OWA-Based Extensions of the Chamberlin–Courant Rule

Abstract

Given a set of voters V, a set of candidates C, and voters’ preferences over the candidates, multiwinner voting rules output a fixed-size subset of candidates (committee). Under the Chamberlin–Courant multiwinner voting rule, one fixes a scoring vector of length |C|, and each voter’s ‘utility’ for a given committee is defined to be the score that she assigns to her most preferred candidate in that committee; the goal is then to find a committee that maximizes the joint utility of all voters. The joint utility is typically identified either with the sum of all voters’ utilities or with the utility of the least satisfied voter, resulting in, respectively, the utilitarian and the egalitarian variant of the Chamberlin–Courant’s rule. For both of these cases, the problem of computing an optimal committee is NP-hard for general preferences, but becomes polynomial-time solvable if voters’ preferences are single-peaked or single-crossing. In this paper, we propose a family of multiwinner voting rules that are based on the concept of ordered weighted average (OWA) and smoothly interpolate between the egalitarian and the utilitarian variants of the Chamberlin–Courant rule. We show that under moderate constraints on the weight vector we can recover many of the algorithmic results known for the egalitarian and the utilitarian version of Chamberlin–Courant’s rule in this more general setting.