This post is in response to Andrew Gelman’s post here. Gelman is a very good political scientist and statistician, but I’ve notice over and over that’s he’s rather overconfident when he has a disagreement with other experts.

(For what it’s worth, I don’t hold that voting is irrational, even if your vote doesn’t count for much.)

I’ll start this post by patting myself on the back. Unlike many political philosophers, when I’m making or relying on a social scientific claim in my work, I’m aware that I’m doing so. I always do a literature search to see what social scientists have to say about that issue. After all, often certain things that most people take for granted–e.g., that candidates can enjoy “mandates”–are in deep disrepute among social scientists. (The AP Government test even asks students, “Why are political scientists skeptical of the mandate theory of elections?”)

At various points in The Ethics of Voting, I needed to discuss the issue of the decisiveness of individual votes. In an election, an individual vote is decisive just in case it breaks a tie. Going into the election, there is some probability p that other voters will be split 50-50, and that your vote will decide the election.

Now, it’s artificial to talk at all about your vote having a chance of being decisive. In any large-scale election, there is a significant counting error (which varies depending on the counting method or technology being used). There are also legal issues that confound the process. If the vote ends up actually being very close, one candidate will probably contest the count, and as a result, it may that lawyers, judges, or bureaucrats end up “deciding” the election ultimately. At any rate, it’s common to treat presidential elections in which the two major candidates differ by only a percentage point as a statistical tie.

I’ve read pretty much every paper published about the probability of being decisive. Most of these papers use what we can call a binomial model or binomial method. To oversimplify: they treat voters as a set of weighted coin flips, and they try to estimate what’s the probability that all the coin tosses will come up exactly half heads and half tails, such that you can then cast the tie-breaking vote. From what I could tell while writing, the Brennan-Lomasky formulae from their 1993 Democracy and Decision were especially influential, though not without criticism. (As a note in a footnote in The Ethics of Voting, some people, such as A. J. Fischer, think their formulae produce too low results.) We can write the Brennan-Lomasky formula for decisiveness as p=f(N,m), when N= the expected number of voters, and m = the anticipated proportional majority. (The anticipated proportional majority is a measure of the lead one candidate has going into the election.) Brennan and Lomasky argue that your probability of being decisive goes down slowly as N increases, but it goes down much more quickly as m deviates from 0.5.  These two variables are interactive. Brennan and Lomasky argue that in a large-scale election, such as a US presidential election, where N is high (in the order of 120 million), then m would need to be extremely close to 0.5, or your probability of being decisive becomes infinitesimal.

In the The Ethics of Voting, I use the Brennan-Lomasky formula as an objection to one argument for the duty to vote. However, I don’t need Brennan and Lomasky to be right, as I introduce a separate refutation in the next chapter.

Gelman has a different method for estimating the probability of being decisive. To oversimplify things a bit, Gelman and his co-authors try to create statistical estimates. We might say their method of estimating is aposterioristic, while most papers on the issue are aprioristic. Perhaps Gelman et al are right and the others are wrong. Still, I haven’t seen Gelman take on these others and explain where and why they go wrong. For instance, Gelman et al give a positive argument for their position here (in a paper published in Economic Inquiry), but they don’t take on, e.g., Brennan and Lomasky and say why Brennan and Lomasky are mistaken. (I expressed this concern in an endnote in The Ethics of Voting.) As far as I can tell, Gelman still has the minority position here. So, it seems a bit self-congratulatory or overconfident for him to call people “innumerate” for using the Brennan-Lomasky formulae (or similar formulae) rather than his method. He may be right, but he’s not obviously right. He thinks that any model that says you have basically no chance of making a difference in a large-scale, not-very-close presidential election must for that very reason be a bad formula. But while Gelman is a serious thinker, that’s a silly objection. On the contrary, it’s interesting if you still do have a serious chance. What makes Gelman’s estimates interesting is that he claims that at least some voters–voters in certain swing states, e.g.–manage to have as high as a 1 in 10 million chance, though others have very little chance. One reason Gelman’s paper on this topic gets so much attention is that breaks with what everyone else says! The other reason, unfortunately, is that many people want Gelman to be right, and we all know how confirmation bias works.

Given the issue of confirmation bias, it’s worth pointing out that I don’t have a dog in this fight. I have pretty much no stake in whether Gelman or Brennan and Lomasky are right. In The Ethics of Voting, I argue A) that there is no moral duty to vote, but B) that if a citizen does vote, she has very stringent duties to vote a particular way. As I say in the introduction of the book: If it turns out that the expected utility (or disutility) of individual votes is high, it makes it harder for me to argue for A but easier for me to argue for B. (In fact, if Gelman’s estimates are right, then I am almost certainly correct that most Americans have a moral duty to abstain from voting, but my argument against the duty to vote would still stand.) However, if it turns out that the expected utility/disutility of individual votes is low, then it makes it easier for me to argue for A, but harder for me to argue for B. In short: If your vote makes a big difference, it’s harder to show that you don’t have a duty to vote, but easier to show that you must either vote well (in good faith, competently, with epistemic justification, etc.) or abstain. If your vote makes little difference, it’s easier to show that you don’t have a duty to vote (because that kills a few, but not all or even the best, of the arguments for the duty to vote), but harder to show that you must vote well.

Gelman says,

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.

That’s just straight up smug. I have no problem with well-earned smugness. But Gelman hasn’t earned it. He’s enjoying some stolen smug here. Given that so many of Gelman’s epistemic peers, and arguably some of his epistemic superiors, dispute this very point, he’s in no position to be so dismissive.

As for Republican voting and income, Caplan has dealt with this here.