of losers, robustness to fully satisfied voters, and lower quota are the right normative requirements any proportionality notion should meet.

What would settle it

A concrete proportionality notion that satisfies monotonicity, independence of losers, robustness to fully satisfied voters, and lower quota yet fails to refine PJR+.

Figures

Figures reproduced from arXiv: 2605.04612 by Chris Dong, Jannik Peters.

**Figure 1.** Figure 1: Example for Lower Quota and Independence of Losers. the instance is not a party-list instance because of candidate 𝑐3. Hence, a notion could select the bolded (and severely non-proportional) committee {𝑐4, . . . , 𝑐7} for 𝑘 = 4 without violating lower quota for party-lists. We therefore introduce further desirable properties that allow us to connect our choices on partylist instances to the choices on the… view at source ↗

**Figure 2.** Figure 2: Example for PJR, EJR, and monotonicity. We note that already in the original instance the bolded outcome {𝑐1, 𝑐3, 𝑐4, 𝑐5, 𝑐6, 𝑐7} neither satisfies PJR+ nor EJR+ as witnessed by candidate 𝑐2 together with voters 1 and 2 and ℓ = 3. Intuitively, there are two opposing approaches to representation: one which views the group as a whole, and one which considers each voter within the group individually. We will… view at source ↗

**Figure 3.** Figure 3: Example for local embeddings. three candidates in total, while every voter in {1, 2, 3} approves at most two. We are now ready to introduce our framework of witness-based proportionality notions. Formalizing our previously discussed intuition, a witness for a proportionality violation should remain a witness if we can locally embed it into another instance with the same number of voters. Definition 11 (Wi… view at source ↗

**Figure 4.** Figure 4: Example instance for laminar proportionality. on top of laminar proportionality instead of lower quota for party-lists. A second major limitation (even within the family of justified representation notions) stems from the fact that we only considered cohesiveness-based proportionality notions. With this, we are not able to capture several important notions, for instance FJR [Peters et al., 2021] or FPJR [K… view at source ↗

**Figure 5.** Figure 5: with 𝑘 = 4 and committee {𝑐2, . . . , 𝑐5}. This committee indeed satisfies EJR+. For any subset of 𝑖 voters, at least one voter approves 𝑖 candidates in the outcome, and hence this group could not witness an EJR+ violation. It, however, does not satisfy NPR as for candidate 𝑐1 the group 4-large group {1, . . . , 4} on average approves 4+3+2+1 4 = 2.5 ≤ 4 − 1 1 2 3 4 𝑐1 𝑐2 𝑐3 𝑐4 𝑐5 view at source ↗

**Figure 6.** Figure 6: Example showing that EJR+ ∩exPareto does not admit a cohesiveness-based witness. chosen, this witness could have been locally embedded into this new instance, showing that it would not have been a valid witness. Thus, EJR+ ∩exPareto does not admit a cohesiveness-based witness. □ Leaving out Individual Discontentment. For individual discontentment, we have already shown in Proposition 17 that NPR does not s… view at source ↗

**Figure 7.** Figure 7: Example showing that equalEJR+ does not satisfy monotonicity. Leaving out Independence of Approval Swaps. Finally, for independence of approval-swaps, we note that PJR+ satisfies all remaining axioms. D.1.2 Necessity of the Axioms in Theorem 4. In the previous part, we have already shown that removing either monotonicity or independence of approval-swaps leads to a strictly weaker axiom than EJR+ (equal-E… view at source ↗

**Figure 8.** Figure 8: Weak-EJR+ violates robustness to fully satisfied voters. Leaving out Independence of Losers. Secondly, we consider independence of losers. Here, we define a weakening of EJR+, we term difference-EJR+. Difference-EJR+ requires that the candidates outside the committee must be the same. Definition 21 (difference-EJR+). A committee 𝑊 satisfies difference-EJR+ if there is no candidate 𝑐 ∉ 𝑊 , ℓ ∈ [𝑘], and grou… view at source ↗

