Justified Representation in Approval-Based Committee Voting

TL;DR

This paper introduces justified representation (JR) and extended justified representation (EJR) axioms for approval-based committee voting, analyzing their compatibility with various voting rules and exploring related computational issues.

Contribution

It defines JR and EJR axioms, proves PAV satisfies EJR while other rules do not, and examines the computational complexity of related problems.

Findings

PAV always provides EJR, unlike RAV.

JR can be satisfied by some committees, but not always by common rules.

EJR characterizes PAV within weighted PAV rules.

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 fixed-size 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.