Voting can be rational, and don’t believe innumerate arguments that claim otherwise

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.


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.

50 Responses to Voting can be rational, and don’t believe innumerate arguments that claim otherwise

  1. Julia October 4, 2012 at 11:39 am #

    I don’t have the time right not to write out a full response, but I’m always thought the dabate about wether voting is worth it for individuals has a lot in common with the debate over vaccinating children. One child who isn’t vaccinated against the measles is protected by all the other children’s vaccinations, but if none of them were vaccinated, there would be problems. Similarly, one person not voting may have close to null impact, but if everyone stopped…

    Essentially, the whole is more than the sum of its parts.

    • Andrew Gelman October 4, 2012 at 12:03 pm #


      Yes, but to Mangu-Ward, the cure is worse than the disease. One reason she wants lower voter turnout is that this would delegitimize democracy, and she’s coming from a perspective where she wants government to have less power and less prestige. That’s why I didn’t want to get into a dispute on the political theory, I wanted to focus on the mathematics rather than the political values.

      • dL October 5, 2012 at 6:46 am #

        Yes, there is a distinction between Rational Ignorance and a payoff matrix for decisive voting for a preferred candidate(i.e., selfish voting).

        Mangu-Ward made an erroneous argument that voting wasn’t rational because the expected payoff for selfish voting was essentially zero. That’s really not a correct argument, in part because the opportunity cost of selfish voting is near zero.

        Rational Ignorance is that informed voting for the “best” candidate is a public goods problem. The opportunity cost of making an informed decision about the best candidate is not compensated by the expected benefit of voting for that best candidate.
        A primary distinction is that there definitely is an opportunity cost involved in making an informed decision about the “best” candidate and/or the “best” policy.

        My theory of liberal democracy is that democracy only serves the role of an accountability mechanism. There is an “incentive-incompatibility problem” of collective choice to abide by the “unanimity” of the so-called social contact. In plain terms, this simply means that control of the government begins where unanimity ends. In the end, we are reliant on democracy as an accountability mechanism to correct or constrain collective choice actions. Unfortunately, democracy is an unreliable mechanism in this regard.

        Case in point: The United States government recently declared WikiLeaks an “Enemy of the State,” on par with al-Qaeda. This is behavior of an authoritarian regime. Combine this with perpetual war, a massive spy infrastructure and corporate bailouts and you have a government that in no way resembles a “liberal democracy.” There needs to be a fundamental correction. Fat chance by voting for either twiddle-dee or twiddle-dum.

        You yourself, with all your training as a political scientist, seem particularly unconcerned about this. Indeed, you seem particularly concerned instead about “legitimacy.” I would contend this classifies you as an example of a selfish voter.

        To the extent I would get involved in politics it would be to contribute to the likes of the ACLU, EFF and WiiLeaks(unfortunately, this would have to be done via a crypto-currency) in an attempt to enforce some degree of collective choice constraint on those who actually control the government(those who have control ain’t me and it’s not “we”).

        Voting be damned…

    • Deen October 4, 2012 at 4:31 pm #

      Funny you mention vaccines since for a particular type of vaccine (say a weakened live variant) choosing not to vaccinate yourself may allow you to become susceptible to a vaccine derived infection. Vaccines of that type become most dangerous only when the wild types are almost eradicated – no competition while – but not quite yet. In that case choosing not to vaccinate yourself is not only foolish but dangerous.

    • Andre Kenji October 6, 2012 at 3:28 pm #

      No, it has nothing to do with vaccination. One can argue that´s it´s useless to increase the number of voters because that will bring uninformed voters to the polls(That´s the problem with mandatory voting). On the other hand, the objective of elections is not choose the “best” candidate – a good political system with good institutions and good laws should work even with the crappiest of the candidates.

      If a certain group of people does not vote or vote in smaller numbers then it´s more possible that politicians will ignore this group of people. It´s no wonder that the United States has a safety net biased toward seniors, that votes in a larger number than the young. The Jim Crow Laws were enacted in part because only white people were allowed to vote in the South.

  2. PBR October 4, 2012 at 12:06 pm #

    She might be right that elections are a bad idea? Really? Sure social scientists often prefer to abstain from normative judgements, but that seems to me like bending over backward.

    Anyway, sorry to zero in on the only sentence of your post I disagreed with. Otherwise, great points!

