#### Graduation Year

2012

#### Document Type

Open Access Senior Thesis

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### Reader 1

Francis Edward Su

#### Reader 2

Nicholas Pippenger

#### Terms of Use & License Information

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

#### Rights Information

© Patrick Eschenfeldt

#### Abstract

Under approval voting, every voter may vote for any number of canditates. To model approval voting, we let a *political spectrum* be the set of all possible political positions, and let each voter have a subset of the spectrum that they approve, called an *approval region*. The fraction of all voters who approve the most popular position is the *agreement proportion* for the society. We consider voting in societies whose political spectrum is modeled by $d$-dimensional space ($\mathbb{R}^d$) with approval regions defined by axis-parallel boxes. For such societies, we first consider a Tur&#aacute;n-type problem, attempting to find the maximum agreement between pairs of voters for a society with a given level of overall agreement. We prove a lower bound on this maximum agreement and find in the literature a proof that the lower bound is optimal. By this result we find that for sufficiently large $n$, any $n$-voter box society in $\mathbb{R}^d$ where at least $\alpha\binom{n}{2}$ pairs of voters agree on some position must have a position contained in $\beta n$ approval regions, where $\alpha = 1-(1-\beta)^2/d$. We also consider an extension of this problem involving projections of approval regions to axes. Finally we consider the question of $(k,m)$-agreeable box societies, where a society is said to be *$(k, m)$-agreeable* if among every $m$ voters, some $k$ approve a common position. In the $m = 2k - 1$ case, we use methods from graph theory to prove that the agreement proportion is at least $(2d)^{-1}$ for any integer $k \ge 2.$

#### Recommended Citation

Eschenfeldt, Patrick, "Approval Voting in Box Societies" (2012). *HMC Senior Theses*. 36.

https://scholarship.claremont.edu/hmc_theses/36