# Representative Voting

## Table of Contents

## 1. Introduction

Current voting systems are broken, and people argue about ways to solve it. Many talk about about ranked-choice
voting or other ballot-systems, but I argue that the real problem in voting has to do with game theory principles.
In this article I endorse a system that has been tried out before, but has been forgotten: *random representation*. I
argue that it has game theoretic foundations that make it superior to other kinds of voting systems.

### 1.1. The Model

Let us assume that there is a small probability that you can swing the election \[ \rho \], and a cost to voting; that
is to say, it takes some amount of time, which has opportunity cost associated with it to vote, which we
will call \[ \alpha \]. Let us assume that there is a high *reward* in swinging the vote; that is to say, if you were
the one that swings the vote, your vote is worth some high monetary value. Let \[ \beta \] be the median price of swinging.
Let \[ n \] be the number of people voting, and let the weight of each vote be equal between all participants.
Let the choice of candidate between all voters be binary; voting for one candidate mutually excludes you from
voting for another, and there are two candidates (this is to simply the model; you will see that this does not
lose generality). Then, let us model the expected value of voting for singular individuals.

For a given person, the probability that your vote swings (or at least ties) depends on the probability that \[ x = \frac{n - 1}{2} \], where $ x \[ is the number of people that vote for your candidate. The probability density function for the probability that \] m \[ people vote for your candidate we'll call \] f $$. We will assume it is binomial, and you might expect it to be closer to 50/50 most of the time, but that is pretty hard to model. We will therefore compensate by modeling it more accurately afterwards. For now, we assume all participants have a 50% chance to pick either candidate.

\begin{align*} \end{align*}