    • Andrew Gelman October 4, 2012 at 12:15 pm #


      Just to clarify: I disagree with Mangu-Ward on that one. But I’m not expert on political theory; I don’t think I’d be adding much to my blog post by setting out my ill-formed ideas in that area!

  3. Jim Fearon October 4, 2012 at 12:16 pm #

    I’m sure I should read the paper where you probably address this question, but …

    So you would be willing to pay on the order $10,000,000 to be decisive in the presidential election? (more if you are risk averse) Or at any rate, an incredibly large sum for any non-completely trivial estimate of the dollar cost in time and effort of voting. Do you think most people would?

    • Andrew Gelman October 4, 2012 at 2:16 pm #


      Yes, please read the linked paper. We discuss how the numbers work out, and how a stable turnout rate arises if there is a distribution of utilities of voting in the population.

      • Jim Fearon October 4, 2012 at 2:30 pm #

        But what about the answer to this question? Doesn’t your argument require a pretty ridiculous level of altruism? If you want to explain turnout as instrumentally rational, you have to think that the value individuals put on Obama vs Romney is really massive (because your chance of tipping things are so small). Your point is to say that people put weight on other people’s welfare or the collective good, so this can be very large. But doesn’t that imply that if given the option to pay to decide the election by yourself, you would be willing to pay a fantastical amount? Is that plausible?

        I’m not saying that people don’t vote with a mind to the collective good, but that this is unlikely to make voting (in most cases) purely instrumentally rational. Also, if you think people are motivated by a general desire for the collective good, why not assume that they can be motivated to vote just because they think it’s the right thing to do? (which seems to me like the obviously right answer for most of this puzzle anyway.)

        • Andrew Gelman October 4, 2012 at 2:42 pm #


          We have some numbers on page 297 of our paper and some math on pages 298-300. That said, our calculations, like any of this sort of utility calculations in political behavior or international relations or anywhere else, are inevitably suggestive rather than precise. People vote because they think it’s the right thing to do, but it also can be subjectively instrumentally rational. Why does this matter? For one thing, just as it can be rational for a swing-state resident with strong opinions to go and vote, it can be rational for a campaign to try to change the voting or turnout decision of one potential voter: even if there’s only a 1 in 10 million chance of this decision making a difference, it can be worth it. And people respond to such appeals. In sections 3 and 4 of our article, we discuss supporting evidence for our theory, and in section 5 we discuss the compatibility of rational-choice and more traditional psychological models for voting. I think these different explanations are complementary, not competing. Even though I think voting in the presidential election can be instrumentally rational (although not for me! I live in New York), I don’t think the rational argument is why people vote. People vote for all sorts of motivations, but the rationality of the individual vote underlies many of the other reasons, for example people being motivated by active campaigns.

          • Larry Bartels October 4, 2012 at 3:14 pm #

            I’m not sure how “the rationality of the individual vote underlies many of the other reasons, for example, people being motivated by active campaigns.” Do you suppose that people who vote expressively cannot be mobilized by campaign appeals? Or is “rationality” here not just “instrumental rationality”?
            Comparing the turnout rates in swing states and uncompetitive states should be sufficient to suggest that most voters are motivated by duty or expressive values rather than instrumental rationality, however altruistic that may be (or else that they are wildly wrong about their likelihood of being pivotal). Of course, they could be pivotal in some lower-level election, but that is also unlikely, and many of them don’t bother to vote in those races when there is no presidential election on the ballot.

            • Ted Brader October 4, 2012 at 3:57 pm #

              Andrew, I did take a (very hasty) look at the source material. It’s an interesting approach, but, to the extent I found it compelling, it was in seeing your dollar value attached to the preferred outcome as more figurative than literal (more psychic benefits than material) — and thus expressive voting by another name (at least if expressive voting includes not just officially voicing a preference, but also the satisfaction in seeing that preference realized). But perhaps you would insist the person’s utility in your model is based on material outcomes (just for others as well as self)? If so, then doesn’t your model assume a level of other-regardingness such that people value benefits to all other individuals in the society equally to benefits to themselves? I believe in the existence of other-regarding motives, but I’m not sure this fits the typical motivational profile of my fellow Americans.

