On the Probability of Being Decisive

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.


  • If an election is won by one vote, which voter cast the deciding vote? Answer: ALL OF THEM that cast their vote for the winner. Because if ANY of those voters changed their vote, the other candidate would win. Now that doesn’t change the fact and the accompanying math that vanishingly few elections are that close. But a vote is a vote. (BTW, the same logic works for “the winning shot” in a basketball game, or whatever.)

  • Andrew Gelman


    Thanks for the feedback. In brief reply:

    1. Setting aside any views you might have about my overconfidence, the claim of “10 to the −2,650th power” is indeed innumerate. This can be seen in many ways. For example, several presidential elections have been decided by less than a million votes, so a number of the order 1 in a million can’t be too far off, for a voter in a swing state in a close national election. For another data point, in a sample of 20,000 congressional elections, a few were within 10 votes of being tied and about 500 were within 1000 votes of being tied. This suggests an average probability of a tie vote of about 1/80,000 in any randomly selected congressional election. It’s hard to see how the probability of a tie could be of order 10^-5 for congressional elections and then become 10^-2650 for presidential elections.

    Yes, I accept that a particular mathematical model gives a probability of something like 10^-2650. My quarrel is not with the mathematics but with the application of this number to the real world.

    2. My post was not intended to be smug. I do not enjoy seeing innumeracy in the world. I do not know what is an epistemic peer or an epistemic superior (as you put it in your post), but I think the claim of 10^-2650 makes no sense, no matter who makes the claim.

    3. You write, “I haven’t seen Gelman take on these others and explain where and why they go wrong.” My colleagues and I actually have carefully made these explanations, however they were not in the articles I linked to in my blog so I can see how you would not have been aware of them. Here are two articles in which we explain the flaws in certain probability calculations of the probability of a decisive vote: here and here. The first paper appeared in the journal Statistical Science and is full of probability models, the second paper appeared in the British Journal of Political Science and has empirical data. We also wrote a more general paper on the estimation of small probabilities (using presidential elections as our central example) here in the Journal of the American Statistical Association.

    4. You write that “it’s artificial to talk at all about your vote having a chance of being decisive.” Please see the appendix on page 674 of the British Journal of Political Science linked to above. There we explain how, even though there is no sharp dividing line between winning and losing (because of disputed votes, recounts, etc.), we can still use the probability of a tie election to calculate the probability of a decisive vote. This is a question that’s come up a lot, so we went to the trouble to go through all the steps carefully.

    5. At the end of your post, you link to a post by Bryan Caplan who reproduces a finding from our book that richer people are more likely to vote Republican. Caplan notes that the relation between income and Republican voting is lower than many people would expect. The mistake I was referring to earlier was Mangu-Ward writing, “Rich people are not more likely to vote Republican,” which is incorrect (as can be seen, for example, in the graph that Bryan posted and you linked to). Perhaps it is unfair of me to refer to that statement by Mangu-Ward as innumerate rather than simply referring to it as a mistake on her part. The reason I consider it innumerate is that it is easily available information. I think there is an element of innumeracy in making a false statement about numbers that can be checked. I would have seen it as more numerate for her to have written something like “I seem to recall hearing that rich people are not more likely to vote Republican” (or, of course, something more informative such as “Yes, rich people are more likely to vote Republican, but not by as much as you might think”).

    • Can you really not see that you and Brennan are answering different statistical questions? Brennan is calculating the chance of it being EXACTLY 50/50 because that’s the only way a single vote would actually change the outcome. You can’t arbitrarily decide that if it’s between 10 votes then it’s decisive and expect to get the same results. Even if you assume the true proportion is exactly 50/50, because some proportion of voters aren’t going to make it to the voting booth for completely random reasons, the chances of the outcome being within X votes grows exponentially with X because of the binomial nature of individual voters. So of course you’re orders of magnitude off from each other; that’s what it means for the growth to be exponential. And how exactly did you arrive at 80,000 in the above analysis?

      • Andrew Gelman


        I am also calculating the probability of it being exactly 50/50. Or the probability that it is within +/- 10 votes of being tied, and then dividing that probability by 20, which is the same thing. You can read my papers for the details. For now, just believe me that I know all about the binomial distribution and probability theory!

