Beyond Lower Quota: Avoiding Overrepresentation in Multi-Winner Voting

Anton Baychkov; Jan Maly; Jannik Peters; Martin Lackner; Oliviero Nardi

arxiv: 2606.19968 · v1 · pith:VN5UDAT2new · submitted 2026-06-18 · 💻 cs.GT

Beyond Lower Quota: Avoiding Overrepresentation in Multi-Winner Voting

Anton Baychkov , Martin Lackner , Jan Maly , Oliviero Nardi , Jannik Peters This is my paper

Pith reviewed 2026-06-26 15:26 UTC · model grok-4.3

classification 💻 cs.GT

keywords multi-winner votingapproval votingproportionalityoverrepresentationThiele rulesapportionmentjustifiable upper quota

0 comments

The pith

Justifiable upper quota axiom identifies Adams-AV as the unique composite Thiele rule that avoids overrepresentation.

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

The paper formulates justifiable upper quota (JUQ) to capture the requirement that no group receives more seats than its proportional share in approval-based multi-winner elections. It defines composite Thiele rules as a broad generalization of existing Thiele rules and proves that Adams-AV, which extends the classical Adams apportionment method, is the only member of this class that meets JUQ. The work also supplies a polynomial-time algorithm satisfying the axiom and introduces justified near quota to handle both under- and overrepresentation simultaneously. These results give concrete rules that respect upper bounds on representation while remaining computationally feasible.

Core claim

We formulate a strong and appealing axiom for avoiding overrepresentation, called justifiable upper quota (JUQ). We introduce a generalization of Thiele rules, composite Thiele rules, and characterize the unique rule in this class satisfying our axiom. This rule, Adams-AV, which naturally extends Adams' apportionment method, has not been studied before. Additionally, we introduce a polynomial-time rule that satisfies JUQ and justified near quota, an axiom that balances avoiding under- and overrepresentation.

What carries the argument

Composite Thiele rules, a generalization of Thiele rules whose scoring vectors can be composed across different numbers of approved candidates, with the JUQ axiom serving as the selector that isolates Adams-AV.

If this is right

Adams-AV produces committees in which no group exceeds its justifiable upper quota.
The polynomial-time rule offers an efficient alternative that still meets JUQ without requiring the full composite Thiele structure.
Justified near quota characterizes the unique Thiele rule that simultaneously respects both lower and upper quota bounds.
The new axioms are compatible with existing notions such as EJR+ in some cases.

Where Pith is reading between the lines

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

Rules outside the composite Thiele class may or may not satisfy JUQ, opening the question of whether stronger uniqueness results exist beyond this family.
The poly-time rule could serve as a practical default when computational speed matters more than the exact Thiele scoring structure.
The extension of classical apportionment methods suggests that similar upper-quota ideas could transfer to other proportional representation settings such as party-list systems.

Load-bearing premise

The uniqueness result holds only when attention is restricted to composite Thiele rules.

What would settle it

A concrete approval profile in which some other composite Thiele rule also satisfies JUQ, or an instance where Adams-AV itself produces a committee that violates JUQ.

Figures

Figures reproduced from arXiv: 2606.19968 by Anton Baychkov, Jan Maly, Jannik Peters, Martin Lackner, Oliviero Nardi.

**Figure 1.** Figure 1: Two groups with the same number of voters. The main goal of our paper is to look at the other side of the coin, which has not received as much attention: overrepresentation. In short, this happens when a group of voters receives a fraction of the winning committee that is much larger than what its size would warrant. To illustrate this, consider the example shown in [PITH_FULL_IMAGE:figures/full_fig_p00… view at source ↗