              • Andrew Gelman October 4, 2012 at 4:01 pm #


                Yes, I’d buy the “expressive voting by another name” description.

            • Andrew Gelman October 4, 2012 at 4:01 pm #


              Indeed, I’m careful to say that voting “can” be rational, not that rationality is the prime reason for people to vote, even in swing states.

              • Theorist October 4, 2012 at 10:23 pm #

                You’ve still never answered Fearon’s question above (glad to see there are game theorists about). Your argument boils down to: the value of providing the decisive vote is extremely large, while the probability your vote will be decisive is extremely small and the cost of voting is small. This means that you should also be willing to pay a very large amount to directly influence the outcome, which seems to not be the case.

                • Andrew Gelman October 4, 2012 at 10:29 pm #


                  Please read the linked paper. The answer is that the outcome of the election really is a big deal, given its implications for the world. The purpose of the above blog post was to criticize the innumeracy of Mangu-Ward’s article. There is no point in me trying to explain in a series of blog comments what my coauthors and I carefully wrote in an article published several years ago.

                  • Theorist October 5, 2012 at 12:57 am #

                    I have now read the article. The point several of us are trying to make still stands. You just pick a number and ask us to imagine that: 1) it represents the value of the “right” president and 2) that the amount by which voters discount the summed values is not so large as to make voting non-rational.

                    Your calculations are perfectly correct so far as they go in th points about electorate size but what I and others are pointing out is that assumption 2 implies things that do not appear reasonable. Say we accept your $1000 number in the short piece (and asssume risk neutrality) and the conclusion that you should then vote if you value $60,000 in benefits to others more than you value the time you spend voting. Suppose voting costs you as little as $10, what you’re suggesting then is that voters have at worst a 6000:1 altruism trade off and should be willing to spend $50 million to buy an election with certainty. No one believes this is true. Now, you can try to beg off with a story about declining marginal utility for charity, in which case you’d at least have an answer, but constantly pointing to just how big you’ve assumed the value to be isn’t convincing.

                    Your entire argument boils down to assuming the expected value of an outcome is ENORMOUS and then pointing out that an enormous number multiplied by a small number can still be quite large. The trouble is that we all agree (roughly) that the small number is quite small, but no one else finds your enormous number convincing, given that your only derivation involves the fact that you can get an enormous number by multiplying a large one and a medium one.

                    If instead of starting with a value like $1000, you start with an equally made up one like $0.10, then your argument falls part. Realistically, even if people are altruistic, they are also risk averse, uncertain about how electoral choices influence welfare, and highly inclined to discount the future. This makes your assumption about how highly they value changing an outcome highly suspect. Now, I may be wrong and you may be right but you’ve presented no evidence whatsoever. I know that if I had $50 million, I wouldn’t be willing to spend it to reverse the outcome of Presidential elections where I think the “wrong” guy won. When I think about it, I’d be willing to hypothetically spend about a few thousand dollars of my own money to reverse the 2000 election, which is orders of magnitude off what your assumptions suggest I should be willing to pay, and I’d guess my figure is higher than most people’s.

                    • Andrew Gelman October 5, 2012 at 1:36 am #


                      1. You write that “no one else” finds my analysis convincing. That’s not true. According to Google Scholar, our paper has been cited 118 times. Not all those citations are by my colleagues and myself, nor do all the citations by others represent disagreement with us. I respect that you do not find our argument convincing, but not everyone agrees with you either!

                      2. I am skeptical about your thought experiments (at least, to the extent they apply to people other than you). I think it is difficult to assess the amount of money someone might spend to singlehandedly reverse an election outcome, because this is a choice that none of us (Sandra Day O’Connor aside) will ever have. To the extent that we believe that the election of our preferred candidate would give benefits to the general population, I think it makes complete sense to note that those benefits scale with N, just as the probability of a decisive vote scales with 1/N, and I find the numbers in our paper to be reasonable. As we discuss in our article, I think it makes sense to speak of voting as rational (in some settings) without implying that rational decision making is the reason why people turn out to vote.

                      3. We do provide evidence in our paper. It is qualitative not quantitative evidence but it is evidence nonetheless. One bit of evidence is that many people will send in campaign contributions such as $20 to a candidate. This is consistent with the idea that swinging an election can be important.

                      In any case, I respect your disagreement. I can’t convince everyone, I just want to get the idea out there and allow people to make their own judgments.

