A Quantitative Analysis of Multi-Winner Rules

To choose a suitable multi-winner voting rule is a hard and ambiguous task. Depending on the context, it varies widely what constitutes the choice of an "optimal" subset of alternatives. In this paper, we offer a new perspective on measuring the quality of such subsets and---consequently---of multi-winner rules. We provide a quantitative analysis using methods from the theory of approximation algorithms and estimate how well multi-winner rules approximate two extreme objectives: a representation criterion defined via the Approval Chamberlin--Courant rule and a utilitarian criterion defined via Multi-winner Approval Voting. With both theoretical and experimental methods we classify multi-winner rules in terms of their quantitative alignment with these two opposing objectives. Our results provide fundamental information about the nature of multi-winner rules, and in particular about the necessary tradeoffs when choosing such a rule.