read the original abstract

Recently, in the social choice literature, much attention has been given to the question of avoiding underrepresentation in approval-based multi-winner voting. In this paper, we explore the largely overlooked complementary question of avoiding overrepresentation. This has not been explored systematically, despite being a desirable property with concrete applications. Intuitively, overrepresentation happens when a group determines a disproportionately large part of the committee, thereby exceeding the group's quota. We formulate a strong and appealing axiom for avoiding overrepresentation, called justifiable upper quota (JUQ). We introduce a generalization of Thiele rules, composite Thiele rules, and characterize the unique rule in this class satisfying our axiom. This rule, Adams-AV, which naturally extends Adams' apportionment method, has not been studied before. Additionally, we introduce a polynomial-time rule that satisfies JUQ. Furthermore, we introduce justified near quota, an axiom that balances avoiding under- and overrepresentation. It characterizes the unique Thiele rule extending the Sainte-Lagu\"e apportionment method. Finally, we analyze the compatibility of our axioms with established proportionality notions such as EJR+.

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

Adams-AV is the unique composite Thiele rule satisfying the new JUQ axiom, with a near-quota variant also characterized.

read the letter

Adams-AV is the unique composite Thiele rule satisfying the new JUQ axiom for avoiding overrepresentation. The paper also pins down a Sainte-Laguë extension via justified near quota.

The work fills a clear gap by formalizing an upper-quota condition that complements the heavy focus on lower quotas and EJR-style axioms. Defining the composite Thiele class is a clean generalization, and linking the rules to Adams and Sainte-Laguë apportionment methods is a sensible move. The EJR+ compatibility checks and the mention of a polynomial-time rule add practical value.

The characterization is explicitly scoped to the composite Thiele class, so it does not overclaim uniqueness across all possible rules. That keeps the result self-contained. The abstract gives no proof sketches, which leaves the derivations unverified from the summary alone, but the stated claims show no internal contradictions or circularity.

This is for social choice researchers working on axiomatic properties of approval-based multi-winner rules. Readers who care about proportionality axioms and characterizations will find the new definitions and results useful. It deserves peer review because the scope is honest and the topic addresses a real, previously understudied side of the problem.

Referee Report

0 major / 2 minor

Summary. The paper introduces the justifiable upper quota (JUQ) axiom for avoiding overrepresentation in approval-based multi-winner voting. It defines composite Thiele rules as a generalization of Thiele rules, characterizes Adams-AV (an extension of Adams' apportionment method) as the unique rule in this class satisfying JUQ, presents a polynomial-time rule satisfying JUQ, introduces the justified near quota axiom characterizing the unique Thiele rule extending Sainte-Laguë, and analyzes compatibility of these axioms with EJR+.

Significance. If the characterizations hold, the paper contributes a systematic treatment of overrepresentation, a previously under-explored complement to lower-quota concerns in multi-winner voting. The introduction of composite Thiele rules and the explicit uniqueness result for Adams-AV within this class, together with the new justified near quota axiom, provide precise tools for selecting proportional rules. The work builds directly on Thiele and apportionment literature without circularity and offers concrete algorithmic and axiomatic advances.

minor comments (2)

[§3] §3 (definition of composite Thiele rules): an explicit small example showing how a composite rule differs from a standard Thiele rule would improve readability of the generalization.
The polynomial-time rule satisfying JUQ is mentioned in the abstract and introduction but lacks a name or forward reference to its formal definition; adding one would aid navigation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our paper, the recognition of its contributions to the study of overrepresentation in approval-based multi-winner voting, and the recommendation of minor revision. The referee's assessment accurately reflects the introduction of the JUQ axiom, the composite Thiele rules framework, the uniqueness characterization of Adams-AV, the polynomial-time rule, the justified near quota axiom, and the compatibility analysis with EJR+.