                    • matt w October 5, 2012 at 10:28 am #

                      “I think it is difficult to assess the amount of money someone might spend to singlehandedly reverse an election outcome, because this is a choice that none of us (Sandra Day O’Connor aside) will ever have.”

                      This is a good point. There’s a related (I think) problem with the formulation of the original “Would you spend $10 million to change the outcome of the election?” problem, which is that it doesn’t take into account the decreasing marginal utility of money, or rather the increasing marginal disutility of spending money. I don’t have $10 million, so the marginal disutility of spending $10 million is effectively infinite for me — no matter how much utility spending $10 million would give me, I can’t do it.

                      This means that we can’t blithely take the estimated monetary cost of voting and divide it by the probability of affecting the election in order to arrive at the value people place on changing the election. You’d have to take the estimated utility cost of voting, divide it by the probability of affecting the election, and convert that back into money.

                      For what it’s worth, Sheldon Adelson has spent more that $10 million for considerably less than a 100% chance of changing the election outcome. But the marginal disutility of spending $10 million to him is probably less than a million times the marginal disutility of standing in line for an hour.

                      (I also think that a conception of rationality that fails to deal with collective action problems is ipso facto flawed, but that’s not really germane to the problem.)

  4. R Barron October 4, 2012 at 12:44 pm #

    I often use the analogy in these discussions that casting the vote that wins the election is probably about as likely as firing the bullet that wins the war. But does that mean the efforts of an individual soldiers are wasted? In other words, voting will always be irrational from an individualist standpoint, but is highly rational from a view.

    • R Barron October 4, 2012 at 12:45 pm #

      That was supposed to say “from a collectivist view”

      • dL October 5, 2012 at 6:08 am #

        In WWI, only 15-20% of American soldiers actually fired their weapons in combat. Today, the ratio is pretty close to 100%. So, a century of the progressive state gives us a much greater efficiency regarding the willingness of soldiers to participate in mass murder.

        No doubt, some things definitely require the collectivist action of the State.

        The point: piss poor analogy

  5. Joe Ura October 4, 2012 at 12:58 pm #

    Voting isn’t casino gambling. There is no fixed probability of being the decisive voter. My choice about whether or not to vote is a strategic decision that is made conditional on my expectations about the behavior of others, which are condition on their expectations of my behavior. If no one is expected to vote, incentives for any one person to turnout are huge. Once one person votes, there are large incentives for any one person who prefers the other candidate (in a two party race) to vote. And so on. Substantial voter turnout is consistent with rational individual behavior in (mixed strategy) equilibrium without recourse to arguments based an assumption of massive difference in individual utility related to the outcome of the election.

    • Larry Bartels October 4, 2012 at 3:15 pm #

      “And so on” seems to be doing a great deal of work here.

      • Joe Ura October 4, 2012 at 5:37 pm #

        Fair enough. My note also blurred the jump from pure strategies to mixed. The point remains, though, that there can be instrumentally rational voter turnout without making seemingly (to me) unrealistic assumptions about the value a probabilistically pivotal voter places on her more preferred candidate defeating her less preferred candidate.

  6. will October 4, 2012 at 1:53 pm #

    The people arguing on Twitter over not voting for a preferred candidate (Friedersdorf for instance) are more likely to sway the election than the average Joe, having as they do an audience of thousands of potentially persuadable people.

  7. david mizner October 4, 2012 at 4:41 pm #

    Math aside….

    People don’t go to the polls because they’ll think their vote with be decisive, or because they think what they do will influence what other people do, but because they want to do their part. They don’t want to be free-riders like the oh-so-rational economists who let others do the work for them. Likewise, whether or not recycle doesn’t make a difference to the environment but we recycle because we’d rather be part of the solution than part of the problem. That’s the essence of social responsibility. If that’s irrational, I’m glad to be irrational.

    • idiot October 4, 2012 at 5:27 pm #

      Doesn’t social responsibility require thinking through the consequences of potential actions and doing things that are actually productive or worthwhile? It’s not a choice between voting or not voting, it’s a choice between voting or cleaning up litter or saying a kind word to someone or teaching someone about politics or getting 10 other people to vote, or…

  8. Will October 5, 2012 at 12:18 am #

    Seriously? Were Riker and Ordeshook inumerate? Is that why their paper following much the same logic has been cited so extensively? You may disagree with Mangu-Wards analysis and you do indeed have some logic on your side but that doesn’t make your opponent an idiot for embracing a very influential position from the literature. I read your linked paper and I found your analysis silly and unconvincing – should I go around calling you illiterate?

