Why your ballot isn't meaningless.

A mathematician's guide to the news.
Oct. 11 2004 1:29 PM

Vote!

Why your ballot isn't as meaningless as you think.

Let's say, for the sake of argument, you were among the 72,000 people who participated in the Guinness-certified world's largest chicken dance in Canfield, Ohio, in 1996. You probably feel pretty proud. But according to Slate's Stephen E. Landsburg, you shouldn't. After all, unless a previous chicken dance for 71,999 were on the books, your participation made no difference; the record would have fallen whether or not you'd shown up.

Landsburg is arguing against voting, not chicken-dancing: Your presidential vote, he says, "will never matter unless the election in your state is within one vote of a dead-even tie." That, of course, is extremely unlikely. So, the negligible chance of casting the deciding ballot is outweighed by the small but certain costs of voting, like the gas you'll use and the time you'll spend.

And yet people vote anyway, by the millions. Political scientists call this conundrum "the paradox of voting," and you could stay up half the night (I just did) reading research literature on the subject. Why do people vote when it's so unlikely to matter? Maybe because the pleasurable feeling of doing one's duty offsets the cost of gas. Maybe because people have an interest in their candidate not just winning but winning by as large a margin as possible. Maybe because we're motivated to avoid even small possibilities of regret—the regret that those Al Gore supporters who sat out Florida in 2000 surely feel, whether economists think they're being rational or not.

But let's stick to mathematics. Suppose we grant to Landsburg that voting carries a certain cost and that your vote should be considered worthwhile only if it decides the election. Everyone can agree that's unlikely—but how unlikely? Landsburg first proposes
modeling voters in a state, say Florida, as 6,000,000 coin-flippers, each choosing George Bush with some probability p and John Kerry with probability 1-p. For instance, if p is 1/2, 1-p is also 1/2; each voter has an equal chance of selecting Bush or Kerry. As you might expect, the odds of a tied outcome are not bad—about 1 in 3,100, as Landsburg computes.

But p might not be 1/2, and even a tiny bias in voter preference can make a tie exceedingly unlikely. For instance, if p = .51, the chance of a tie drops to 1 in 101046, a probability so small as to be effectively zero. (Here's Landsburg's computation.) Your vote is not going to count.

So, are we back to Landsburg's discouraging conclusion that voting is most often a waste of time? Not quite, because it's impossible to know in advance what proportion of your fellow Floridians are planning to vote for Bush. If you knew p was exactly 1/2, you'd be sure to get out and vote. If you knew Bush held a 51 percent advantage, you'd be foolish to bother. But you don't know, and without that knowledge you can't reason as Landsburg wants you to.

You don't know, but you can guess. A Sept. 29 poll of 704 Florida voters by CNN/USA has Bush leading Kerry 52-43. For simplicity, let's dump the still-undecideds and third-party enthusiasts and say that, among 669 randomly selected likely Florida voters, 366 supported Bush and 303 Kerry, a 55-45 margin in Bush's favor. If forced to make a guess, we might expect 55 percent of Florida voters to favor Bush. But how confident should we be that our guess is right? In particular, how likely is it that the real proportion of Bush votes in the state is very close to 50 percent?

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The inconvenient truth is that the poll alone can't tell you. If, for instance, a poll in Massachusetts showed a 10-point Bush lead, we'd still think Bush was behind, though we might rate the race closer than we did previously. Our best guess about the true state of things represents a compromise between our prior intuitions and the poll results.

The mathematical method by which this compromise is hammered out is called Bayesian inference. The computations involved, though elementary, are a bit tedious to include here, but stats fans can find more in the accompanying computations page. Let's suppose we start out with the (somewhat unrealistic) belief that the true vote count for Bush in Florida is equally likely to be any number between zero and 6,000,000. Given the 52-43 poll result, the Bayesian computation puts the chance of a tie at about 1 in 5 million. If the polls were exactly even, the chance would go up to 1 in 300,000. Those still aren't fantastic odds, but both beat the 1-in-120 million chance of winning Powerball by a mile. * Suddenly voting seems a lot more justifiable.

Even if your vote helps swing Florida, Florida might not swing the election. But if the electoral vote is sufficiently close, many states could be in a position to affect the national outcome. You know that if 538 fewer Bush votes had been counted in Florida, Al Gore would be president. But did you know that only 1,231,944 more Bob Dole voters, carefully apportioned among Nevada, Kentucky, Arizona, Tennessee, New Mexico, Florida, New Hampshire, Delaware, Ohio, and Pennsylvania, would have given their man the election, despite Clinton's lead of 8 million in the popular vote?

It's precisely this sensitivity to small swings in key states that makes people fume about the Electoral College—that saturates Tampa with campaign ads and volunteers and leaves Los Angeles quiet, that makes elections vulnerable to targeted fraud beforehand and targeted lawsuits afterwards. So, let's take a moment to cheer this one fine feature of our system: It puts many voters in many states on notice that their vote might really count. The state that swings could be your own. So, ignore Landsburg! Take your place in this big majestic chicken dance we call democracy! Vote!

Thanks to John Londregan and Howard Rosenthal for helpful suggestions and pointers to relevant literature.

Correction, Oct. 14, 2004: This article originally stated that the chance of winning the Powerball lottery is 1 in 80 million. It is actually close to 1 in 120 million. (Return to the corrected sentence.)

Jordan Ellenberg is a professor of mathematics at the University of Wisconsin and the author of How Not to Be Wrong. He blogs at Quomodocumque.

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