### Positional Weighted Voting and Linear Algebra

#### Abstract

This paper works off of previous research in the field of mathematics applied to voting theory; specifically, the investigation of the ways to "weight" a voting profile, or, in other words, how to score a particular ballot in a vote. This question is important from a practical standpoint in considering the effectiveness or objectivity of voting procedures. This paper looks at this issue from a completely mathematical standpoint, using primarily linear algebra to analyze the way a weighting procedure affects a vote, and will prove an important result using linear algebra, namely that there are mathematically infinite different ways people can vote that, when coupled with a specific weighting system, could lead to a specific numerical result. However, this particular conclusion is not always applicable to practical voting situations, and therefore this paper shows what these theoretically infinite ways people could vote actually means in a real world scenario. Specifically, given a set of weights and results, there are a specific set of cases where there is a particular, finite set of voting profiles that connect the two. Furthermore,this paper shows how to move between statements we can make about the cardinal (in a number of points assigned to each candidate based on the voting) ranking of candidates to the ordinal ranking of candidates, given a real world setting where certain cardinal designations may be impractical or impossible. While this paper will only consider the mathematics behind this voting theory, the results will certainly be of interest to anyone interested in ensuring that voting accurately represents the interests of the parties involved, as these results will emphasize that the selection of a weight system for a vote is often more important than the actual vote itself.

#### Keywords

Voting Theory

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