  9. Andrew Gelman October 5, 2012 at 12:31 am #


    I didn’t say that Mangu-Ward was an idiot, I just said that her article showed innumeracy in two places, first by reporting with a straight face the claimed probability of “4.77×10 to the −2,650th power” and second by claiming that rich people are not more likely to vote Republican.

  10. joe October 5, 2012 at 3:58 am #

    A few things:

    * The difference with the vaccine case is that not getting a vaccination (or littering, or whatever) makes a actual, real difference in the world. In the voting case, the counter-factual world is the same as our world. If you don’t vote, nothing changes.

    * This ‘but if everyone did it’ fallacy, is just that–a fallacy. In what magical world do we live that my actions suddenly change the actions of everyone else. Yes, if no one voted, really bad things would happen. But you can agree with that and *simultaneously* contend that your vote doesn’t matter.

    * The idea that because the well-educated classes vote should be evidence of rationally is not at all convincing. There was a time in which the well-educated thought slavery was not that big of a deal and the world is flat. If voter turnout is a result of cultural factors, I hardly find that as a convincing reason that one should vote.

    * There is certainly an argument that Kathrine should have kept her mouth shut. If it is a collective action problem, then even if it is irrational to vote, no one should be trying to convince others not to do so. (Though it sounds like she is also referring to people who actually harm the world (in the 1 in a million sense) with uninformed voting.

    * I think your paper discovered why rational-decision theory shouldn’t use linear equations for low-probability/high-impact events. So we see that the social ‘benefit’ is independent of N: B*N/(alpha(N/3) = 3*B/alpha. It’s not hard to reduce this to an absurdity:

    Say you find a magic lamp and a genie pops out. “I’ll give 1 million dollars of wealth for every person living now and until the end of all time (sorry, none for you). I know that all those people sum to 10 trillion. The catch is you have to play my lottery by standing in this line for 5 hours, but you seem like a nice guy, I’ll cut the odds by 1/3rd. What’s the probability you ask? 3.3 trillion-to-one. But don’t worry about that, just check out my rational-choice model, you’re really conferring an expected benefit of 30 million dollars–that’s like a 30 million dollar donation! In fact, if you don’t win, you probably should play at least once a week, I mean you’re giving 30 million dollars to your fellow citizens. I know you think it’s stupid, but look at the math!”

    * The above sounds silly, but remember we had one of these calculations before. CERN was supposed to destroy the world with hyperstable strangelets and all sorts of exotic matter. “We should not do the experiment!” the wackos argued, “because the costs are so high even for such low probabilities–we are talking about the end of the world!!” Well, the problem is rationality has a threshold or is not linear, at these extremely low-probability events.

    There may be reasons to vote, but if you told me that you threw me in 6 million universes and my vote wouldn’t matter, but I should still do it because I am conferring a societal benefit, I wouldn’t say it’s our actions that are wrong, but your math.

    • Andrew Gelman October 5, 2012 at 8:50 am #


      The difference between your story and mine is there is no such thing as a genie but there are such things as economic policy, foreign policy, and social policy. The Democrats and Republicans indeed do differ on many importance issues.

      • joe October 5, 2012 at 1:48 pm #

        So you agree that *IF* the genie did exist it would be irrational to sit in line for 5 hours once a week?

        Philosophers (and even scientists and especially mathematicians) do this *all the time* to properly grasp the limits and concepts of reasoning and theories. In fact, I’m pretty sure if you survey many reductio ad absurdums you’ll find most of them don’t seem quite possible. Yes, maybe god doesn’t exist, maybe you could never have the chance to kill Hitler, and when will you ever be able to change the path of a train to kill 1 instead of 5? These things just don’t happen, that doesn’t mean you can’t make sense of them and assess proper justification. So *if* there was a genie, your math would fail miserably. Sure, there’s no such thing as a genie, but the fact it fails so bad in the hypothetical case is proper justification for a failed model, the burden of proof is on you provide evidence for why we should trust your model at that probability level given the fact that it does eventually fail.

