Phragmén’s Voting Methods and Justified Representation

Brill, Markus; Freeman, Rupert; Janson, Svante; Lackner, Martin

doi:10.1609/aaai.v31i1.10598

Cited by 31 publications

(43 citation statements)

References 11 publications

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“…This voting rule can be seen as an analogue of the Black rule, which is a single-winner rule that outputs a Condorcet winner if one exists and a Borda winner otherwise. Very recently, Brill et al (2017) identified a voting rule that provides both PR and PJR for all values of n and k, namely, a maximization version of the Phragmén's rule, which they refer to as max-Phragmen.…”

Section: Discussionmentioning

confidence: 99%

“…It is then natural to ask if there is a polynomial-time computable voting rule that satisfies PJR for all values of n and k. Interestingly, it turns out that the answer to this question is 'yes': in a very recent paper, Brill et al (2017) describe an approval-based multi-winner rule developed by the Swedish mathematician Lars Edvard Phragmén more than 100 years ago, and show that a sequential variant of this rule, which they refer to as seq-Phragmen, is polynomial-time computable and provides PJR. Another voting rule with this combination of properties is the ODH rule, which has been proposed by Sánchez-Fernández, Fernández, and Fisteus in a recent arXiv preprint .…”

Section: Theorem 5 Consider a Ballot Profilementioning

confidence: 99%

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Proportional Justified Representation

Sánchez-Fernandez

Elkind

Lackner

et al. 2017

AAAI

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The goal of multi-winner elections is to choose a fixed-size committee based on voters’ preferences. An important concern in this setting is representation: large groups of voters with cohesive preferences should be adequately represented by the election winners. Recently, Aziz et al. proposed two axioms that aim to capture this idea: justified representation (JR) and its strengthening extended justified representation (EJR). In this paper, we extend the work of Aziz et al. in several directions. First, we answer an open question of Aziz et al., by showing that Reweighted Approval Voting satisfies JR for k = 3; 4; 5, but fails it for k >= 6. Second, we observe that EJR is incompatible with the Perfect Representation criterion, which is important for many applications of multi-winner voting, and propose a relaxation of EJR, which we call Proportional Justified Representation (PJR). PJR is more demanding than JR, but, unlike EJR, it is compatible with perfect representation, and a committee that provides PJR can be computed in polynomial time if the committee size divides the number of voters. Moreover, just like EJR, PJR can be used to characterize the classic PAV rule in the class of weighted PAV rules. On the other hand, we show that EJR provides stronger guarantees with respect to average voter satisfaction than PJR does.

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Section: Discussionmentioning

confidence: 99%

Section: Theorem 5 Consider a Ballot Profilementioning

confidence: 99%

Proportional Justified Representation

Sánchez-Fernandez

Elkind

Lackner

et al. 2017

AAAI

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“…'s Rules In the late 19th century, Swedish mathematician Lars Edvard Phragmén proposed a load-balancing approach for selecting committees (Phragmén 1894; Janson 2016). Here, we formulate two particularly interesting variants (see Brill et al 2017).…”

Section: Approval-based Multiwinner Election Rulesmentioning

confidence: 99%

Multiwinner Approval Rules as Apportionment Methods

Brill

Laslier

Skowron

2017

AAAI

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We establish a link between multiwinner elections and apportionment problems by showing how approval-based multiwinner election rules can be interpreted as methods of apportionment. We consider several multi-winner rules and observe that some, but not all, of them induce apportionment methods that are well established in the literature and in the actual practice of proportional representation. For instance, we show that Proportional Approval Voting induces the D'Hondt method and that Monroe's rule induces the largest remainder method. We also consider properties of apportionment methods and exhibit multiwinner rules that induce apportionment methods satisfying these properties.

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“…This model captures a number of applications: the candidates could be potential members of a governing body, items to be shown on a seller's homepage, or tunes to be played at a wedding. Accordingly, there is a number of natural voting rules that take approval ballots as their input and output a set of committees that are tied for winning (Kilgour 2010;Brams and Fishburn 2007;LeGrand, Markakis, and Mehta 2007;Aziz et al 2015;Skowron, Faliszewski, and Lang 2016;Sánchez-Fernández, Fernández, and Fisteus 2016;Brill et al 2017). Many of these voting rules attempt to ensure that all groups of voters are fairly represented in the selected committee.…”

Section: Introductionmentioning

confidence: 99%

“…argued that two well-studied approval-based committee selection rules satisfy PJR when the target committee size k divides the number of voters n; one of these rules is polynomial-time computable. Other authors identified two polynomial-time computable rules that satisfy PJR for all values of k and n (Brill et al 2017;Sánchez-Fernández, Fernández, and Fisteus 2016). However, the complexity of checking whether a given committee provides PJR remained open.…”

Section: Introductionmentioning

confidence: 99%

On the Complexity of Extended and Proportional Justified Representation

Aziz

Elkind

Huang

et al. 2018

AAAI

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We consider the problem of selecting a fixed-size committee based on approval ballots. It is desirable to have a committee in which all voters are fairly represented. Aziz et al. (2015a; 2017) proposed an axiom called extended justified representation (EJR), which aims to capture this intuition; subsequently, Sanchez-Fernandez et al. (2017) proposed a weaker variant of this axiom called proportional justified representation (PJR). It was shown that it is coNP-complete to check whether a given committee provides EJR, and it was conjectured that it is hard to find a committee that provides EJR. In contrast, there are polynomial-time computable voting rules that output committees providing PJR, but the complexity of checking whether a given committee provides PJR was an open problem. In this paper, we answer open questions from prior work by showing that EJR and PJR have the same worst-case complexity: we provide two polynomial-time algorithms that output committees providing EJR, yet we show that it is coNP-complete to decide whether a given committee provides PJR. We complement the latter result by fixed-parameter tractability results.

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Phragmén’s Voting Methods and Justified Representation

Cited by 31 publications

References 11 publications

Proportional Justified Representation

Proportional Justified Representation

Multiwinner Approval Rules as Apportionment Methods

On the Complexity of Extended and Proportional Justified Representation

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