Cubic Preferences and the Character Admissibility Problem

TL;DR

This paper introduces a graph-theoretic approach to constructing voter preferences with specific interdependence structures, focusing on cubic preferences derived from hypercube symmetries, with implications for election simulation.

Contribution

It develops a novel method using Hamiltonian paths on hypercube graphs to generate cubic preferences with unique separability properties, expanding the understanding of multidimensional voter preferences.

Findings

Cubic preferences satisfy distinct set-theoretic properties.

The algebraic structure of hypercube symmetries influences preference separability.

New functions are defined to construct cubic preferences.

Abstract

In multiple-question referendum elections, the separability problem occurs when a voter's preferences on some questions or proposals depend on the predicted outcomes of others. The notion of separability formalizes the study of interdependence in multidimensional preferences, and the character admissibility problem deals with the construction of voter preferences with given separability structures. In this paper, we develop a graph theoretic approach to the character admissibilty problem, using Hamiltonian paths to generate voter preferences. We apply this method specifically to the hypercube graph, defining the class of cubic preferences. We then explore how the algebraic structure of the group of symmetries of the hypercube impacts the separability structures exhibited by cubic preferences. We prove that the characters of cubic preferences satisfy set theoretic properties distinct from those produced by previous methods, and we define two functions to construct cubic preferences. Our results have potential applications to experimental work involving election simulation.