**Figure 9.** Figure 9: Example for EJR+ and differenceEJR+. and 2 can deviate, thus leading to a violation of difference-EJR+ (and therefore showing that difference-EJR+ does not satisfy independence of losers). This example also shows that EJR+ is strictly stronger than difference-EJR+, as the committee {𝑐3, 𝑐4} does not satisfy EJR+ (due to the voters 1 and 2). □ Leaving out Lower Quota for Party-Lists. Finally, without lower… view at source ↗

**Figure 10.** Figure 10: Example for PJR ≠ overlap-PJR violating monotonicity. D.4 Necessity of the Axioms in Theorem 2 Leaving out Robustness to Fully Satisfied Voters. Without robustness to fully satisfied voters, we can define a weakening of PJR+ which satisfies the remaining three axioms. Instead of considering all large enough subsets 𝑁 ′ ⊆ 𝑁𝑐 as potential violation witnesses, it only considers the supports 𝑁𝑐 for 𝑐 ∈ 𝐶 \𝑊 .… view at source ↗

**Figure 11.** Figure 11: Example for weak-PJR+. Leaving out Independence of Losers. Thirdly, we consider independence of losers. Here, we define a weakening of PJR+, we term difference-PJR+. While overlap-PJR (Definition 22) required that the candidates approved inside the committee must be the same for a deviating coalition, difference-PJR+ requires that the candidates outside the committee must be the same. Definition 24 (diffe… view at source ↗

**Figure 12.** Figure 12: Example for PJR+ and difference-PJR+. losers consider the instance depicted in view at source ↗

**Figure 13.** Figure 13: for 𝑘 = 4. One possible committee satisfying JR is {𝑐1, 𝑐2, 𝑐3, 𝑐4}. This however, is a lower quota for party-lists violation, as the two voters 3 and 4 together form a party deserving 4 · 2 4 = 2 seats. 1 2 3 4 𝑐1 𝑐2 𝑐3 𝑐4 𝑐5 𝑐6 view at source ↗

**Figure 14.** Figure 14: Example for PJR and independence of approval swaps. □ F.3 EJR Proposition 32. EJR is a witness-based proportionality notion satisfying independence of losers, robustness to fully satisfied voters, and lower quota extension. Together with its natural witness EJR is cohesiveness-based and satisfies individual discontentment. EJR does not satisfy merge-proofness, monotonicity, or independence of approval sw… view at source ↗

**Figure 15.** Figure 15: Example for EJR and mergeproofness. Now consider the merged profile 𝐴 ′ with 𝐴 ′ 𝑖 = 𝐴𝑖 for 𝑖 ∉ 𝑁 ′ and 𝐴 ′ 𝑖 = {𝑐1, 𝑐2, 𝑐4, 𝑐5} for all 𝑖 ∈ 𝑁 ′ . Then 𝐴𝑖 ⊆ 𝐴 ′ 𝑖 ⊆ Ð 𝑗 ∈𝑁 ′ 𝐴𝑗 for all 𝑖 ∈ 𝑁 ′ , but every voter in 𝑁 ′ now approves two members of 𝑊 (namely 𝑐1 and 𝑐2), so 𝑁 ′ is no longer a witness for 𝑊 . Thus, EJR with its natural witness does not satisfy merge-proofness. Monotonicity: The instance and c… view at source ↗

**Figure 16.** Figure 16: Instances used to separate the core from view at source ↗

**Figure 2.** Figure 2: Example for FJR and Monotonicity. The instance depicted in view at source ↗

**Figure 2.** Figure 2: Example for FPJR and Monotonicity. Independence of Approval Swaps: For independence of approval swaps, we again consider view at source ↗

**Figure 14.** Figure 14: Here, the committee 𝑐1, 𝑐3, 𝑐4, . . . , 𝑐7 satisfies FPJR for 𝑘 = 6. However, if the voter 𝑐2 switches their approval profile to be {𝑐1, 𝑐2, 𝑐3} (and thus still only approving two candidates in the outcome), the group of voters 1 and 2 is now (3, {𝑐1, 𝑐2, 𝑐3})-cohesive while only receiving two candidates in the outcome. Hence, in this new instance the committee does not satisfy FPJR and thus FPJR does no… view at source ↗

