One Way to Think about Voting

Feb 20, 2020
#random #math


It can be difficult to motivate oneself to vote in elections when the voter base is large because, in the grand scheme of things, one vote doesn't matter. However, if a lot of people operate under this mindset then it does matter, since the group of people who end up voting might not be representative of the population as a whole. I'd like to propose a different way of thinking about voting in order to counteract this mindset. Specifically, what I'm proposing is meant to be a helpful philosophy, not a rigorous (or even accurate) mathematical model.

The main idea is to believe that individual votes are not independent. If you do vote, then there is a higher chance that people like you will also vote. The opposite is true if you don't. Therefore, the effect of your vote is magnified by the group of people you belong to.

More formally, let \( G \) be the population that is eligible to vote in an election. Partition \( G \) into disjoint groups \( g_1, g_2, \cdots g_n \) of "similar" people. By "similar", I mean that people in the same group have the same probability of voting. Now, a critical assumption to make is that "similar" people will also vote for the same candidate. I think this is a reasonable assumption because I believe that both a person's likelihood of voting and political preference are strongly determined by their background[1].

For each group \( g_i \), let \( p_i \) be the probability that someone in \( g_i \) votes. We'll have an estimate of \( p_i \) based on historical data, but conceptually we can think of \( p_i \) as being described by an unknown probability distribution due to uncertainty.

Finally, let \( g_i \) be the group that you are a part of, and \( v \) be the indicator variable representing whether you vote (\( 1 \)) or not (\( 0 \)). By Bayes' theorem, \[ Pr(p_i = p | v = 1) = \frac{Pr(v = 1 | p_i = p)Pr(p_i = p)}{Pr(v = 1)} \] The right hand side simplifies to \[ \frac{p \cdot Pr(p_i = p)}{ \mathbb{E}[p_i] }\] because of the definition of \( v \). This implies that \( Pr(p_i = p | v = 1) \) grows with respect to \( Pr(p_i = p) \) as \( p \) increases, which means \( \mathbb{E}[p_i | v = 1] > \mathbb{E}[p_i] \). Similarly, \( \mathbb{E}[p_i | v = 0] < \mathbb{E}[p_i] \). Because the amount of people from group \( g_i \) expected to vote is \( \mathbb{E}[p_i] \cdot | g_i | \), if you vote then you are effectively increasing the number of votes for your desired candidate by \( (\mathbb{E}[p_i | v = 1] - \mathbb{E}[p_i | v = 0]) \cdot | g_i | \), which is probably much bigger than \( 1 \).

This approach scales because the size of your group \( |g_i| \) will be roughly proportional to the population \( |G| \) that is participating in the election, so it can be applied to elections of all sizes, local and national.

Footnotes

  1. For example, I consider myself to belong to a group consisting of Asian-American college students who grew up in California. To a first approximation of political views, almost all of us would probably vote for the Democrat candidate. Using a very rough back-of-the-envelope calculation, there are at least 30,000 people in this group who are eligible to vote. If the fact that I vote increases expected probability (discussed in the second to last paragraph) by 1%, then I am effectively contributing 300 votes to my desired candidate. Back to text

Acknowledgements

Thanks to Ganesh for helping me proofread this post!





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