# Approval Voting in Product Societies

• Published in 2017
In the collection
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.

### BibTeX entry

@article{ApprovalVotinginProductSocieties,
title = {Approval Voting in Product Societies},
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.},
url = {http://arxiv.org/abs/1703.09870v1 http://arxiv.org/pdf/1703.09870v1},
author = {Kristen Mazur and Mutiara Sondjaja and Matthew Wright and Carolyn Yarnall},
comment = {},
urldate = {2017-03-30},
archivePrefix = {arXiv},
eprint = {1703.09870},
primaryClass = {math.CO},
year = 2017,
collections = {protocols-and-strategies}
}