Justified representation in approval-based committee voting

Abstract

We consider approval-based committee voting, i.e. the setting where each voter approves a subset of candidates, and these votes are then used to select a fixedsize set of winners (committee). We propose a natural axiom for this setting, which we call justified representation (JR). This axiom requires that if a large enough group of voters exhibits agreement by supporting the same candidate, then at least one voter in this group has an approved candidate in the winning committee. We show that for every list of ballots it is possible to select a committee that provides JR. However, it turns out that several prominent approval-based voting rules may fail to output such a committee. In particular, while Proportional Approval Voting (PAV) always outputs a committee that provides JR, Reweighted Approval Voting (RAV), a tractable approximation to PAV, does not have this property. We then introduce a stronger version of the JR axiom, which we call extended justified representation (EJR), and show that PAV satisfies EJR, while other rules we consider do not; indeed, EJR can be used to characterize PAV within the class of weighted PAV rules. We also consider several other questions related to JR and EJR, including the relationship between JR/EJR and core stability, and the complexity of the associated algorithmic problems.

A Representative Committee by Approval Balloting

“…Within this approach Aziz et al (2015) study an interesting property. Justified representation is a weak condition establishing that if a subset of voters of a considerable size agree on at least one candidate, the opinion of someone in this group should be taken into account.…”

“…Justified representation is a weak condition establishing that if a subset of voters of a considerable size agree on at least one candidate, the opinion of someone in this group should be taken into account. AV, SAV and PAV voting rules do not fulfil JR (see Aziz et al (2015)). Aziz et al (2015) use an algorithm (Greedy Approval Voting (GAV)) selecting a committee providing justified representation, proving in this way that such a committee always exists.…”

“…Aziz et al (2015) use an algorithm (Greedy Approval Voting (GAV)) selecting a committee providing justified representation, proving in this way that such a committee always exists. Aziz et al (2015) and Sánchez-Fernández et al (2016) analyze other, more restrictive, types of representation properties but as Sánchez-Fernández et al (2016) claim, "justified representation is a very interesting axiom, and that it must be a necessary requirement (although not a sufficient one) for considering the use of an approval-based multi-winner voting rule when it is desired that the winning set represents the different opinions or preferences of the agents involved in the election".…”

“…In recent works, Aziz et al (2015), Aziz and Walsh (2014) and Sánchez-Fernández et al (2016) study new properties related to representativeness: Justified Representation (JR), Extended Justified Representation (EJR), Proportional Justified Representation (PJR), among others. Justified Representation is the weakest of these conditions and it should be a necessary requirement when representation of the different opinions of the voters is important.…”

“…Justified Representation ensures that a large group of voters with shared preferences should have at least one representative allocated to them. Aziz et al (2015) show that most of the standard approval-based multi-winners rules do not satisfy JR and, for some of them, they propose natural modifications to overcome this lack of representation.…”