In this video, we’ll focus on p — the probability your vote matters.
In part 1, we explained why there are only two scenarios where your vote matters.
This can be tricky to understand. So let’s run through the explanation again, but this time from a slightly different angle. In part 1, we counted all votes except yours. This time, we’ll count all votes including yours.
If there’s an exact tie, then your vote matters. Actually, every vote matters. I repeat: Every vote matters. That’s because if any single voter had stayed at home, the result would instead have been a win.
Now. Suppose instead the winner wins by exactly one vote. Then every single one of the winner’s votes matters. That’s because if any one of the winner’s voters had stayed at home, the result would instead have been an exact tie. In contrast, none of the loser’s votes matters. That’s because if any one of the loser’s voters had stayed at home, the result would still have been exactly the same.
Now. A single vote matters only in these two scenarios. In every other scenario, no single vote matters. For example, in the UK’s 2017 General Election, the constituency of North East Fife was won by exactly two votes. Yet even in this incredibly close election, no single vote mattered. That’s because if any single voter had stayed at home, the result would still have been exactly the same.
Altogether then, your vote matters only in two highly-unlikely scenarios. And so p — the probability your vote matters — is tiny. How tiny, exactly?
One study found that in the 2008 US presidential election, the probability your vote would matter was as high as 1 in 6M in New Mexico, and as low as 1 in 20B in Oklahoma and 1 in 500B in DC.
A similar study found that in 2016, the probability your vote would matter was as high as 1 in 1M in New Hampshire and Colorado, and as low as 1 in 30B in Wyoming and Oklahoma. In this study, the researchers didn’t even bother with DC. They simply assumed there was zero chance your vote would matter in DC.
Now. Here’s a morbid comparison to illustrate how absurdly tiny some of these odds are. The US averages about one fatality per 100M vehicle miles traveled. Equivalently, for every mile driven, the chance of a fatality is roughly 1 in 100M. And so in Wyoming or Oklahoma, a voter is 300 times more likely to kill someone while driving a mile to the polling station, than to have her vote matter!
If p is indeed this tiny, then how could voting possibly be rational?
We’ll now present three arguments as to why p might not be quite so tiny — and hence why voting might be rational.
Argument 1. In smaller elections, p is bigger. We’ll skip the math, but obviously, the fewer the voters, the more likely it is that your vote matters. We’ve been focusing heavily on the US presidential election. But most elections are far smaller.
For example, one study found that historically, for US Congressional elections, 1 in 89,000 votes mattered. And for US state legislator elections, 1 in 15,000 mattered.
If p is as high as 1 in 89 or 15,000, then voting could easily be rational.
Next. Argument 2. There may be multiple issues on the ballot. For example, in 2016, the ballot in Santa Barbara county, California contained 25 different issues!
Voters could vote on the marquee matchup. But they could also vote on 6 other offices; 1 countywide measure; and 17 statewide propositions, of which one concerned the adult industry and two concerned plastic bags.
If there’s only one thing to vote for, the probability your vote matters is nearly zero. But if there are multiple things to vote for, the probability your ballot as a whole matters will be considerably larger. In which case, a trip to the polling station might be perfectly rational after all.
Now. Arguments 1 and 2 are solid. In many cases, they can explain why voting is perfectly rational. Unfortunately, they do not apply to large elections where there’s only one issue on the ballot. In such elections, the puzzle of voting remains.
Let’s now move on to Argument 3. The social effect. When voting, you can drag along to the polling station friends and family who would not otherwise have voted. More generally, simply by voting, you are in a better position to preach, nag, and shame others into voting.
And in this age of polarization and self-segregation, anyone you call a friend or family will probably vote the same as you.
Suppose that by voting, you manage to persuade 9 otherwise-non-voters to vote. Then the probability your vote matters will have been magnified tenfold.
Now. The social effect is real, but probably small. By voting and making a big deal about it, Katy Perry might be able to persuade many thousands of otherwise-non-voters to vote. But most people aren’t Katy Perry. Most people will have trouble persuading even one otherwise-non-voter to vote. And so for most people, the social effect is probably small.
So far, we’ve been assuming that although your vote is unlikely to matter, there are nonetheless two definite scenarios where it matters.
This may be true in theory. But in practice, your vote may not matter even in these two scenarios. In practice, vote counting is a messy and inexact affair. Which means that whenever the vote is really close, the powers-that-be may find some excuse to interfere and decide the election to their liking.
This was most vividly illustrated in the 2000 US presidential election. For this and more, stay tuned for the next video!
 Or in econ jargon, “the probability your vote is pivotal or decisive”.
 Gelman, Silver, and Edlin (2012), “What is the Probability Your Vote Will Make a Difference?”, Economic Inquiry (PDF). The publication actually doesn’t have the precise numbers (it only has graphs). I requested the dataset with precise numbers from Gelman and he kindly acquiesced: see probs.csv. I have rounded off the odds to one significant figure.
 According to the Fatality Analysis Reporting System (FARS) Encyclopedia, in 2014, there were 1.08 fatalities per 100 million vehicle miles traveled.
 Of course, one could argue that a sober and responsible citizen driving to the polling station on election day is much less likely than the average US driver to cause a fatality. But even if it were the case that you’re a hundred-fold less likely than the US average to cause a fatality, you’d still be thrice as likely to kill someone while driving a mile, than that your vote matters in Wyoming or Oklahoma.
The Indian general elections are bigger, but no single contest in India comes close to the US presidential election. For example, in the 2014 Indian general election, the highest voter turnout for any Lok Sabha constituency was 1,620,397 in Malkajgiri, Andhra Pradesh (source).
A counterargument might be that thanks to the convoluted electoral college system, the US presidential election isn’t a single direct election and voters in each state merely vote for their state’s electors. And so the US presidential election cannot be considered the world’s largest election. If we admit this counterargument, then the world’s single largest election is probably the Brazilian presidential elections.
 For some reason, Argument 2 has been largely neglected by students of the paradox of voting. The only academic study I can recall coming across that explicitly accounts for the fact that ballots sometimes contain multiple issues is Fauvelle-Aymar & François, 2015, “Mobilization, cost of voting and turnout – a natural randomized experiment with double elections”, Public Choice (PDF).