The Voter Basis and the Admissibility of Tree Characters

TL;DR

This paper introduces a linear algebraic method to construct preference orders with specific separability properties, using a voter basis, and characterizes which tree-structured collections of subsets can arise as separable sets.

Contribution

It develops a novel linear algebraic technique to realize preference orders with prescribed separability characters, especially for tree-structured collections.

Findings

A voter basis induces preference orders with desirable separability properties.

Preference orders with a given tree-structured character can be constructed using the voter basis.

The method provides a systematic way to analyze the admissibility of tree characters in preference modeling.

Abstract

When making simultaneous decisions, our preference for the outcomes on one subset can depend on the outcomes on a disjoint subset. In referendum elections, this gives rise to the separability problem, where a voter must predict the outcome of one proposal when casting their vote on another. A set $S \subset [n]$ is separable for preference order $\succeq$ when our ranking of outcomes on $S$ is independent of outcomes on its complement $[n]-S$. The admissibility problem asks which characters $\mathcal{C} \subset \mathcal{P}([n])$ can arise as the collection of separable subsets for some preference order. We introduce a linear algebraic technique to construct preference orders with desired characters. Each vector in our $2^n$-dimensional voter basis induces a simple preference ordering with nice separability properties. Given any collection $\mathcal{C} \subset \mathcal{P}([n])$ whose subset lattice has a tree structure, we use the voter basis to construct a preference order with character $\mathcal{C}$.