# Approval Voting in Product Societies

## TL;DR

This paper investigates conditions for agreement in approval voting when voters make multiple decisions simultaneously, extending previous one-dimensional models to multi-dimensional spectra like cylinders and tori.

## Contribution

It introduces a general lower bound on agreement in two-dimensional voting societies and analyzes specific cases for cylindrical and toroidal spectra.

## Key findings

- Established a lower bound for voter agreement in two-dimensional societies.
- Analyzed approval voting scenarios on cylindrical and toroidal spectra.
- Extended single-issue agreement results to multi-dimensional decision spaces.

## Abstract

In approval voting, individuals vote for all platforms that they find acceptable. In this situation it is natural to ask: When is agreement possible? What conditions guarantee that some fraction of the voters agree on even a single platform? Berg et. al. found such conditions when voters are asked to make a decision on a single issue that can be represented on a linear spectrum. In particular, they showed that if two out of every three voters agree on a platform, there is a platform that is acceptable to a majority of the voters. Hardin developed an analogous result when the issue can be represented on a circular spectrum. We examine scenarios in which voters must make two decisions simultaneously. For example, if voters must decide on the day of the week to hold a meeting and the length of the meeting, then the space of possible options forms a cylindrical spectrum. Previous results do not apply to these multi-dimensional voting societies because a voter's preference on one issue often impacts their preference on another. We present a general lower bound on agreement in a two-dimensional voting society, and then examine specific results for societies whose spectra are cylinders and tori.