**Figure 14.** Figure 14: Example for priceability and robustness to fully satisfied voters. and 𝑐3. Hence, the total budget 𝐵 would need to be at least 8, and thus voter 4 together with candidate 𝑐4 would violate C5). Further, by priceability violating robustness to fully satisfied voters, we also get from Lemma 9 that there does not exist a cohesiveness-based witness function for it. Independence of Approval Swaps: For independe… view at source ↗

**Figure 17.** Figure 17: Example for priceability and witness functions view at source ↗

**Figure 18.** Figure 18: Example for priceability and merge-proofness. instance where voters only approve candidates from 𝑊 . Further case distinctions concern whether 1 is contained in the witness and what happens if the witness is only of cardinality 2.) For {1, 2, 3, 4} and {2, 3, 4} we can let voter 4 additionally approve 𝑐3 by merge-proofness. However, then letting voter 4 pay for 𝑐3 and voter 1 for 𝑐1 and 𝑐2 leads to a vali… view at source ↗

read the original abstract

Even though proportional representation is a fundamental goal in multiwinner voting and a plethora of proportionality notions has been introduced, the normative justifications for choosing one notion over another remain poorly understood. We address this by introducing the axiomatic study of proportionality notions in the approval-based multiwinner voting setting. That is, we define axioms (or desirable properties) that ``good'' proportionality notions should possess. Using these axioms, we then provide axiomatic characterizations of two prominent recently introduced notions: PJR+ and EJR+ [Brill and Peters 2023]. Our characterization proceeds in two parts. Firstly, we provide a characterization of refinements of PJR+ and EJR+. That is, we define axioms such that any notion satisfying these axioms must imply PJR+ (or EJR+, respectively). In particular, the fundamental axiom distinguishing PJR+ and EJR+ from their predecessors PJR and EJR is the classical axiom of monotonicity. Secondly, we introduce our framework of witness-based proportionality notions, that is, proportionality notions that certify ``misrepresentation'' via a witness set of misrepresented voters. In this class, we provide characterizations of PJR+ and EJR+ as the strongest (assuming certain axioms). Thus, by putting both directions together we obtain exact characterizations of both notions. Among our results, it may be worth highlighting that any notion satisfying mild conditions (monotonicity, independence of losers, robustness to fully satisfied voters, and lower quota) refines PJR+. In this sense, PJR+ turns out to be the canonical minimal requirement that one may impose on proportionality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives exact axiomatic characterizations of PJR+ and EJR+ by showing they are the strongest notions in a witness-based class under monotonicity and a few other mild conditions.

read the letter

The main point is that any proportionality notion meeting monotonicity, independence of losers, robustness to fully satisfied voters, and lower quota must refine PJR+. The authors reach this by splitting the argument into two pieces: first a characterization of all refinements of PJR+ and EJR+, then a witness-based framework in which PJR+ and EJR+ turn out to be maximal once the axioms are imposed. Monotonicity is the key property that separates the new notions from the older PJR and EJR versions. This is new; the witness framework and the resulting exact characterizations do not appear in the earlier literature on these rules. The work is useful because it supplies a reusable way to compare proportionality definitions without having to invent new ones from scratch each time. The implication that PJR+ is the weakest notion satisfying the listed conditions is clean and directly addresses the question of why one might adopt it over prior alternatives. The proofs follow the standard pattern of showing both directions of the characterization, and the definitions are stated without circularity or hidden parameters. One limitation is that the normative status of the axioms is asserted rather than derived from first principles or tested against concrete election data. Different applications may reasonably disagree on whether monotonicity should be required, which limits how far the “canonical” claim travels outside the axiomatic setting. The paper is written for people already working in computational social choice and multiwinner voting. A reader who wants to understand the design space of proportionality axioms will find the distinctions clear and the framework reusable. It is technically grounded enough to deserve a serious referee, even if some readers will want to debate the choice of axioms. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The paper introduces axioms for 'good' proportionality notions in approval-based multiwinner voting and provides exact axiomatic characterizations of PJR+ and EJR+. It proceeds in two parts: (i) axioms (including monotonicity as the distinguisher from PJR/EJR) such that any satisfying notion must refine PJR+ (resp. EJR+), and (ii) a witness-based framework in which PJR+ and EJR+ are maximal under the axioms. A highlighted result is that any notion satisfying monotonicity, independence of losers, robustness to fully satisfied voters, and lower quota refines PJR+.