Circularity Check

0 steps flagged

No significant circularity; uniqueness result is self-contained within introduced class

full rationale

The paper introduces the class of composite Thiele rules and then proves a characterization theorem identifying the unique member of that class satisfying the new JUQ axiom as Adams-AV. This is a standard mathematical uniqueness result inside an explicitly delimited family; it does not reduce the target rule or axiom to fitted parameters, prior self-citations, or definitional equivalence. Background citations to Thiele methods and apportionment are treated as external context rather than load-bearing premises for the central claim. No equations or steps in the provided abstract or reader summary exhibit the forbidden patterns of self-definition, fitted-input prediction, or imported uniqueness.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical properties of voting rules and apportionment methods plus the domain assumption that attention is restricted to composite Thiele rules. No free parameters or invented entities are introduced.

axioms (2)

domain assumption Composite Thiele rules form the appropriate class for uniqueness characterization of JUQ
The uniqueness result is stated only inside this class; the paper does not justify why other rules are excluded.
standard math Standard properties of Thiele rules and apportionment methods carry over to the composite generalization
Invoked when extending Adams and Sainte-Laguë methods.

pith-pipeline@v0.9.1-grok · 5735 in / 1304 out tokens · 21328 ms · 2026-06-26T15:26:08.118369+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 1 canonical work pages

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Justified Representation in Approval-Based Committee Voting.Social Choice and Welfare48, 2 (2017), 461–485. M. Balinski and H. P. Young. 2001.Fair Representation: Meeting the Ideal of One Man, One Vote(2nd ed.). Brookings Institution Press. N. Boehmer, M. Brill, A. Cevallos, J. Gehrlein, L. Sánchez-Fernández, and U. Schmidt-Kraepelin

2017
[2]

InProceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI)

Approval-Based Committee Voting in Practice: A Case Study of (Over-)Representation in the Polkadot Blockchain. InProceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI). 9519–9527. F. Brandt, V. Conitzer, U. Endriss, J. Lang, and A. Procaccia (Eds.). 2016.Handbook of Computational Social Choice. Cambridge University Press. M. Brill, S. F...

2016
[3]

Phragmén’s Voting Methods and Justified Representation.Mathematical Programming203, 1–2 (2024), 47–76. M. Brill, J. Israel, E. Micha, and J. Peters

2024
[4]

Individual Representation in Approval-Based Committee Voting.Social Choice and Welfare64, 1–2 (2025), 69–96. M. Brill, J.-F. Laslier, and P. Skowron

2025
[5]

Multiwinner Approval Rules as Apportionment Methods.Journal of Theoretical Politics30, 3 (2018), 358–382. M. Brill and J. Peters

2018
[6]

Full version arXiv:2302.01989 [cs.GT]. M. Brill and J. Peters

arXiv
[7]

InProceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

New Combinatorial Insights for Monotone Apportionment. InProceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 1308–1328. A. Cevallos and A. Stewart

2025
[8]

arXiv:1611.08826 [math.HO] https://arxiv.org/abs/1611.08826 C

Phragmén’s and Thiele’s Election Methods. arXiv:1611.08826 [math.HO] https://arxiv.org/abs/1611.08826 C. Jerrett and E. Anshelevich

Pith/arXiv arXiv
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arXiv:2510.21039 [cs.GT] https://arxiv.org/abs/2510.21039 Y

Low Cost, Fair, and Representative Committees in a Metric Space. arXiv:2510.21039 [cs.GT] https://arxiv.org/abs/2510.21039 Y. H. Kalayci, D. Kempe, and J. Liu

arXiv
[10]

Presented at the 16th Multidisciplinary Workshop on Advances in Preference Handling (M-PREF 2025)

Avoiding Overrepresentation: Upper Quota Axioms for Committee Voting. Presented at the 16th Multidisciplinary Workshop on Advances in Preference Handling (M-PREF 2025). M. Lackner, P. Regner, and B. Krenn

2025
[11]

Journal of Open Source Software8, 81 (2023),

abcvoting: A Python package for approval-based multi-winner voting rules. Journal of Open Source Software8, 81 (2023),

2023
[12]

Lackner and P

https://doi.org/10.21105/joss.04880 M. Lackner and P. Skowron

work page doi:10.21105/joss.04880
[13]

