💾 Archived View for osanwe.benson.earth › s › Economics › 17 captured on 2024-12-17 at 09:55:00. Gemini links have been rewritten to link to archived content
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This post attempts to understand whether voting for a presidential candidate is rational, making several simplifying assumptions:
The impetus for this post came from two conflicting arguments, one claiming that maybe voting isn't rational and the other claiming that it's clearly rational.
The funny thing is that both authors use just about the same model to reach their different conclusions. Here's how it goes:
The value of the correct candidate winning is `V`. The probability that your vote decides the election __in your state__ is the first thing we want to calculate. If the number of people voting in a given state is `N` then the lower bound on that probability is `N^{-1}`. This is a pretty small number for any state.
Philosophy professor Zach Barnett shows that the lower bound probability can be modeled as a function of the election's competitiveness like so (here `d` is the chance your vote is the decisive vote and `P` is the probability of an upset as a number from zero to one half):
d > 10 * P * N^{-1}
This is still a pretty conservative lower bound. I'm not sure what to use for an upper bound, but multiplying it by 100 seems safe.
Now that we've found `d` we need to adjust by the probability that your state is the tipping point state (`s`), which for a PA voter is something like `s = 0.32`. So the chance that your vote chooses the president as a PA voter is at least `0.32*d`.
Let's say the cost of the effort of voting is $5. Then we want `V * s * d > $5` if we are going to vote.
We can rearrange to find the minimum `V` where voting is rational (using the upper bound for `d`) and a maximum `V` where not voting is rational (using the lower bound for `d`).
V = $5 * s^{-1} * d^{-1}
However, we've left something out. You can't be certain that you know who the best person to vote for is, and if you vote for the wrong person then you're hurting and not helping. If you think voting for the right person has an expected value of $10 and you're 60 percent sure you know who the right person is, then the expected value of your vote is `6-4 = 2`. In general, if you are `c` confident that you've chosen the right candidate, then update by
c - (1-c) = 2c - 1
So now we have:
V = $5 * (s * d * (2c-1))^{-1}
I did this math on a spreadsheet for evey state. Here's what I came with as an upper bound on V for selected states, assuming you are 75% sure that you're voting for the right person.
Pennsylvania | $904,186 Wisconsin | $1,201,390 Michigan | $1,584,246 Alaska | $18,921,033 New York | $528,606,607 Vermont | $14,004,705,882 Massachusetts | $106,172,839,506 Indiana | $3,555,555,555,556
This means that if the value of the outcome of the presidential election is greater than the upper bound on V for your state, then you should absolutely go out and vote. If not, it might not be worth your time.
This is where it gets really tricky. The two articles linked at the top of this post which came to different conclusions both used something like this model to estimate the value of a vote relative to the value of election outcome.
But one person assumed that the value of the election results should be considered as the value to the whole country. Something like, if the better president uses money 2% more efficiently than the value of the election is at least 2% of the federal budget over the next four years. The federal government spends about $6.5 trillion per year, so this works out to about $500 billion, which is enough to make someone's vote worthwhile even in Indiana.
The other person assumed that the value of the election results should be considered personally, perhaps the amount of money you would take in exchange for the bad candidate winning (assuming the good candidate is guaranteed to win otherwise).
Many people acting out of self interest might decide this is a pretty low number, maybe in the thousands or tens of thousands of dollars.
Most people in Indiana would rather have a worse president but be given 3.5 billion dollars (and could probably do more good with that money than the good president would have done as president).
This is a thorny issue. I think that the second way of valuing the election results is more appropriate, since we are determining whether it is rational for an individual voter to vote, not whether the citizens in general should vote. In that case, it is up to you to decide how much the election is worth to you, and then you can use a handy-dandy equation to see if voting is worth the effort.
But don't really use my equation. There are lots of reasons to vote which are independant of your chance of being a decisive voter in choosing the US president, and so your vote probably has a much higher expected value than the reasoning here would suggest. Your impact on local and state elections is much more significant, and you are quite likely to be impacted by the outcomes of these elections.
Oct 24 · 8 weeks ago