Significance. If the characterizations hold, the work supplies normative justifications for selecting among proportionality notions, establishing PJR+ as the canonical minimal requirement under mild conditions. The witness-based framework and the two-part exact characterization (refinement plus maximality) are technical strengths that clarify relationships in a crowded literature. The approach is internally consistent and relies on standard axiomatic methods without circularity or unstated assumptions.

minor comments (2)

[Introduction] The abstract and introduction could include a short concrete example showing how monotonicity separates PJR+ from PJR on a small instance.
[Section 4] Notation for witness sets in the witness-based framework (around the definitions of PJR+ and EJR+) would benefit from an additional running example to aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging review of our manuscript. We are pleased that the referee finds the axiomatic characterizations of PJR+ and EJR+ to be technically strong and normatively useful, and we appreciate the recommendation to accept.

Circularity Check

0 steps flagged

Minor self-citation to prior definition of PJR+/EJR+; central axiomatic implications remain independent

full rationale

The paper introduces fresh axioms (monotonicity, independence of losers, robustness to fully satisfied voters, lower quota) and proves that any proportionality notion satisfying them refines PJR+. This is a direct implication from the stated axioms rather than a reduction to fitted parameters, self-referential definitions, or unverified self-citations. The citation to Brill and Peters 2023 merely supplies the target notions being characterized; the load-bearing proofs (characterization of refinements and witness-based maximality) are developed and verified within the current manuscript. No step equates a derived claim to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions from social choice theory plus newly introduced axioms (monotonicity, independence of losers, robustness to fully satisfied voters, lower quota, witness-based certification). No free parameters or invented entities are used.

axioms (2)

domain assumption Monotonicity is a desirable property for proportionality notions
Used to distinguish PJR+ and EJR+ from their predecessors PJR and EJR
domain assumption Independence of losers, robustness to fully satisfied voters, and lower quota are appropriate mild conditions
Invoked to show that any notion satisfying them refines PJR+

pith-pipeline@v0.9.0 · 5589 in / 1205 out tokens · 69707 ms · 2026-05-08T17:00:49.604572+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

300 extracted references · 28 canonical work pages

[1]

Semi-random impossibilities of condorcet criterion

Lirong Xia , booktitle = proc #. Semi-random Impossibilities of Condorcet Criterion , year =. doi:10.1609/aaai.v37i5.25727 , pages =

work page doi:10.1609/aaai.v37i5.25727
[2]

A quantitative version of the gibbard--satterthwaite theorem for three alternatives

Ehud Friedgut and Gil Kalai and Nathan Keller and Noam Nissan , date-added =. A Quantitative Version of the Gibbard--Satterthwaite Theorem for Three Alternatives , volume =. doi:10.1137/090756740 , journal =

work page doi:10.1137/090756740
[3]

The Impact of a Coalition: Assessing the Likelihood of Voter Influence in Large Elections

Lirong Xia , booktitle = proc #. The Impact of a Coalition: Assessing the Likelihood of Voter Influence in Large Elections. , year =
[4]

Explaining Tournament Solutions with Minimal Supports , year =

Cl. Explaining Tournament Solutions with Minimal Supports , year =
[5]

Winner robustness via swap- and shift-bribery: Parameterized counting complexity and experiments

Niclas Boehmer and Robert Bredereck and Piotr Faliszewski and Rolf Niedermeier , booktitle = proc #. Winner Robustness via Swap- and Shift-Bribery: Parameterized Counting Complexity and Experiments. , year =. doi:10.24963/ijcai.2021/8 , pages =

work page doi:10.24963/ijcai.2021/8 2021
[6]

A calculus for computing structured justifications for election outcomes

Arthur Boixel and Ulle Endriss and Ronald de Haan , booktitle = proc #. A Calculus for Computing Structured Justifications for Election Outcomes , year =. doi:10.1609/aaai.v36i5.20414 , pages =

work page doi:10.1609/aaai.v36i5.20414
[7]

A Graph-Based Algorithm for the Automated Justification of Collective Decisions , url =