Artificial Intelligence288 (2020), 103366

Utilitarian Welfare and Representation Guarantees of Approval-Based Multiwinner Rules. Artificial Intelligence288 (2020), 103366. M. Lackner and P. Skowron

2020
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Consistent Approval-Based Multi-Winner Rules.Journal of Economic Theory192 (2021), 105173. M. Lackner and P. Skowron. 2023.Multi-Winner Voting with Approval Preferences. Springer. Anton Baychkov, Martin Lackner, Jan Maly, Oliviero Nardi, and Jannik Peters20 J. P. Mayberry

2021
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Quota Methods for Congressional Apportionment are Still Non-Unique.Proceedings of the National Academy of Sciences (PNAS)75, 8 (1978), 3537–3539. D. Peters, G. Pierczyński, and P. Skowron

1978
[16]

InProceedings of the 21st ACM Conference on Economics and Computation (ACM-EC)

Proportionality and the Limits of Welfarism. InProceedings of the 21st ACM Conference on Economics and Computation (ACM-EC). ACM Press, 793–794. Full version arXiv:1911.11747 [cs.GT]. A. D. Procaccia, J. S. Rosenschein, and A. Zohar

arXiv 1911
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On the Complexity of Achieving Proportional Representation.Social Choice and Welfare30 (2008), 353–362. F. Pukelsheim. 2014.Proportional Representation: Apportionment Methods and Their Applications. Springer. J. Rawls. 1971.A Theory of Justice. Harvard University Press. L. Sánchez-Fernández, E. Elkind, M. Lackner, N. Fernández, J. A. Fisteus, P. Basanta V...

2008
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The Maximin Support Method: An Extension of the D’Hondt Method to Approval-Based Multiwinner Elections.Mathematical Programming203, 1–2 (2024), 107–134. A. K. Sen. 2017.Collective Choice and Social Welfare(expanded ed.). Penguin. P. K. Skowron, P. Faliszewski, and J. Lang

2024
[19]

Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation.Artificial Intelligence241 (2016), 191–216. T. N. Thiele

2016
[20]

Social Choice Scoring Functions.SIAM J. Appl. Math.28, 4 (1975), 824–838. Anton Baychkov, Martin Lackner, Jan Maly, Oliviero Nardi, and Jannik Peters21 A Omitted Proofs and Definitions Observation 1.For apportionment instances, EJR+ is equivalent to apportionment lower quota. Proof. Let (𝑨, 𝑘) be an apportionment instance with parties𝑃1, . . . , 𝑃𝑡, and l...

1975
[21]

From the remaining candidates, in order to satisfy EJR+16 we must select: • At least 3 candidates from each of {𝑐1,

In order to obtain 𝑊𝑗 ∗, UQER removes ˆ𝑐1 − ˆ𝑐6, as these are the UQ-violating candidates with fewest supporters, and terminates. From the remaining candidates, in order to satisfy EJR+16 we must select: • At least 3 candidates from each of {𝑐1, . . . , 𝑐6}, {𝑐7, . . . , 𝑐12} and {𝑐13, . . . , 𝑐18}. This holds as each group of three voters among the first...

2020
[22]

Otherwise, we compute 𝛼(𝑐)for each𝑐∈𝐶 𝑟 as 𝛼(𝑐)=min    𝛼∈R| ∑︁ 𝑖∈𝑁(𝑐) min(𝛼, 𝑏 𝑟 (𝑖))=1   

If 𝐶𝑟 is empty, the rule terminates and returns𝑊𝑟 . Otherwise, we compute 𝛼(𝑐)for each𝑐∈𝐶 𝑟 as 𝛼(𝑐)=min    𝛼∈R| ∑︁ 𝑖∈𝑁(𝑐) min(𝛼, 𝑏 𝑟 (𝑖))=1    . 16Or even the weaker proportionality notion of EJR. Anton Baychkov, Martin Lackner, Jan Maly, Oliviero Nardi, and Jannik Peters27 We select a candidate 𝑐∗ with minimal 𝛼(𝑐 ∗) and set 𝑏𝑟+1 (𝑖)=max( 0...

