Your Vote Doesn’t Count

Taking costs and benefits into account, each person would then decide whether to vote. If a voter thought she would benefit equally from both alternatives, she might decide not to pay the costs of voting and stay home. Downs called this rational abstention—it makes sense for some people not to participate because they literally think, “There’s not a dime’s worth of difference between the two.” Conversely, people who believe one alternative is much better than the other are likely to care a lot more about the outcome and would therefore be more likely to stand up and be counted, even if the costs of voting are very high. Those Ohio voters soaked to the bone are just one example of such highly motivated individuals.

But does this really explain why people vote, especially when they may also think that they cannot influence the outcome? Do they simply calculate the benefits and costs and make a choice?

Actually, it’s more complicated than that. William Riker, a tremendously influential political scientist at the University of Rochester in the 1960s and 1970s, pointed out that Downs had overlooked the important fact that not just one voter makes this decision but millions. Thus, to determine the value of voting, we need to decide not just who we like better but also the probability that our action—our vote—will help that person win. Calculating this probability may seem like an impossible task because there are so many possible outcomes. Obama could beat McCain by 3 million votes. Or he could beat him by 2,999,999, or he could lose to McCain by 1,345,267. Or… There are literally millions of possible outcomes.

Of course, there is only one circumstance in which an individual vote matters. And that is when we expect an exact tie. To see why this is true, ask yourself what you would do if you could look into a crystal ball and see that Obama would win the election by 3 million votes. What effect would your vote have on the outcome? Absolutely none. You could change the margin to 2,999,999 or to 3,000,001, and either way Obama still wins. Notice that the same reasoning is true even for very close elections. No doubt some citizens of Florida felt regret about not voting in 2000 when they learned that George W. Bush had won the state (and therefore the whole election) by 537 votes. But even here, the best a single voter could do would be to change the margin to 536 or to 538, neither of which would have changed the outcome.

So what is the probability of an exact tie? One way of looking at this is to assume that any outcome is equally possible. Suppose 100 million people vote for Obama or McCain. McCain could win 100 million to 0. Or he could win 99,999,999 to 1. Or he could win 99,999,998 to 2. You get the idea. Counting all these up, there are 100 million different possible outcomes, and only one will be a tie. Because roughly 100 million people vote in U.S. presidential elections, that would mean that the probability of a tie is about 1 in 100 million. 

The exact probability is obviously much more complicated than this, as it is unlikely that Obama or McCain would win every single vote cast. Close elections are probably more likely than landslides. So instead of theorizing about the probability of a tie, we could study lots and lots of real elections to see how often a tie happens. In one survey of 16,577 U.S. elections for the House and Senate over the past hundred years, not one of them yielded a tie. The closest was an election for the representative for New York’s 36th congressional district in 1910, when the Democratic candidate won by a single vote, 20,685 to 20,684. However, a subsequent recount in that election found a mathematical error that greatly increased the margin, meaning there are actually no examples of single-vote wins.

In this survey of elections, the average number of voters per election was about one hundred thousand. This is far fewer than the millions who turn out for a national election, and therefore we would expect the odds of a tie in a national election to be even lower. However, calculating this probability is not easy. U.S. presidential elections are complicated because they are not decided by the popular vote. Instead, each state has a number of electors it sends to an electoral college to choose the president. Bigger states get more electors, and most states award all their electors to the candidate who wins the popular vote in their state. As a result, it is possible to win a few big states by a narrow margin and gain the presidency by winning the electoral college vote while losing the popular vote (as George W. Bush did in 2000). Taking all of these complications into account in one big statistical model, political scientists Andrew Gelman, Gary King, and John Boscardin used real data from one hundred years of presidential elections to model the vote within each state and the effect this would have on electoral college votes. Their model showed that the chance of a tie in any state changing the state’s electoral college vote and hence the outcome was about 1 in 10 million.

So let’s go back to the original question posed by Anthony Downs. Suppose you were deciding whether to vote in the 2008 election. When, given all this, does it make rational sense to vote?