Oliviero Nardi and Arthur Boixel and Ulle Endriss , booktitle = proc #. A Graph-Based Algorithm for the Automated Justification of Collective Decisions , url =
[8]

Automated Justification of Collective Decisions via Constraint Solving

Arthur Boixel and Ulle Endriss , booktitle = proc #. Automated Justification of Collective Decisions via Constraint Solving. , url =
[9]

Explainability in mechanism design: Recent advances and the road ahead

Sharadhi Alape Suryanarayana and David Sarne and Sarit Kraus , booktitle =. Explainability in Mechanism Design: Recent Advances and the Road Ahead , year =. doi:10.1007/978-3-031-20614-6_21 , pages =

work page doi:10.1007/978-3-031-20614-6_21
[10]

On Minimal Achievable Quotas in Multiwinner Voting , year =

Patrick Becker and Fabian Frank , booktitle = proc #. On Minimal Achievable Quotas in Multiwinner Voting , year =
[11]

Understanding the Impact of Proportionality in Approval-Based Multiwinner Elections , year =

Niclas Boehmer and Lara Glessen and Jannik Peters , booktitle = proc #. Understanding the Impact of Proportionality in Approval-Based Multiwinner Elections , year =
[12]

Discrete Budget Aggregation: Truthfulness and Proportionality , year =

Ulrike Schmidt-Kraepelin and Warut Suksompong and Markus Utke , booktitle = proc #. Discrete Budget Aggregation: Truthfulness and Proportionality , year =. doi:10.24963/ijcai.2025/450 , pages =

work page doi:10.24963/ijcai.2025/450 2025
[13]

Axiomatic Characterization of Committee Scoring Rules , volume =

Piotr Skowron and Piotr Faliszewski and Arkadii Slinko , date-added =. Axiomatic Characterization of Committee Scoring Rules , volume =. 2019 , bdsk-url-1 =. doi:10.1016/j.jet.2018.12.011 , journal =

work page doi:10.1016/j.jet.2018.12.011 2019
[14]

Singles monotonicity and stability in one-to-one matching problems , volume =

Yoichi Kasajima and Manabu Toda , date-added =. Singles monotonicity and stability in one-to-one matching problems , volume =. 2024 , bdsk-url-1 =. doi:10.1016/j.geb.2023.11.001 , journal =

work page doi:10.1016/j.geb.2023.11.001 2024
[15]

An axiomatization of the core of cooperative games without side payments , volume =

Bezalel Peleg , date-added =. An axiomatization of the core of cooperative games without side payments , volume =. 1985 , bdsk-url-1 =. doi:10.1016/0304-4068(85)90020-5 , journal =

work page doi:10.1016/0304-4068(85)90020-5 1985
[16]

Consistency and characterization of the core of two-sided matching problems , volume =

Hiroo Sasaki and Manabu Toda , date-added =. Consistency and characterization of the core of two-sided matching problems , volume =. 1992 , bdsk-url-1 =. doi:10.1016/0022-0531(92)90078-V , journal =

work page doi:10.1016/0022-0531(92)90078-v 1992
[17]

An axiomatic characterization of

Florian Brandl and Felix Brandt , date-added =. An axiomatic characterization of. 2024 , bdsk-url-1 =. doi:10.3982/TE5825 , journal =

work page doi:10.3982/te5825 2024
[18]

Avoiding Overrepresentation: Upper Quota Axioms for Committee Voting , url =

Martin Lackner and Oliviero Nardi , date-added =. Avoiding Overrepresentation: Upper Quota Axioms for Committee Voting , url =. 2025 , bdsk-url-1 =

2025
[19]

Proportional representation in rank aggregation

Proportional Representation in Rank Aggregation , year =. arXiv , author =:2508.16177 , primaryclass =

work page arXiv
[20]

arXiv , author =:2502.02815 , primaryclass =

Exploring Relations among Fairness Notions in Discrete Fair Division , year =. arXiv , author =:2502.02815 , primaryclass =

work page arXiv
[21]

On the Edge of Core (Non-)emptiness: An Automated Reasoning Approach to Approval-based Multi-winner Voting

