A Quantitative Analysis of Multi-Winner Rules

To choose a multi-winner rule, i.e., a voting rule that selects a subset of k alternatives based on preferences of a certain population, is a hard and ambiguous task. Depending on the context, it varies widely what constitutes an “optimal” committee. In this paper, we offer a new perspective to measure the quality of committees and—consequently—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: diversity as captured by the (Approval) Chamberlin–Courant rule (CC) and individual excellence as captured by Approval Voting (AV). With both theoretical and experimental methods we establish a classification of multi-winner rules in terms of their quantitative alignment with these two opposing objectives.