Why it can be rational to vote

I think I can best do my civic duty by running this one every Election Day, just like Art Buchwald on Thanksgiving. . . .

With a national election coming up, and with the publicity at its maximum, now is a good time to ask, is it rational for you to vote? And, by extension, wass it worth your while to pay attention to whatever the candidates and party leaders have been saying for the year or so? With a chance of casting a decisive vote that is comparable to the chance of winning the lottery, what is the gain from being a good citizen and casting your vote?

The short answer is, quite a lot. First the bad news. With 100 million voters, your chance that your vote will be decisive–even if the national election is predicted to be reasonably close–is, at best, 1 in a million in a battleground district and much less in a noncompetitive district such as where I live. (The calculation is based on the chance that your district’s vote will be exactly tied, along with the chance that your district’s electoral vote is necessary for one party or the other to take control of a house of congress. Both these conditions are necessary for your vote to be decisive.) So voting doesn’t seem like such a good investment.

But here’s the good news. If your vote is decisive, it will make a difference for 300 million people. If you think your preferred candidate could bring the equivalent of a $50 improvement in the quality of life to the average American–not an implausible hope, given the size of the Federal budget and the impact of decisions in foreign policy, health, the courts, and other areas–you’re now buying a $1.5 billion lottery ticket. With this payoff, a 1 in 10 million chance of being decisive isn’t bad odds.

And many people do see it that way. Surveys show that voters choose based on who they think will do better for the country as a whole, rather than their personal betterment. Indeed, when it comes to voting, it is irrational to be selfish, but if you care how others are affected, it’s a smart calculation to cast your ballot, because the returns to voting are so high for everyone if you are decisive. Voting and vote choice (including related actions such as the decision to gather information in order to make an informed vote) are rational in large elections only to the extent that voters are not selfish.

That’s also the reason for contributing money to a candidate: Large contributions, or contributions to local elections, could conceivably be justified as providing access or the opportunity to directly influence policy. But small-dollar contributions to national elections, like voting, can be better motivated by the possibility of large social benefit than by any direct benefit to you. Such civically motivated behavior is consistent with both small and large anonymous contributions to charity.

The social benefit from voting also explains the declining response rates in opinion polls. In the 1950s, when mass opinion polling was rare, we would argue that it was more rational to respond to a survey than to vote in an election: for example, as one of 1000 respondents to a Gallup poll, there was a real chance that your response could noticeably affect the poll numbers (for example, changing a poll result from 49% to 50%). Nowadays, polls are so common that a telephone poll was done recently to estimate how often individuals are surveyed (the answer was about once per year). It is thus unlikely that a response to a single survey will have much impact.

So, yes, if you are in a district or state that might be close, it is rational to vote.

For further details, see our articles in Rationality and Society and The Economist’s Voice.

I’d like to add one more thing. You’ve all heard about low voter turnout in America, but, among well-educated, older white people, turnout is around 90% in presidential elections. Some economists treat this as a source of amusement–and, sure, I’d be the first to admit that well-educated, older white people have done a lot of damage to this country–but it’s a funny thing . . . Usually economists tend not to question the actions of this particular demographic. I’m not saying that the high turnout of these people (e.g., me) is evidence that voting is rational. But I would hope that it would cause some economists to think twice before characterizing voting as irrational or laughable.

(And, no, it’s not true that “the closer an election is, the more likely that its outcome will be taken out of the voters’ hands.” See the appendix on the last page of this article for a full explanation, with calculus!)

18 thoughts on “Why it can be rational to vote

    • Simon,

      Thanks for the pointer. It’s a bit annoying, though, that he links to my paper while attributing it to my coauthor alone!

      Also, I don’t at all understand why he talks about the probability of a simultaneous tie in several states. A tie in one state is all that’s needed.

    • That could be. There are a lot of acts that are rational and unethical. Actually, I’d describe rationality and ethics as two orthogonal axes. One can use rational or irrational means to pursue ethical or unethical ends.