Ratip Emin Berker and Emanuel Tewolde and Vincent Conitzer and Mingyu Guo and Marijn Heule and Lirong Xia , booktitle = proc #. On the Edge of Core (Non-)emptiness: An Automated Reasoning Approach to Approval-based Multi-winner Voting. , year =
[22]

Lee and Toby Walsh , date-added =

Haris Aziz and Alexander Lam and Barton E. Lee and Toby Walsh , date-added =. Proportionality-based fairness and strategyproofness in the facility location problem , volume =. Journal of Mathematical Economics , pages =. 2025 , bdsk-file-1 =

2025
[23]

Casey and Edith Elkind , booktitle = proc #

Matthew M. Casey and Edith Elkind , booktitle = proc #. Justified Representation: From Hare to Droop , year =
[24]

Proportional Representation in Practice: Quantifying Proportionality in Ordinal Elections , year =

Tuva Bardal and Markus Brill and David McCune and Jannik Peters , date-added =. Proportional Representation in Practice: Quantifying Proportionality in Ordinal Elections , year =
[25]

Robust Committee Voting, or The Other Side of Representation , year =

Gregory Kehne and Ulrike Schmidt-Kraepelin and Krzysztof Sornat , booktitle = proc #. Robust Committee Voting, or The Other Side of Representation , year =. doi:10.1145/3736252.3742676 , pages =

work page doi:10.1145/3736252.3742676
[26]

The core in Participatory Budgeting can be empty , volume =

Jan Maly , date-added =. The core in Participatory Budgeting can be empty , volume =. Economics Letters , pages =
[27]

Barbaro and A

S. Barbaro and A. Specht , date-added =. Condorcet Method, Independence of Irrelevant Alternatives, and the Size of the Bundestag , volume =. German Politics , number =
[28]

H. W. Lenstra , date-added =. Integer programming with a fixed number of variables , volume =. Mathematics of Operations Research , pages =
[29]

D. Bil. Geometric Network Creation Games , volume =. SIAM Journal of Discrete Mathematics , number =
[30]

McTavish and B

S.Kadekodi and H. McTavish and B. Ustun , booktitle = proc #. Selective Preference Aggregation , year =
[31]

Schulze , date-added =

M. Schulze , date-added =. The Schulze Method of Voting , year =
[32]

F. E. Kızılkaya and D. Kempe , booktitle = proc #. k -ApprovalVeto: A Spectrum of Voting Rules Balancing Metric Distortion and Minority Protection , year =
[33]

Delemazure and Rupert Freeman and J

T. Delemazure and Rupert Freeman and J. Lang and Jean-Fran. Reallocating Wasted Votes in Proportional Parliamentary Elections with Thresholds , year =
[34]

Dutta and R

R. Dutta and R. Kumar and S. Borkotokey , date-added =. How to choose a compatible committee? , volume =. Public Choice , pages =. 2024 , bdsk-file-1 =

2024
[35]

Diss and C

M. Diss and C. G. Gassi and E. Kamwa , date-added =. On the price of diversity for multiwinner elections under (weakly) separable scoring rules , year =. European Journal of Operational Research , note =
[36]

Proportional Multiwinner Voting with Dynamic Candidate Sets , url =

Chris Dong and Jannik Peters , booktitle = proc #. Proportional Multiwinner Voting with Dynamic Candidate Sets , url =. 2025 , bdsk-url-1 =

2025
[37]

Procaccia , booktitle = proc #

Niclas Boehmer and Sara Fish and Ariel D. Procaccia , booktitle = proc #. Generative Social Choice: The Next Generation , url =. 2025 , bdsk-url-1 =

2025
[38]

The Core of Approval-Based Committee Elections with Few Seats , year =

Dominik Peters , booktitle = proc #. The Core of Approval-Based Committee Elections with Few Seats , year =. doi:10.24963/ijcai.2025/447 , pages =

work page doi:10.24963/ijcai.2025/447 2025
[39]

Committees and Equilibria: Multiwinner Approval Voting Through the Lens of Budgeting Games , year =

Adrian Haret and Sophie Klumper and Jan Maly and Guido Sch. Committees and Equilibria: Multiwinner Approval Voting Through the Lens of Budgeting Games , year =. doi:10.1145/3670865.3673484 , pages =

work page doi:10.1145/3670865.3673484
[40]

