Justified representation in approval-based committee voting

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 (\(\mathrm {JR}\)). 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 \(\mathrm {JR}\). However, it turns out that several prominent approval-based voting rules may fail to output such a committee. In particular, while Proportional Approval Voting (\(\mathrm {PAV}\)) always outputs a committee that provides \(\mathrm {JR}\), Sequential Proportional Approval Voting (\(\mathrm {SeqPAV}\)), which is a tractable approximation to \(\mathrm {PAV}\), does not have this property. We then introduce a stronger version of the \(\mathrm {JR}\) axiom, which we call extended justified representation (\(\mathrm {EJR}\)), and show that \(\mathrm {PAV}\) satisfies \(\mathrm {EJR}\), while other rules we consider do not; indeed, \(\mathrm {EJR}\) can be used to characterize \(\mathrm {PAV}\) within the class of weighted \(\mathrm {PAV}\) rules. We also consider several other questions related to \(\mathrm {JR}\) and \(\mathrm {EJR}\), including the relationship between \(\mathrm {JR}\)/\(\mathrm {EJR}\) and core stability, and the complexity of the associated computational problems.