        Say I’m Einstein, thinking about the special theory of relativity. I have a model that comes to a conclusion that E=MC2. But to come to that conclusion I have to evoke the idea of a person flying in a ship at the speed of light. (He really thought this by the way.) This is an impossibility, but I’m not going to say, “That’s just silly, speed of light cars don’t exist, in fact my very theory proves that any object traveling at c is an impossibility. Forget this, I need to test my model in more reasonable ways.” I’m sorry, but ‘our model only works for situations where everyone has agreed beforehand that the entities under consideration pass the test of metaphysical realism’ is a gaping flaw. There are numerous models (economic, physical, chemical) that scale fine (without absurdities) into impossible regimes and those that do break down, we have proper justification for believing why it is to be trusted near those regimes (e.g experimental evidence), but all you have is a crude model.

        But it turns out it’s very easy to modify the reductio to accommodate your unreasonable requirement. What you are essentially saying with your model is that low-outcome-probability actions can be rational (in terms of rational-choice) if the social benefit (B*N) is large enough, as long as it doesn’t fall below our opportunity cost (scaled by a discount). The reason why your model seems to work at the low end and fail at the high is because it contradicts with our intuition. Now, intuition is a horribly unreliable, but in this case by careful selection of numbers we can make our intuition so strong that we *at least* have to question how we are defining ‘rationality’ in these extremely-low-probability regimes (is rationality really ‘let me show you this LINEAR equation’?), and this provides ample evidence to place the burden of proof on you to prove why it works at 1-in-six million since it doesn’t work at 1-in-10 trillion. So scratch the genie. The people of the world suddenly experience a massive love for the united states and they collectively want to funnel all their economic resources to every citizen of the US, BUT only if you play the 5 hour stand-in-line-lottery. The odds? 1/n or about 6.9 billion-to-one. But it’s ok you’re making a donation to society of ~150K. Come’on this has to be worth it to you at any reasonable alpha. Or we can picture a future where there are 10 trillion individuals across the galaxy, two candidates and any significant difference between the two (the odds: 1-in-10 trillion). Or take your case, but assume we vote for president every day, or every week, now let’s go convince the public that they should keep standing in line for 5 hours because the societal benefit is so high. Or we can change it from the other direction: Say person A uses your model and calculates a scaled $1000 societal benefit, I’ll donate $500 to society if A doens’t vote (or how about I find their discount and offer just $1 under). (It least it seems like you need a term which accounts for real opportunity costs at the high end.)

        These really are absurdities. I know you have a vested interest in your model, after all it is yours, and so it’s harder for you to see the absurdity, but that doesn’t mean it’s not absurd. So what we have is a conflict between the intuition of most people and your model. It definitely could be that our intuitions are wrong, they often are. But in this case the math is so shaky and the intuition can be scaled to absurd proportions that you really have an uphill battle trying to make this convincing even at the current levels, let alone for populations eons in the future.

  11. David Karger October 5, 2012 at 10:08 am #

    Regarding innumeracy number 1, without doing any math myself, I’m going to guess that Brennan’s calculation was based on estimating the probability of a tie when you flip several million 50.5%-biased coins. The stated number may be correct for that experiment, though of course it doesn’t have much to do with the actual situation being discussed.

    More importantly, I think the focus, in the discussion of rationality, on multiplying the low odds of impact by the large value of having an impact, misses an equally important issue: the *repeated game* aspect of the voting system. Sure, it may have no impact for me to vote this year. But if (say) rational votes decide *as a block* not to vote, then in future years politicians can discount the entire block of rational voters. (Wait, maybe that’s what they’re already doing!)

    • Andrew Gelman November 25, 2012 at 5:33 pm #


      I agree. Also there’s a repeated game aspect for each individual, in that voting is a habit: once someone’s voted, he or she is likely to continue voting in future elections.

  12. Acilius October 5, 2012 at 10:49 am #

    I think a couple of things are upside down in this discussion. The first, of course, is the relative weight of a scholarly publication as compared with a series of blog posts. Obviously people who disagree with your analysis should read your article and review the literature that engages with it.

