(Cross-posted from elsewhere; I thought some readers here might like it too. Please excuse the unusually informal style.)

Short answer: Actually, yes!

Slightly longer answer: Yes, if the election is close. You'll never get to know that your vote was decisive, but one vote can substantially change the odds on Election Day nonetheless. Even if the election is a foregone conclusion (or if you don't care about the major candidates), the same reasoning applies to third parties- there are thresholds that really matter to them, and if they reach those now they have a significantly better chance in the next election. And finally, local elections matter in the long run just as state or nation elections do. So, in most cases, voting is rational if you care about the outcome.

Full answer: Welcome! This is a nonpartisan analysis, written by a math PhD, of when and how a single vote matters in a large election. I've got a table of contents below; please skip to whatever section interests you most. And feel free to share this!

Sections:

1. How can a single vote matter in a huge election?

2. What if I know it's not going to be close?

3. Do local elections matter?

In what follows, I'm going to be assuming an American-style voting system (first-past-the-post, for you voting-system buffs), but most of what I say carries over to other voting systems found around the world.

1. How can a single vote matter in a huge election?

To answer this, let's imagine a different voting system. In the land of Erewhon, voters cast their ballots for president just as they do here; but instead of decreeing that the candidate with the most votes is the winner, each vote is turned into a lottery ball, and one is chosen at random to determine the next president.

While this system has its drawbacks (they get fringe candidates elected every so often, and they've had to outlaw write-in campaigns to prevent every voter from simply voting themselves in), the citizens of Erewhon agree on its main advantage: every one of them knows that their vote counts, that they increase the chance of their candidate winning by just that much.

I'm going to argue that if you would bother to vote if an election were held in Erewhon, then you should also vote—for the same reason—if the same election were held the normal way, and if it looked like the outcome might be close. That is, your effect on the outcome is about equivalent if the pre-election polls fall within the margin of error (about ±3% for each candidate, or a margin of less than 6% between two candidates, for an electorate of millions). And if the polls are nearly tied, then voting in our system might have the same effect as voting hundreds or even thousands of times in Erewhon's system!

This seems counterintuitive, because we imagine that the votes of everyone else are "locked in" somehow, and that we're only deciding whether to add ours to the pile- in which case, the only way that it could matter is in the event that it makes or breaks an exact tie. And not only are those exceedingly rare (the largest such example I could find was a recent Massachusetts election for state representative, in which 13,500 ballots were cast), but if the initial count for a massive election did show a margin of one or zero, we would be headed for an interminable recount. (See, for instance, the 2008 Minnesota Senate election, which was decided by about 300 votes; the extensive recount delayed the winner's inauguration by about six months.) What this messes with, though, isn't your chance of helping decide the election, but your chance of knowing that you helped decide the election.

That is, in modern elections, there's not a perfect sharp boundary between "Candidate A wins" and "Candidate B wins", but a gigantic muddle; and if A is narrowly ahead, then every additional vote for A represents a greater chance that the recount will eventually end, or end earlier, or not be contested at all; while every vote for candidate B means a greater chance that there will be a recount, or that it will go on longer, or that it might turn out victorious for B after all. You'd never know that your vote, which changed the lead from 412 to 413, was the straw that broke the proverbial camel's back and led the other candidate to concede, but at some point that's what happens.

And even more significantly, we can't consider everyone else's votes as "locked in". If you had the ability to re-play Election Day over and over, the chaotic dynamics of everyday life would affect the vote totals. Someone makes a light about to turn red, and so she ends up at the next light behind a bumper sticker that infuriates her, and so she remembers to vote. Or someone else bumps into an old friend, and starts a conversation, and then he realizes he doesn't have enough of his lunch break left to get to the polling place anyhow. It's the butterfly effect in action, and when multiplied by millions of people it leads to fluctuations in the hundreds or thousands. (The exact mechanism for this estimate is the Central Limit Theorem.) And what that means is that the margin isn't a fixed number pending your decision to vote; it's a mix of different possibilities, and your decision changes the odds as surely as adding a lottery ball to the Erewhon election does.

Of course, if one candidate is polling 10 points ahead of the other, then those fluctuations will be irrelevant. If the margin is closer, then it just might have an effect; remember that polls have several sources of error in them, so you can't be exactly sure of the margin before voting actually happens. And in those cases where the actual vote turns out to be extremely close, like Minnesota in 2008, your choice to vote swings those odds just as if you'd poured in thousands of Erewhonian lottery balls with your candidate's name on them. (That is, it changes those odds by a fraction of a percent, but the ability to nudge the odds by that much in an electorate of millions is pretty impressive!)

2. What if I know it's not going to be close?

If the difference between the top candidates is outside the combined margin of error (7 points or more in a big election), then it's true that your vote won't affect the outcome of the current election, but it can matter greatly for the next one. This especially holds when there's a third party you prefer to the current major parties, but there are other relevant reasons to vote even if that's not the case.

First, about third parties: the conventional wisdom that they can't win is an outright falsehood. We tend to forget that Ross Perot nearly became President: he held a six-point lead in June 1992 over both Bush and Clinton, and despite his candidacy crashing and burning later on (for reasons more personal than systemic), he still won nearly 20% of the popular vote in the end. In addition, there are two current Senators who were elected as independents.

For a third party that's not polling well enough yet, votes now matter for future elections, and there are several significant thresholds. If they get rounded up to 1% on Election Day, then they get mentioned in election coverage. (Next, getting rounded up to 2% sounds much more impressive than getting rounded down to 1%.) If they get to 5%, then they can qualify for FEC matching funds, and double their ability to reach voters next time. And 10% is a significant number for media exposure (as well as invitations to the main debates). Higher than 20%, and we're talking about a viable candidate; there's a runaway dynamic in three-candidate races where as soon as the third party looks viable, any voters from the other parties who prefer the third party will suddenly switch (now that they're no longer worried about supporting an obvious loser), and suddenly the third party becomes the front-runner. (This kind of behavior is common in game theory in what's known as a coordination problem.)

Even if there's not a third party you prefer to the main ones, the margin of victory now helps determine what sort of candidates the parties select next time. If one party wins an election by more than 10% (this round number has a psychological hold on people, so it's what usually counts as a "landslide victory"), then next time that party is more likely to nominate a less moderate candidate, while the defeated party is likely to nominate a more moderate candidate. (Also, it goes without saying that you should be voting in primaries as well!)

All of these thresholds can hinge just as easily on a few hundred votes, and so your vote can matter greatly. You just need to care about what happens two, or four, or six years from now.

3. Do local elections matter?

Most certainly. First, your vote affects the odds more strongly in a local election. (And, of course, there can be margins of zero or one votes, without recounts, in local elections!) Secondly, there are a number of important issues decided at the local level, both directly (tax and bond issues) and through representatives (especially local school boards). And finally, many big names in politics start out in local elections: the current President began as an Illinois state senator, and the current Vice President began as a city councilman.

It's more difficult to do research at the local level. Project VoteSmart may have data on some of the candidates (including the voting record of incumbents, public statements, and response to a questionnaire about their positions). Otherwise, your local newspaper may be the best bet (short of going to campaign events yourself). But a little bit of research can go a long way here.

(Addendum: If I'd been composing this for the Less Wrong crowd, I'd have also noted that the decisions of people similar to you should be correlated, which adds another multiplier to the effectiveness of voting. I might like I bumper sticker that says "I'm a timeless decision theorist, therefore I vote!")

(Second addendum: I'm mindful of the dangers of posting this within a few days of a major election. I've done my very best to keep from mindkilling language; I hope you do the same in the comments.)