2023
[23]

B.3 Relationships with Other Proportionality Notions We define some additional proportionality notions for multi-winner voting

Instance with𝑘=𝑛=15. B.3 Relationships with Other Proportionality Notions We define some additional proportionality notions for multi-winner voting. Definition 12 ([Peters et al ., 2021]).Given an integer ℓ∈N >0 and a set of candidates 𝐶 ′ ⊆𝐶 , a group of voters 𝑁 ′ ⊆𝑁 is weakly (ℓ, 𝐶 ′)-cohesive if |𝑁 ′ | ≥ |𝐶 ′ | · 𝑛/𝑘 and if for all 𝑖∈𝑁 ′ it holds that...

2021
[24]

C Counterexamples for Other Rules Satisfying JUQ or JNQ In this section, we present several counterexamples for familiar multi-winner voting rules for JUQ and JNQ

Example instance with𝑘=3. C Counterexamples for Other Rules Satisfying JUQ or JNQ In this section, we present several counterexamples for familiar multi-winner voting rules for JUQ and JNQ. We tested all the rules present in the abcvoting Python package [Lackner et al., 2023] at the time of writing, as well as some additional rules of interest. For the sa...

2023
[25]

[2023]) can select{𝑎, 𝑐}in this instance

(in the formulation given by Brill et al. [2023]) can select{𝑎, 𝑐}in this instance. C.2 JNQ Example 1𝑚=7,𝑛=5,𝑘=3. 𝐴1 ={𝑏, 𝑒, 𝑓}, 𝐴2 ={𝑓}, 𝐴3 ={𝑎}, 𝐴4 ={𝑎, 𝑏, 𝑐, 𝑔}, 𝐴5 ={𝑔}. In this instance {𝑎, 𝑏, 𝑓} does not satisfy JNQ. We can swap𝑏 with 𝑔. The quota of 𝑁(𝑏) is 6/5, while the quota of voter5is 3/5. The rules AV, SAV, CC, LEXCC, GEOM2, SEQPAV, REVSEQPAV...

2023

[1] [1]

Justified Representation in Approval-Based Committee Voting.Social Choice and Welfare48, 2 (2017), 461–485. M. Balinski and H. P. Young. 2001.Fair Representation: Meeting the Ideal of One Man, One Vote(2nd ed.). Brookings Institution Press. N. Boehmer, M. Brill, A. Cevallos, J. Gehrlein, L. Sánchez-Fernández, and U. Schmidt-Kraepelin

2017

[2] [2]

InProceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI)

Approval-Based Committee Voting in Practice: A Case Study of (Over-)Representation in the Polkadot Blockchain. InProceedings of the 38th AAAI Conference on Artificial Intelligence (AAAI). 9519–9527. F. Brandt, V. Conitzer, U. Endriss, J. Lang, and A. Procaccia (Eds.). 2016.Handbook of Computational Social Choice. Cambridge University Press. M. Brill, S. F...

2016

[3] [3]

Phragmén’s Voting Methods and Justified Representation.Mathematical Programming203, 1–2 (2024), 47–76. M. Brill, J. Israel, E. Micha, and J. Peters

2024

[4] [4]

Individual Representation in Approval-Based Committee Voting.Social Choice and Welfare64, 1–2 (2025), 69–96. M. Brill, J.-F. Laslier, and P. Skowron

2025

[5] [5]

Multiwinner Approval Rules as Apportionment Methods.Journal of Theoretical Politics30, 3 (2018), 358–382. M. Brill and J. Peters

2018

[6] [6]

Full version arXiv:2302.01989 [cs.GT]. M. Brill and J. Peters

arXiv

[7] [7]

InProceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

New Combinatorial Insights for Monotone Apportionment. InProceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 1308–1328. A. Cevallos and A. Stewart

