M A ] 12 S ep 2 01 6 Justified Representation in Approval-Based Committee Voting

We consider approval-based committee voting, i.e. the sett ing where each voter approves a subset of candidates, and these votes are th en used to select a fixedsize set of winners (committee). We propose a natural axiom f or this setting, which we calljustified representation (JR) . This axiom requires that if a large enough group of voters exhibits agreement by supporting the same candida te, then at least one voter in this group has an approved candidate in the winning commit tee. We show that for every list of ballots it is possible to select a committee tha t providesJR. However, it turns out that several prominent approval-based voting r ules may fail to output such a committee. In particular, while Proportional Approv al Voting (PAV) always outputs a committee that provides JR, Reweighted Approval Voting ( RAV), a tractable approximation toPAV, does not have this property. We then introduce a stronger version of theJRaxiom, which we callextended justified representation (EJR) , and show thatPAV satisfiesEJR, while other rules we consider do not; indeed, EJRcan be used to characterize PAVwithin the class of weighted PAV rules. We also consider several other questions related to JR and EJR, including the relationship between JR/EJRand core stability, and the complexity of the associated alg orithmic problems.