Clone Structures in Voters' Preferences

TL;DR

This paper investigates the properties and structures of clone sets in voters' preferences, providing axiomatic characterizations, hierarchical insights, and algorithms for simplifying elections to single-peaked or single-crossing forms.

Contribution

It offers a novel axiomatic framework for clone structures, analyzes their hierarchy, and develops algorithms for collapsing clones to achieve specific election properties.

Findings

Clone structures have a well-defined hierarchical structure.

A polynomial-time algorithm is provided for collapsing clones in single-peaked elections.

Finding minimal clone collapses in single-crossing elections is NP-hard.

Abstract

In elections, a set of candidates ranked consecutively (though possibly in different order) by all voters is called a clone set, and its members are called clones. A clone structure is a family of all clone sets of a given election. In this paper we study properties of clone structures. In particular, we give an axiomatic characterization of clone structures, show their hierarchical structure, and analyze clone structures in single-peaked and single-crossing elections. We give a polynomial-time algorithm that finds a minimal collection of clones that need to be collapsed for an election to become single-peaked, and we show that this problem is NP-hard for single-crossing elections.