Justified Representation in Approval-Based Committee Voting

We consider approval-based committee voting in which agents approve a subset of candidates and based on the approvals, a set of winners is selected. We propose a natural axiom called justified representation that is a strengthening of unanimity. The axiom ensures that a group of agents that is large enough should have at least one approved candidate in the winning set. The concept has some better computational, fairness, and proportionality properties than some of the previous approaches for proportional voting using approval ballots. We show that some standard approval-based multi-winner rules --- AV (approval voting), SAV (satisfaction approval voting) and MAV (minimax approval voting) --- do not satisfy justified representation. We suggest natural modifications of MAV as well as PAV (proportional approval voting) that satisfy justified representation. We then present a simple linear-time algorithm that satisfies justified representation and constitutes a reasonable approval-based rule in its own right. Finally, we show how justified representation can be used to formulate other attractive approval-based multi-winner rules.