Reconfiguring Proportional Committees , year =

Chris Dong and Fabian Frank and Jannik Peters and Warut Suksompong , date-added =. Reconfiguring Proportional Committees , year =
[41]

Goel and M

A. Goel and M. Goyak and Kamesh Munagala , booktitle = proc #. Metric Distortion of Small-group Deliberation , year =
[42]

van Laarhoven and E

T. van Laarhoven and E. Marchiori , date-added =. Axioms for Graph Clustering Quality Functions , volume =. Journal of Machine Learning Research , pages =. 2014 , bdsk-file-1 =

2014
[43]

Cohen-Addad and V

V. Cohen-Addad and V. Kanade and F. Mallmann-Trent , booktitle = proc #. Clustering Redemption--. 2018 , bdsk-file-1 =

2018
[44]

Sack and N

D. Sack and N. Freier and A. Marx and H. J. Lietzmann and E. Blank and M. Wunder and E. Himmelsbach and S. McDonnell and V. Margreiter , date-added =. B
[45]

Revel and Niclas Boehmer and R

M. Revel and Niclas Boehmer and R. Colley and Markus Brill and Piotr Faliszewski and Edith Elkind , booktitle = proc #. Selecting Representative Bodies: An Axiomatic View , year =
[46]

A. Y. Kondratev and A. S. Nesterov , date-added =. Measuring majority power and veto power of voting rules , volume =. Public Choice , pages =. 2020 , bdsk-file-1 =

2020
[47]

Bedaywi and B

M. Bedaywi and B. Flanigan and M. Latifian and Nisarg Shah , booktitle = proc #. The Distortion of Public-Spirited Participatory Budgeting , year =
[48]

Selecting Interlacing Committees , url =

Chris Dong and Martin Bullinger and Tomasz W. Selecting Interlacing Committees , url =. 2025 , bdsk-file-1 =

2025
[49]

Nguyen and H

T. Nguyen and H. Song , date-added =. Approximate Core of Participatory Budgeting via Lindahl Equilibrium , year =
[50]

Hajiaghayi and M

M. Hajiaghayi and M. Khani and D. Panigrahi and M. Springer , booktitle = proc #. Online Algorithms for the Santa Claus Problem , year =
[51]

Wagner and R

J. Wagner and R. Meir , date-added =. Distribution Aggregation via Continuous Thiele's Rules , year =
[52]

Tideman , booktitle =

N. Tideman , booktitle =. Multiple-winner voting rules , year =
[53]

Green-Armytage and N

J. Green-Armytage and N. Tideman , date-added =. Selecting the runoff pair , volume =. Public Choice , pages =
[54]

S. J. Brams and R. F. Potthoff , date-added =. The paradox of grading systems , volume =. Public Choice , number =. 2015 , bdsk-file-1 =

2015
[55]

Proportional Clustering, the -Plurality Problem, and Metric Distortion , year =

Leon Kellerhals and Jannik Peters , date-added =. Proportional Clustering, the -Plurality Problem, and Metric Distortion , year =
[56]

Representative Ranking for Deliberation in the Public Sphere , url =

Manon Revel and Smitha Milli and Tyler Lu and Jamelle Watson-Daniels and Maximilian Nickel , booktitle = proc #. Representative Ranking for Deliberation in the Public Sphere , url =. 2025 , bdsk-file-1 =

2025
[57]

Papasotiropoulos and O

G. Papasotiropoulos and O. Skibski and Piotr Skowron and Tomasz W. Proportional Selection in Networks , year =
[58]

Full Proportional Justified Representation , url =

Yusuf Hakan Kalayci and Jiasen Liu and David Kempe , booktitle = proc #. Full Proportional Justified Representation , url =. 2025 , bdsk-file-1 =

2025
[59]

Proportional Representation in Matching Markets: Selecting Multiple Matchings under Dichotomous Preferences , volume =

Niclas Boehmer and Markus Brill and Ulrike Schmidt-Kraepelin , date-added =. Proportional Representation in Matching Markets: Selecting Multiple Matchings under Dichotomous Preferences , volume =. Social Choice and Welfare , keywords =. 2025 , bdsk-file-1 =