    Also, I’d say that while instrumental voting may be sometimes be more rational than your opponents argue, it is almost never as rational as expressive voting. Considering how thorough the post-election analyses are that advocacy groups, campaigns, parties, and other organizations sponsor these days, even the tiniest bloc of voters is likely to be noticed and accounted for in future electoral strategizing.

  13. DavidT October 5, 2012 at 11:24 am #

    I live in Illinois, which is a safe Obama state, and which has no gubernatorial or US Senate elections this year. Not only that but I live in a safe Democratic congressional district, and safe Democratic state legislative districts (for both the state senate and state house of representatives).

    So why am I voting? Well, for one thing it is hardly a burden. I will be voting early, at my local library, which I like to go to anyway. There are unlikely to be long lines (there weren’t last time when I voted early) and it’s even a chance to socialize a bit–chat about the weather or whatever with the Democratic and Republican election judges present and sometimes with neighbors I meet there.

    Now of course even if voting were harder and more unpleasant, I would no doubt do it for the usual “sense of social responsibility” reasons. But I think that the “why do people vote?” questions are too quick to assume that voting is always a burden, and must therefore either find some sort of instrumental justification or else be irrational. Really, it’s no burden at all for me. I don’t have anything better to do with the few minutes it takes me.

  14. Joel October 5, 2012 at 4:43 pm #

    There’s an inherent problem with asserting “rational” voting in the context of a decision theoretic model, when the act is clearly strategic. What we have here is a debate between two sides, both of whom ignore the strategic interaction, over the appropriate way of calculating an expected utility.

    Palfrey and Rosenthal (1983) show that when accounting for the strategic nature of the interaction there are equilibria to a complete information voting game that feature high turnout, high costs of voting, and purely selfish utility functions (it is interesting how their 1985 paper modifies this insight under incomplete information). In short, one can explain rational voting without appeals to altruism or questionable assumptions about what one side or another winning means for the country.

    • Andrew Gelman October 5, 2012 at 8:52 pm #


      Given that people do express sociotropic reasons for voting, and that it is more rational to choose one’s vote based on such reasons (that factor of N=300 million), I see no particular reason to explain voting using “purely selfish utility functions.” You can do so if you want but it doesn’t make much sense to me.

      • Joel October 7, 2012 at 2:27 pm #

        For what it’s worth, the proof from Palfrey and Rosenthal trivially generalizes to the sociotropic preferences that interest you. The basic point is simply that once we see voting as strategic, the paradox of participation ceases to be a paradox at all.

  15. Andrea Cohen October 7, 2012 at 4:28 pm #

    It’s interesting to see that someone advocates not voting in an election. As you pointed out, the chances of someone’s vote being decisive is 1 in 80,000 for congressional elections, or 1 in 10 million for the Presidential elections, but that doesn’t mean that people shouldn’t vote. Voting should be a civic duty. Although people think that their vote may not matter, especially in a non-swing state, everyone should still vote. Voting shows that you care about American politics and what is best for your country. Plus every vote is important and can make a difference, just look at the Florida in the 2000 elections. Bush didn’t win Florida by that much and a couple hundred votes could have changed the elections. If more people had come out and voted, America could have had a very different future. The real problem though is that many don’t place enough importance on voting, they don’t think it really matters. But we have to remember that not too long ago, not everyone could vote and voting to those who couldn’t was seen as a very big thing. We have to honor that and remember that we can make a difference if we do exercise our right to vote.

  16. Peter Bridgman October 9, 2012 at 4:57 pm #

    I cannot vote in the Obama-Romney race. I am neither resident in, nor citizen or the USA.

    In Australia, where I can and do, with pride, vote, voting is not a statistical issue, but a legal and civic duty: it is compulsory to enrol to vote and compulsary to vote in all elections for all three levels of government. The ides of compulsion is very decisive among Aamericans in my experience. I lived in the US during the Reagan-Carter election and was touring during the 07 primary season. Talk of compulsion was greeted both times, almost 30 years apart, with horror!

    But why should this be? Voter turnout in Australia ranges from high 80% to high 90% figures. (The law is a poor behaviour change tool, eh?). Recusants are liable to a fine unless a reasonable excuse is provided (eg I was resident in the US during the 1980 Australian Federal election and unable to arrange for absentee voting).

    The net result is that the populace knows that the elected government is the result of what most of the overwhelming majority of the voters chose. (Apologies for vague non-probabilistic language.) Ronald Reagan won 51% of 53% of the eligible voters: almost 3/4 of eligible adults opposed or couldn’t have been bother.

