The short answer is, quite a lot. First the bad news. With over 100 million voters, your chance that your vote will be decisive in the presidential election is, at best, 1 in a million in a battleground state and much less in a noncompetitive state such as New York. The calculation is based on the chance that your state’s vote will be exactly tied, along with the chance that your state’s electoral votes are necessary for one party or the other to secure an electoral vote majority. Both these conditions are necessary for your vote to be decisive. So voting might at first not seem like such a good use of your time.

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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 $100 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 $30 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 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.

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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 1,000 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 percent to 50 percent). 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 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.

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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 percent 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 (like 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!)

But what if you live somewhere — such as New York or California or Kansas or Alabama — that will almost certainly not be a swing state? Then, yes, there’s essentially zero chance your vote will be decisive in the election. In the highly unlikely event that your state is tied so that your vote would swing it, the national election would be so lopsided that your state wouldn’t be needed for an electoral college coalition. That’s why we estimate the probability that your vote would swing the election to be less than 1 in 1 billion in the four states mentioned above.

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Even in those states, though, I’d still recommend you cast a vote for president, if you care about the election and you think it’s important for the general good if your candidate wins. Why? Because the election could be close, and there’s a small chance that your vote could determine the winner of the popular vote. The popular-vote winner doesn’t count for anything technically, but it does give some legitimacy.

Or, your vote might be enough to cause a change in the rounded popular vote, for example changing the outcome from 50/50 (to the nearest percentage point) to 51/49. Or enough to make the vote margin in 2016 exceed President Obama’s margin over Mitt Romney in 2012. Any of these can affect perceptions of legitimacy and mandates.