Bounded Single-Peaked Width and Proportional Representation

This paper is devoted to the proportional representation (PR) problem when the preferences are clustered single-peaked. PR is a "multi-winner" election problem, that we study in Chamberlin and Courant's scheme [6]. We define clustered single-peakedness as a form of single-peakedness with respect to clusters of candidates, i.e. subsets of candidates that are consecutive (in arbitrary order) in the preferences of all voters. We show that the PR problem becomes polynomial when the size of the largest cluster of candidates (width) is bounded. Furthermore, we establish the polynomiality of determining the single-peaked width of a preference profile (minimum width for a partition of candidates into clusters compatible with clustered single-peakedness) when the preferences are narcissistic (i.e., every candidate is the most preferred one for some voter).