2025
[60]

The Squared Kemeny Rule for Averaging Rankings , year =

Patrick Lederer and Dominik Peters and Tomasz W. The Squared Kemeny Rule for Averaging Rankings , year =
[61]

Enelow and M

J. Enelow and M. Hinisch , date-added =. On Plott's pairwise symmetry condition for majority rule equilibrium , volume =. Public Choice , number =
[62]

Igarashi and N

A. Igarashi and N. Kamiyama and Warut Suksompong and S. M. Yuen , date-added =. Reachability of Fair Allocations via Sequential Exchanges , volume =. Algorithmica , number =
[63]

Obraztsova and Edith Elkind and Piotr Faliszewski , booktitle = proc #

S. Obraztsova and Edith Elkind and Piotr Faliszewski , booktitle = proc #. On swap convexity of voting rules , year =
[64]

Obraztsova and Edith Elkind and Piotr Faliszewski and Arkadii Slinko , booktitle = proc #

S. Obraztsova and Edith Elkind and Piotr Faliszewski and Arkadii Slinko , booktitle = proc #. On swap-distance geometry of voting rules , year =
[65]

Lassota and A

A. Lassota and A. Vetta and B. von Stengel , date-added =. The Condorcet Dimension of Metric Spaces , year =
[66]

Khoban , date-added =

Z. Khoban , date-added =. Interpretative Interactions: An Argument for Descriptive Representation in Deliberative Mini-publics , volume =. Representation , number =
[67]

Proportionality for ranked voting, in theory and practice , year =

Gerdus Benad. Proportionality for ranked voting, in theory and practice , year =
[68]

Kleindessner and P

M. Kleindessner and P. Awasthi and J. Morgenstern , booktitle = proc #. Fair k -Center Clustering for Data Summarization , year =
[69]

Uhde and N

A. Uhde and N. Schlicker and D. P. Wallach and M. Hassenzahl , booktitle =. Fairness and Decision-making in Collaborative Shift Scheduling Systems , year =
[70]

Siththaranjan and C

A. Siththaranjan and C. Laidlaw and D. Hadfield-Menell , booktitle =. Distributional Preference Learning:
[71]

Papasotiropoulos and S

G. Papasotiropoulos and S. Z. Pishbin and O. Skibski and Piotr Skowron and Tomasz W. Method of Equal Shares with Bounded Overspending , year =
[72]

Wolbeck , date-added =

L. Wolbeck , date-added =. Fairness Aspects in Personnel Scheduling , year =
[73]

Proportional multiwinner voting with dynamic candidate sets , year =

Chris Dong and Jannik Peters , date-added =. Proportional multiwinner voting with dynamic candidate sets , year =
[74]

What Do We Elect Committees For?

Piotr Skowron , booktitle = proc #. What Do We Elect Committees For?. 2015 , bdsk-file-1 =

2015
[75]

Austen-Smith , date-added =

D. Austen-Smith , date-added =. Sincere voting in models of legislative elections , volume =. Social Choice and Welfare , number =
[76]

Sugden , date-added =

R. Sugden , date-added =. Free Association and the Theory of Proportional Representation , volume =. American Political Science Review , keywords =. 1984 , bdsk-file-1 =

1984
[77]

Broadening the research agenda for computational social choice: Multiple preference profiles and multiple solutions , year =

Niclas Boehmer and Rolf Niedermeier , booktitle = proc #. Broadening the research agenda for computational social choice: Multiple preference profiles and multiple solutions , year =
[78]

M. M. Islam and S. Vahidi and B. Schieber and S. Promoting Fairness and Priority in k -Winners Selection Using. Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD '24) , date-added =
[79]

Aronov and M

B. Aronov and M. de Berg and J. Gudmundsson and M. Horton , date-added =. On beta -plurality points in spatial voting games , volume =. 2021 , bdsk-file-1 =

2021
[80]

Filtser and O

A. Filtser and O. Filtser , date-added =. Plurality in Spatial Voting Games with Constant , year =. Discrete & Computational Geometry , pages =

Showing first 80 references.