Abstract
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.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: diversity as captured by the Approval Chamberlin--Courant rule and individual excellence as captured by 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.
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