2025

[8] [8]

arXiv:1611.08826 [math.HO] https://arxiv.org/abs/1611.08826 C

Phragmén’s and Thiele’s Election Methods. arXiv:1611.08826 [math.HO] https://arxiv.org/abs/1611.08826 C. Jerrett and E. Anshelevich

Pith/arXiv arXiv

[9] [9]

arXiv:2510.21039 [cs.GT] https://arxiv.org/abs/2510.21039 Y

Low Cost, Fair, and Representative Committees in a Metric Space. arXiv:2510.21039 [cs.GT] https://arxiv.org/abs/2510.21039 Y. H. Kalayci, D. Kempe, and J. Liu

arXiv

[10] [10]

Presented at the 16th Multidisciplinary Workshop on Advances in Preference Handling (M-PREF 2025)

Avoiding Overrepresentation: Upper Quota Axioms for Committee Voting. Presented at the 16th Multidisciplinary Workshop on Advances in Preference Handling (M-PREF 2025). M. Lackner, P. Regner, and B. Krenn

2025

[11] [11]

Journal of Open Source Software8, 81 (2023),

abcvoting: A Python package for approval-based multi-winner voting rules. Journal of Open Source Software8, 81 (2023),

2023

[12] [12]

Lackner and P

https://doi.org/10.21105/joss.04880 M. Lackner and P. Skowron

work page doi:10.21105/joss.04880

[13] [13]

Artificial Intelligence288 (2020), 103366

Utilitarian Welfare and Representation Guarantees of Approval-Based Multiwinner Rules. Artificial Intelligence288 (2020), 103366. M. Lackner and P. Skowron

2020

[14] [14]

Consistent Approval-Based Multi-Winner Rules.Journal of Economic Theory192 (2021), 105173. M. Lackner and P. Skowron. 2023.Multi-Winner Voting with Approval Preferences. Springer. Anton Baychkov, Martin Lackner, Jan Maly, Oliviero Nardi, and Jannik Peters20 J. P. Mayberry

2021

[15] [15]

Quota Methods for Congressional Apportionment are Still Non-Unique.Proceedings of the National Academy of Sciences (PNAS)75, 8 (1978), 3537–3539. D. Peters, G. Pierczyński, and P. Skowron

1978

[16] [16]

InProceedings of the 21st ACM Conference on Economics and Computation (ACM-EC)

Proportionality and the Limits of Welfarism. InProceedings of the 21st ACM Conference on Economics and Computation (ACM-EC). ACM Press, 793–794. Full version arXiv:1911.11747 [cs.GT]. A. D. Procaccia, J. S. Rosenschein, and A. Zohar

arXiv 1911

[17] [17]

On the Complexity of Achieving Proportional Representation.Social Choice and Welfare30 (2008), 353–362. F. Pukelsheim. 2014.Proportional Representation: Apportionment Methods and Their Applications. Springer. J. Rawls. 1971.A Theory of Justice. Harvard University Press. L. Sánchez-Fernández, E. Elkind, M. Lackner, N. Fernández, J. A. Fisteus, P. Basanta V...

2008

[18] [18]

The Maximin Support Method: An Extension of the D’Hondt Method to Approval-Based Multiwinner Elections.Mathematical Programming203, 1–2 (2024), 107–134. A. K. Sen. 2017.Collective Choice and Social Welfare(expanded ed.). Penguin. P. K. Skowron, P. Faliszewski, and J. Lang

2024

[19] [19]

Finding a Collective Set of Items: From Proportional Multirepresentation to Group Recommendation.Artificial Intelligence241 (2016), 191–216. T. N. Thiele

2016

[20] [20]

Social Choice Scoring Functions.SIAM J. Appl. Math.28, 4 (1975), 824–838. Anton Baychkov, Martin Lackner, Jan Maly, Oliviero Nardi, and Jannik Peters21 A Omitted Proofs and Definitions Observation 1.For apportionment instances, EJR+ is equivalent to apportionment lower quota. Proof. Let (𝑨, 𝑘) be an apportionment instance with parties𝑃1, . . . , 𝑃𝑡, and l...

