Voting for Committees in Agreeable Societies

We examine the following voting situation. A committee of $k$ people is to be formed from a pool of n candidates. The voters selecting the committee will submit a list of $j$ candidates that they would prefer to be on the committee. We assume that $j \leq k < n$. For a chosen committee, a given voter is said to be satisfied by that committee if her submitted list of $j$ candidates is a subset of that committee. We examine how popular is the most popular committee. In particular, we show there is always a committee that satisfies a certain fraction of the voters and examine what characteristics of the voter data will increase that fraction.Comment: 11 pages; to appear in Contemporary Mathematic

Voting in Agreeable Societies

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Exploring Agreeability in Tree Societies

Let S be a collection of convex sets in Rd with the property that any subcollection of d − 1 sets has a nonempty intersection. Helly’s Theorem states that ∩s∈S S is nonempty. In a forthcoming paper, Berg et al. (Forthcoming) interpret the one dimensional version of Helly’s Theorem in the context of voting in a society. They look at the effect that different intersection properties have on the proportion of a society that must agree on some point or issue. In general, we define a society as some underlying space X and a collection S of convex sets on the space. A society is (k, m)-agreeable if every m-element subset of S has a k-element subset with a nonempty intersection. The agreement number of a society is the size of the largest subset of S with a nonempty intersection. In my work I focus on the case where X is a tree and the convex sets in S are subtrees. I have developed a reduction method that makes these tree societies more tractable. In particular, I have used this method to show that the agreement number of (2, m)-agreeable tree societies is at least 1 |S | and 3 that the agreement number of (k, k + 1)-agreeable tree societies is at least |S|−1

On (2,3)-agreeable Box Societies

The notion of $(k,m)$-agreeable society was introduced by Deborah Berg et al.: a family of convex subsets of $\R^d$ is called $(k,m)$-agreeable if any subfamily of size $m$ contains at least one non-empty $k$-fold intersection. In that paper, the $(k,m)$-agreeability of a convex family was shown to imply the existence of a subfamily of size $\beta n$ with non-empty intersection, where $n$ is the size of the original family and $\beta\in[0,1]$ is an explicit constant depending only on $k,m$ and $d$. The quantity $\beta(k,m,d)$ is called the minimal \emph{agreement proportion} for a $(k,m)$-agreeable family in $\R^d$. If we only assume that the sets are convex, simple examples show that $\beta=0$ for $(k,m)$-agreeable families in $\R^d$ where $k<d$. In this paper, we introduce new techniques to find positive lower bounds when restricting our attention to families of $d$-boxes, i.e. cuboids with sides parallel to the coordinates hyperplanes. We derive explicit formulas for the first non-trivial case: the case of $(2,3)$-agreeable families of $d$-boxes with $d\geq 2$.Comment: 15 pages, 10 figure

Approval Voting in Box Societies

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.

Connections Between Voting Theory and Graph Theory

Mathematical concepts have aided the progression of many different fields of study. Math is not only helpful in science and engineering, but also in the humanities and social sciences. Therefore, it seemed quite natural to apply my preliminary work with set intersections to voting theory, and that application has helped to focus my thesis. Rather than studying set intersections in general, I am attempting to study set intersections and what they mean in a voting situation. This can lead to better ways to model preferences and to predict which campaign platforms will be most popular. Because I feel that allowing people to only vote for one candidate results in a loss of too much information, I consider approval voting, where people can vote for as many platforms as they like

Voter Compatibility In Interval Societies

In an interval society, voters are represented by intervals on the real line, corresponding to their approval sets on a linear political spectrum. I imagine the society to be a representative democracy, and ask how to choose members of the society as representatives. Following work in mathematical psychology by Coombs and others, I develop a measure of the compatibility (political similarity) of two voters. I use this measure to determine the popularity of each voter as a candidate. I then establish local “agreeability” conditions and attempt to find a lower bound for the popularity of the best candidate. Other results about certain special societies are also obtaine