      • I don’t think that rationality and ethics are completely orthogonal. That might be true in a single-stage single-player decision making process but it is not necessarily true in multistage multi-player games. That is because on the last ones unethical behavior might involve reputation costs. And it all depends on what exactly the player aims to achieve with his decision. One of his objectives might be to behave morally. In this case rationality and ethics are again not orthogonal.

        • Yes, good point. Also there is an interaction, in the sense that, if one has moral goals, it can be more moral to pursue them purposefully and rationally. Conversely, pursuing immoral goals rationally seems particularly immoral.

        • That is an excellent point. Additionally, pursing immoral goals rationally seems to imply random, seemingly random and/or deterministic deviations from immorality. But that the same is not true when one has moral goals. What do you think?

  1. Hi Andrew, I just read your 2004 blog “why it is rational to vote” and the linked paper at columbia.edu. I have to dispute your statement that the probability of decisive vote is proportional to 1/n. On the simplest model it is more like 1/sqrt(n). So in an election with 100 million voters the chance of a numerical tie is more like 1 in 10,000 not 1 in 10 million.

    I just calculated this, and the result surprised me, but see what you think. Assume that each of n voters is equally likely to vote for each of two candidates. What is the chance that exactly n/2 votes are cast for each? This is just the chance of getting n/2 heads in n tosses of a fair coin and is just n!/(2^n.((n/2)!)^2). Using Stirling’s approximation this simplifies to sqrt(2/n.pi). (This is easy to verify with an example. For 100 tosses of a coin we have 100!/(2^100.50!^2) = 0.07958. The formula gives 0.07978).

    Is the assumption of random equiprobable voting reasonable? It seems the natural assumption for a voter to make if they have no information on how other people will vote. Additional information could make the probability of a tie go either way. Eg if the polls are 60/40 the probability of a tie on a random model is reduced. However if of 100 votes there are 40 declared “tails” voters and 40 declared “heads” voters, and only 20 swing voters assumed to be voting at random then the probability of tie is increased. So it seems a reasonable first approximation for a close election. Perhaps it is not so irrational to vote, even just on the basis of personal gain, after all?

    • Gareth,

      No, the model you have is not an appropriate model for uncertainty in an election. Put it this way: suppose voters have probability p of voting for the Republican. Then all that matters is uncertainty in p, and that needs a forecast distribution. You are assuming p=1/2 which unrealistic. See our Stat Sci paper and our BJPS paper for more.

  2. Yes, I just clicked through to that paper as you were typing! I still think it is interesting that the probability of a tie can be as high as 1/sqrt(n) which a lot different to 1/n. I will read the whole paper later. cheers.

    • OK, I have read your B.J.Pol.S paper and you have convinced me. I am a physicist by training and I like simple models, but empirical evidence always trumps theory. (I have recently been dabbling in genetics and that has led me to an interest in Bayesian statistics, which explains why I am up at midnight (UK time), drunk in charge of an R interpreter and bothering statisticians on the internet)

      I always like to understand, when a simple model is not right, why it is not right. So I particularly like your demonstration that if you randomize districts across states you do get a sqrt(n) rule. I think there is an important political point here: If the 38 “Vermont sized districts” of California were entirely independent, then what would be the use of lumping them into “California”? It is because Californians have a lot in common that they do not obey the sqrt(n) law. A lot of people miss the fact that effective democratic representation depends on a common sense of identity. That needs to work at all levels of representation, and that is why the USA works as a political union and the EU probably never will. Americans all identify as American. But there just is not a sense of European identity that can persuade individual nations to accept measures they do not like for the common good.