1975

[21] [21]

From the remaining candidates, in order to satisfy EJR+16 we must select: • At least 3 candidates from each of {𝑐1,

In order to obtain 𝑊𝑗 ∗, UQER removes ˆ𝑐1 − ˆ𝑐6, as these are the UQ-violating candidates with fewest supporters, and terminates. From the remaining candidates, in order to satisfy EJR+16 we must select: • At least 3 candidates from each of {𝑐1, . . . , 𝑐6}, {𝑐7, . . . , 𝑐12} and {𝑐13, . . . , 𝑐18}. This holds as each group of three voters among the first...

2020

[22] [22]

Otherwise, we compute 𝛼(𝑐)for each𝑐∈𝐶 𝑟 as 𝛼(𝑐)=min    𝛼∈R| ∑︁ 𝑖∈𝑁(𝑐) min(𝛼, 𝑏 𝑟 (𝑖))=1   

If 𝐶𝑟 is empty, the rule terminates and returns𝑊𝑟 . Otherwise, we compute 𝛼(𝑐)for each𝑐∈𝐶 𝑟 as 𝛼(𝑐)=min    𝛼∈R| ∑︁ 𝑖∈𝑁(𝑐) min(𝛼, 𝑏 𝑟 (𝑖))=1    . 16Or even the weaker proportionality notion of EJR. Anton Baychkov, Martin Lackner, Jan Maly, Oliviero Nardi, and Jannik Peters27 We select a candidate 𝑐∗ with minimal 𝛼(𝑐 ∗) and set 𝑏𝑟+1 (𝑖)=max( 0...

2023

[23] [23]

B.3 Relationships with Other Proportionality Notions We define some additional proportionality notions for multi-winner voting

Instance with𝑘=𝑛=15. B.3 Relationships with Other Proportionality Notions We define some additional proportionality notions for multi-winner voting. Definition 12 ([Peters et al ., 2021]).Given an integer ℓ∈N >0 and a set of candidates 𝐶 ′ ⊆𝐶 , a group of voters 𝑁 ′ ⊆𝑁 is weakly (ℓ, 𝐶 ′)-cohesive if |𝑁 ′ | ≥ |𝐶 ′ | · 𝑛/𝑘 and if for all 𝑖∈𝑁 ′ it holds that...

2021

[24] [24]

C Counterexamples for Other Rules Satisfying JUQ or JNQ In this section, we present several counterexamples for familiar multi-winner voting rules for JUQ and JNQ

Example instance with𝑘=3. C Counterexamples for Other Rules Satisfying JUQ or JNQ In this section, we present several counterexamples for familiar multi-winner voting rules for JUQ and JNQ. We tested all the rules present in the abcvoting Python package [Lackner et al., 2023] at the time of writing, as well as some additional rules of interest. For the sa...

2023

[25] [25]

[2023]) can select{𝑎, 𝑐}in this instance

(in the formulation given by Brill et al. [2023]) can select{𝑎, 𝑐}in this instance. C.2 JNQ Example 1𝑚=7,𝑛=5,𝑘=3. 𝐴1 ={𝑏, 𝑒, 𝑓}, 𝐴2 ={𝑓}, 𝐴3 ={𝑎}, 𝐴4 ={𝑎, 𝑏, 𝑐, 𝑔}, 𝐴5 ={𝑔}. In this instance {𝑎, 𝑏, 𝑓} does not satisfy JNQ. We can swap𝑏 with 𝑔. The quota of 𝑁(𝑏) is 6/5, while the quota of voter5is 3/5. The rules AV, SAV, CC, LEXCC, GEOM2, SEQPAV, REVSEQPAV...

2023