    The occasional talk of a shift to voluntary voting in Australia is invariably driven by perceived political advantage, suppositional assumptions such as wealth, capacity and preference.

    Here, every vote counts because it is a bolster to the strength of the body politic and the probability of “my” vote determining the outcome of an election is put in its proper context of elections being collective decision making, not mere individual determinations.

    (As an aside someone for some reason polled Aussies on your presidential race. The figures favour Obama 80-9, the balance presumably undecided. Turnout would not be a problem for Obama either!)

  17. Suresh October 12, 2012 at 8:37 pm #

    Perhaps I missed it above if it was discussed (apologies), and one commenter did allude to it, but it seems to me that something is missing in this analysis. If a group of people collectively decides not to vote because their vote does not count, and if this group disproportionately belongs to one camp, then even an almost-certain contest starts to become a swing-contest, and the probability of the individual vote being important goes up. Surely this probability of a mass drop-out should come into play in the analysis ? And perhaps once that is taken into account, then people realize that they cannot ask others to vote if they do not themselves vote… they cannot rely on the other supporters of their position to make sure the eventual decision turns in their favor. There is probably yet another reason, which is that an overwhelming turnout and majority may lead to more confidence that decisions taken later by the elected officials enjoy mass confidence….

  18. Bill Kittredge October 14, 2012 at 9:41 pm #

    I think most of this misses the point. ‘Rational’ voting depends on the definition of ‘rational’. That definition is not well settled nor, from reading the posts, consistently applied in the discussion. In its economic sense, it has a much more limited meaning than when used in other contexts. Simon pointed out that, even if one were to assume the very limited definition, most decisions do not lend themselves to rationality.

    The second point I’d make involves the choices available in our two party system. Even assuming rationality in a limited sense, any voter is faced with severe compromise and uncertainty. Compromise because it is unlikely that any candidate reflects the views of the individual voter precisely. Uncertainty because at the time of voting, one cannot know what situations will be faced by the candidate once in office. Conditions change, perspectives change. No one was calling for banking reform during the boom, it was only after the collapse that this surfaced as an issue.

    In short, I think the discussion of rationality with respect to humans is irrelevant because we are not rational beings.

    • Andrew Gelman November 25, 2012 at 5:31 pm #


      You write, “the discussion of rationality with respect to humans is irrelevant because we are not rational beings.” Humans are not exclusively rational, but rationality is part of who we are. We are sometimes rational, and rationality is also a baseline that we can use to judge our actions.

  19. Henry Moore November 25, 2012 at 4:54 pm #

    “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.”

    They can be concerned all they want. The problem is that they are often wrong. Does the average voter (or, perhaps, any voter, myself included) have the slightest clue what’s best for the country? And is it at all possible that what they are really doing is projecting their self-interest (usually in the very short term, if at all)?

    • Andrew Gelman November 25, 2012 at 5:30 pm #


      You may be right about that. I’m only addressing the individual decision of whether to vote; I’m not addressing the quality of that vote.

  20. peter nelson May 7, 2013 at 1:18 pm #

    The problem with comparing voting, e.g. for President to a lottery – low probability, big payoff – is that in the entire history of elections there has never been a case where one voter determined the outcome. In other words, you’re comparing it to a lottery that no one has ever won in over 200 years since the lottery was established! Real lotteries do have winners. So from my viewpoint the chances of winning the voter lottery are close enough to zero that the consequences of my candidate or party winning are irrelevant. So I fail to see any rational basis for voting in a major race like US President. I do vote in local town elections where winning margins are often a few dozen votes.

    • Andrew Gelman June 18, 2013 at 8:14 am #


      Small probability, big outcome. Regarding your lottery comparison, what is relevant is not the probability that somebody casts a decisive vote, it’s the probability that you cast a decisive vote. The difference with the lottery is that, in the election, if you cast the decisive vote, so did millions of other voters. In a swing state in a somewhat close presidential election, the probability of your vote being decisive is approx 1 in 10 million, which is not zero considering the huge stakes involved.

      That said, if you live in NY or TX, I agree the probability is essentially zero, and there you wouldn’t be voting for president for instrumental reasons, at least not in the general election.