Consistent Approval-Based Multi-Winner Rules

This paper contains an axiomatic study of consistent approval-based multi-winner rules, i.e., voting rules that select a fixed-size group of candidates based on approval ballots. We introduce the class of counting rules, provide an axiomatic characterization of this class and, in particular, show that counting rules are consistent. Building upon this result, we axiomatically characterize three important consistent multi-winner rules: Proportional Approval Voting, Multi-Winner Approval Voting and Approval Chamberlin--Courant. Our results demonstrate the variety of multi-winner rules and the different, orthogonal goals that multi-winner voting rules may pursue.