        • I read the papers and I see what you’re saying now. But doesn’t doing the calculation that way only work if you assume each outcome is equally likely? I agree that the random voter model is an oversimplification, and therefore the binomial calculation that Brennan presents isn’t right. But Brennan’s statement included the presumption that voters were 50.5% in favor of one candidate (I know that statement has to be reinterpreted under a different model, but the idea is the same). If that’s the case, outcomes with the favored candidate having more votes can be significantly more likely than a tie or the opposite outcomes, no?

    • Sean II

      I’m not claiming to be anyone’s epistemic superior or anything, but I would like to point out that all of this is based on two very strange assumptions: first, that presidential candidates are different from each other, and next that presidents actually decide stuff by themselves.

      If I have a 1 in 10,000,000 chance of picking between two dudes who are 98% identical in political function, then what exactly do I have to get psyched about?

      If, on top of that, my winning presidential dude has only a 1 in WHATEVER chance of getting his way in any given messy consensus process….well now you’ve given me odds that make a ninja attack at the polling place seem far more likely than a meaningful vote.

      Gelman says “it can indeed be rational to spend a minutes for that small probability of having a huge effect on the world.”

      That’s really his key position. But even if you accept his math, it doesn’t remotely follow (not from any sophisticated understanding of how the state works) that getting Candidate X elected ever = a huge effect on the world.

      • Andrew Gelman


        In a comparative study, Huber and Stanig found the Democrats and Republicans in the U.S. to be relatively far apart, compared to left and right parties in other countries (see chapter 7 of our Red State Blue State book for details). So I do think there are differences between the parties and the candidates. But I respect that, from your position, the two parties may be very similar. I’m happy to have people argue with me about the politics (I think your position is a minority view but maybe it is correct); here I just want to establish the mathematics. Given the 1 in 10 million chance, you may very well decide that it’s not worth your time to vote. I just don’t want you to think it’s a 1 in 10^-2650 chance!

        • Sean II

          I understand you’re trying to isolate a particular question here, as one must do when publishing papers and such. I’m simply an innumerate (and borderline illiterate) bystander to that dispute, who cares only to find out if the resulting information is useful.

          What you tore from context to examine up close, I want to put back and broadly evaluate.

          In that spirit I don’t think you can separate the electoral math from the probability of achieving a “huge impact”, which – as you rightly point out – is what most people actually want from voting. They don’t want Candidate A to win as an end in itself. They want to change the world, and they see him as a promising means to that end.

          But you’re not selling a 1 in 10 million chance of changing the world. You’re selling a 1 in 10 million chance to win a raffle ticket that in turn presents in 1 in X chance of changing the world.

          Whatever those combined odds may be in any given electoral contest in any given place and time, they are, by definition, less hopeful than 1 in 10 million.

          But you have my sympathy even if you don’t have my agreement. To be accused of smugness by libertarians…that’s something no one should have to endure.

  • good_in_theory

    “The probability of being decisive” seems like a correct answer to a wrong question to me.

    Isn’t something like the probability of not having been decisive much more important? I guess there are a few sides to this.

    On the one hand there’s the chance that your decision to vote was essential to producing a win.
    Then there’s the chance that your decision not to vote produced a loss

    Then there’s the chance that your decision not to vote could have produced a loss (based on the probability of more than the margin of victory for the opposition turning out).

    Then there’s the chance that your decision to vote could have produced a win (based on the probability of less than the margin of victory for the opposition turning out).

    If the margin of victory is 1 vote, every winner-who-voted’s decision to vote is 100% responsible for victory and every loser-who-did-not-vote’s decision not to vote is 100% responsible for defeat.

    Then the question becomes something like “At what prospective margin of victory does one’s contribution to victory/defeat by voting/not-voting become sufficiently negligible, given the returns to victory/defeat and to voting/not-voting”

    Maybe the payoff of the answer to that isn’t much different from the answer to the question “how likely is it that an election will be decided by one vote”. But it seems like a much better question to me.

  • Unlike Keynesianism. voting is one of those things where the aggregate really counts. Your vote does count because if everyone abstains there is no election, or if everyone on one side stays home, the other guy wins. But each individual vote is relatively unimportant.

    • NoTheoryofJurisprudence

      As Brennan has pointed out, that’s true of farming, too. That doesn’t mean you have a duty to farm.

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