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October 04, 2012 12:45 PM Voting Can Be Rational, and Don’t Believe Innumerate Arguments that Claim Otherwise

By Andrew Gelman

In an article entitled “Your Vote Doesn’t Count: Why (almost) everyone should stay home on Election Day,” Katherine Mangu-Ward writes:

Your vote will almost certainly not determine the outcome of any public election… . I’m talking about pure, raw math.

She correctly points out that that congressional elections are tied at a rate of about 1 in 80,000 (this is Exercise 1.5 in Bayesian Data Analysis!) and that the probability of your vote being decisive in a presidential election is on the order of 1 in 10 million—and that’s if you live in a swing state! If you’re voting in New York, the probability that your vote will decide the presidential outcome is basically zero.

Where she goes wrong is in her assumption that, if the probability is 1 in 10 million, you shouldn’t vote. As Aaron, Noah, and I write (here’s the short version and here’s the long version that appeared in the journal Rationality and Society), it can indeed be instrumentally rational to vote for president—if you live in a swing state—because that 1-in-10-million probability is multiplied by the potentially huge impact on world of your preferred candidate winning. Both theory and empirical evidence suggest that people choose whom to vote for not out of self-interest but out of concern for what’s best for the country. (To her credit, Mangu-Ward realizes this, writing, “people do not typically vote in ways that align with their personal material interests.”)

Mangu-Ward might very well argue that most people have no idea what’s best for the country, and maybe she’s right, but from the perspective of a swing-state voter it can indeed be rational to spend a few minutes for that small probability of having a huge effect on the world.

Innumeracy

So far, I’ve merely said that Mangu-Ward is confused, and I can hardly hold this against her; she’s certainly not the first person to see those low probabilities and conclude that voting is a bad idea. I think she got things wrong, but it’s a common area of confusion (which is why we had to publish a research paper on it).

And, in any case, people don’t vote only for rational reasons. I’m planning to vote for president next month even though I live in New York State.

The reason I say that Mangu-Ward’s article is innumerate is because of two things she writes later on. First, this calculation that she attributes to philosopher Jason Brennan:

Assuming a very close election where that candidate is leading in the polls only slightly and a random voter has a 50.5 percent chance of casting a ballot for her, the expected value of a vote for that candidate is $4.77×10 to the −2,650th power.

That’s innumerate. Or, to put it another way, any model that gives numbers like that is a bad model. Mangu-Ward should’ve stuck with the 1-in-10-million number that she got from our research. “10 to the −2,650th power”? C’mon.

And here’s innumeracy #2. Mangu-Ward writes:

Rich people are not more likely to vote Republican.

All the evidence I’ve seen points to richer people being more likely to vote Republican.

Political theory

Mangu-Ward might be right that elections are a bad idea, that maybe our politics should be run differently. I don’t have anything particular to add on that here. But I don’t like to see mathematical mistakes.

P.S. More here.

[Cross-posted at The Monkey Cage]

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Andrew Gelman is a professor of statistics and political science and director of the Applied Statistics Center at Columbia University.
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