      Anyway, I came to your blog on the question of the “voting paradox” – that it seems irrational to bother to vote at all. I like your explanation in terms of total utility – but it seems to me the essence of the paradox is not statistical but logical. The problem is in going from a statistical statement such as “the probability of the election being tied, in the absence of my vote, is very small” to a practical maxim such as “it is not worth my while voting”. It does not matter much what utilitarian calculus is used. The problem is more obvious if you are, say, a democrat sympathizer in a “blue” state. The election is “certain” to go your way, so it is not worth your time turning up to vote. But if every democrat reasoned that way, none of them would vote and the republicans would win. There has to be something wrong with a maxim of practical reasoning which, if everyone adopted it, would refute itself. We seem to be entering the territory of Kant’s categorical imperative: “Act only according to that maxim whereby you can, at the same time, will that it should become a universal law” (Though this is hypothetical imperative, based on the assumption you want a particular candidate to win).

      I think the essence of the problem is an artificial division of the observer from the observed. I assume that everyone else’s vote is predetermined. I don’t know exactly what it is, so I am entitled to assign it a probability distribution. But my vote is still be determined. I can decide to turn up and vote, or not, as a matter of free will. But in reality I am in the same boat as everyone else. If I have not yet decided if it is worthwhile voting or not, then neither has anyone else. And if I can decide not to bother on the basis of a statistical argument, so can everyone else, whatever they may have said to opinion pollsters.

      The reason to vote is: “because I am nobody special, and if I don’t vote, why should anyone?”

      I have always been suspicious of the idea that “rationality” can be defined as trying to maximise some kind of utility function. I think it is more complicated than that.


  3. I brought this up on Marginal Revolution’s recent post but nobody responded and I’m still curious.

    Isn’t it the case that you are as likely to be a -$250 voter as a +$250 voter in this model?

    So, the expected value to vote for each voter across the population is therefore $0?

    You’d need some sort of superior evidence to base your belief that your vote is the positive one?

    This isn’t then an argument that is applicable to people in general? Only to the rare few that are actually right, whomever they might be?

    • Ben:

      Consider two voters, let’s call them Greg (Mankiw) and Paul (Krugman), and let’s suppose they both live in Ohio so they have a reasonable chance of casting a decisive vote. Greg sees a huge expected social gain from a Romney win; Paul sees this gain from an Obama win. From either of their perspective, it makes sense to vote. They have different models of the world.

      • Your vote can be decisive only if everyone else is tied. That gives you no reason not to trust your own judgement. So you should vote.

  4. Andrew: Interesting symmetry! If you are (a) global minded, so you care about everyone’s benefit equally (and equal to your own), and (b) paternalistic, so you believe that *you* know what’s best for everyone, and (c) voting in an election where the outcome matters at the $50 level to everyone, and (d) voting in an election where every one of the N people has a 1/N probability of casting the decisive vote, then your expected utility is pretty much $50, independent of N. That is sweet! Or obviously wrong. I am not sure which. Of course assumption (a) is absurd.

    One question I have though is this: If the election *is* won by one vote, doesn’t *everyone* believe they cast the decisive ballot? And are they all correct? And is that properly factored in? I guess I should read your papers…

    • Nice observation! In fact, it seems like they would be correct in that belief. More generally, if the election is won by N votes, then *everyone* who voted for the winner should believe there was 1 chance in N that they cast the decisive vote.

  5. Your calculation is off by a factor of 10. 50 * 300MM = 15B, not 1.5B.

    More importantly, it falls victim to the same fallacy explained by David Mackay in his energy science book: http://www.inference.phy.cam.ac.uk/withouthotair/c19/page_114.shtml

    In short, you’re aggregating to create a large effect, but then individualizing it to overstate its size. Yeah, your 15B lotto ticket with 1-in-10MM odds seems like it’s net $1500 expected value, but it’s net $1500 aggregated from the entire country. It’s actually worth $5 * 10^-6 per person. Not nearly as attractive.

    • Theory:

      No, it’s not $1500 aggregated from the entire country. The idea is that, if you’re Krugman or Mankiw and have strong feelings that one candidate’s policies are much better than the others, then you’re talking about a general economic benefit that could be worth a lot per person. Casting a decisive vote is very unlikely, but if it happens, you’re talking about making a huge difference in the world.

Comments are closed.