#+title: Representative Voting
#+author: Preston Pan
#+description: What do we do about voter turnout? Voting demographics? Polarization?
#+html_head:
#+html_head:
#+html_head:
#+html_head:
#+html_head:
#+html_head:
#+html_head:
#+html_head:
#+html_head:
#+language: en
#+OPTIONS: broken-links:t
* 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.
** 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*}