# Should You Vote?

## Model

Imagine there were $n$ people who each had a 50-50 chance of voting “yes” and “no”. For sufficiently large numbers of people (e.g. above 30), the probability of a specific person “swinging the election” is given (approximately) by

$$\sqrt{\frac{2}{\pi n}}$$

For instance, for 100 people the probability is roughly 1 in 13. For 1,000,000 people, the probability is about 1 in 1253.

Of course, this model has no real empirical basis. However, by the law of large numbers, we should expect the probability of swinging an election to be proportional to $\sqrt{1/n}$ for sufficiently large groups of people – if, that is, we treat votes as independent events.

Of course, this is all for a specific election where a certain number of people will vote each way. Obviously, in elections where roughly equal people support both candidates the probability of being the swing voter is much higher than in elections where once side has overwhelming support.

## Defense

Probably the most common criticism leveled at examining swing voting is that in real life, any sufficiently close election is decided in the courts rather than in the ballot box. However, this criticism is actually not the fatal blow it appears.

If the courts are in your favor, the “swing” voter now is simply the voter who makes it close enough for the courts to decide. If the courts are not, then the “swing” voter is simply the voter who makes the distance in votes far enough away to not require the courts.

If you assume that the likelihood of winning by (e.g.) 100 votes is more-or-less the same as the likelihood of losing by 100 votes, then the probability of being these new swing voters is roughly the same as that in the simpler model.

Of course, the other criticism is that votes aren’t independent events.

If you think that me voting one way indicates that its more likely for you to vote the same way, then you should conclude that the swing-voter probability shrinks faster than $\sqrt{1/n}$, because we tend to leave the 50% mark faster. If you think that my voting one indicates you’re less likely to, then you should conclude that the probability shrinks slower than $\sqrt{1/n}$, because then voters tend to cause opposing voters who somewhat cancel them out, pushing us toward equilibrium.

## Empirical Evidence

Based on this empirical study, we can conclude that in an extremely close election with roughly 700,000 voters, there is about a 1 in 100,000 chance of swinging an election. Based on this, we can conclude that, in fact, my voting one way generally makes it more likely for you to vote the same way. Indeed, we can estimate that the probability of swinging an election is roughly

$$\frac{1}{n^{0.85}}$$

## Value of Voting

The Democrats and Republicans have very different philosophies. Imagine you think that that the value to society of having the party of your choice in office is equivalent to giving everyone $X$ dollars.

Based on the above study, the probability of the average American swinging the presidential election is about 1-in-60 million. If your time is worth $20 per hour and it takes half-an-hour to vote, that means if you care about all of society equally, you should vote if and only if you think$X$is greater than$600 million.

I think even the cynics who think that the Democrats and Republicans are both practically the same – just political machines trying to win reelection can admit that who runs the country matters more than $2 per capita. That being said, if you are purely selfish, it is clearly not in your interest to vote – unless, of course, you think that you personally will be rewarded by about$600 million by the right party being elected.

This is, of course, all about averages. In some states the probability of swinging the election is as high as 1-in-10 million. In others, as low as 1-in-1 billion. However, it appears clear to me that this difference doesn’t change the underlying conclusion: selfless people should vote; selfish ones should not.