First, you have to put a value on the difference between a McCain presidency and an Obama presidency. One way to arrive at this value is to ask yourself, How much would I be willing to pay to be the only person who gets to choose whether McCain or Obama is president? You can go to the bank and withdraw any amount you like. How much would you hand over to be the kingmaker, the one person who chooses who runs the country for the next four years? One dollar? Ten dollars? One million dollars? When undergraduates answer this question, they usually give amounts of less than $10, which is astonishing since this is probably the greatest value anyone could get for a $10 purchase. However, for the sake of argument, let’s say you think it is a very important decision and you are willing to spend $1,000 of your own money to be the only person who chooses the next president of the United States.

Second, you have to account for the fact that, by voting, you get the opportunity to determine the election’s outcome only when there is an exact tie. Otherwise, the outcome will not change whether you vote or not. So the value of voting is not $1,000; instead, it is a 1 in 10 million chance to obtain the $1,000 value.

Third, and finally, you have to compare your anticipated benefit to the costs of voting. Most people say that the costs of gathering information and going to the polls are not that great, so for convenience let’s assume they are $1. They could be much higher, of course, but they are almost certainly greater than zero.

Hence, now that we have your costs and benefits all worked out, the rational analysis of voting suggests that the decision to vote equates roughly with the decision to pay $1 for a lottery ticket that gives you a 1 in 10 million chance of winning a $1,000 prize. Las Vegas would love to sell these tickets. If they could sell 10 million tickets, they would make $10 million dollars and owe just $1,000 in prize money. But even the most ardent gambler would probably refuse to buy them, knowing that the odds are extremely unfair. The average person would probably need other inducements to buy a ticket, because slot machines, blackjack tables, and roulette wheels all have vastly better odds. Even state lotteries that use funds from ticket sales to provide public services rather than prize money typically offer people millions of dollars in winnings, not thousands, for odds like these. And so we are left with the same puzzle we began with. Why do millions of people vote in spite of these odds and payoffs? What is it about elections that make them different from lotteries?

This rational analysis of voting is extraordinarily depressing for (at least) three reasons.

First, it suggests that the core act of modern democratic government makes absolutely no sense.

Economists would call voting irrational because it violates the preferences of the people who engage in it. For some reason, people decide to vote even though they would not buy a lottery ticket with identical odds, cost, and payoff. Economists typically think that people who vote are making a mistake or that there are other benefits to voting that we have not considered.

For example, Downs himself noted that people might vote in order to fulfill a sense of civic duty or to preserve the right to vote. Later scholars have pointed out that people might vote because they enjoy expressing themselves—in the same way they enjoy expressing themselves when they cheer for their favorite team at a ballgame.

Second, learning about the irrationality of voting actually depresses turnout.

In 1993, Canadian political scientists André Blais and Robert Young gave a ten-minute lecture on the rationality of voting to three of their classes and compared their students’ voting behavior to that of students in seven other classes who did not hear the lecture. Perhaps not surprisingly, the students who heard the lecture were significantly less likely to vote. Meanwhile, back in the United States, on Election Day in 1996, the Lawrence Journal-World published a guest column by Kansas University political scientist Paul Johnson about his reasons for not voting. He outlined the rational argument and noted that because of it he had not voted in the past thirty years. Within days there were several pointed letters to the editor denouncing his opinion and openly calling for his dismissal from the university. Johnson was not fired, but he did register to vote a week later in part to calm the controversy. 

Third, the inability to explain the decision to vote calls into question the rational analysis of all political behavior.

Since we cannot use cost-benefit analysis to explain something as basic as voter turnout, some scholars argue that it makes no sense to apply rationality to other decisions such as who to vote for, whether to run for office, how to bargain with political adversaries, and so on. Instead of making rational choices that account for the costs and benefits of their actions, political actors might be affected by their emotions or by specific contexts that could not be generalized. In 1990, Stanford professor Morris Fiorina (one of William Riker’s students from Rochester) dubbed this perplexing voting problem “the paradox that ate rational choice.”

This is academics’ way of saying that it makes no sense.