Nate Silver, Aaron Edlin, and I estimated the probability that a single vote in any state will be decisive in the presidential election. It was Aaron’s idea, Nate supplied the simulations, and I calculated the probabilities and made the graphs.
Here’s our article describing what we did, here’s the abstract:

One of the motivations for voting is that one vote can make a difference. In a presidential election, the probability that your vote is decisive is equal to the probability that your state is necessary for an electoral college win, times the probability the vote in your state is tied in that event. We compute these probabilities for each state in the 2008 presidential election, using state-by-state election forecasts based on the latest polls. The states where a single vote is most likely to matter are New Mexico, Virginia, New Hampshire, and Colorado, where your vote has an approximate 1 in 10 million chance of determining the national election outcome. On average, a voter in America has a 1 in 60 million chance of being decisive in the presidential election.

and here are some graphs:

There are more graphs if you follow the link to the article.

As Aaron, Noah, and I have discussed, it can be rational to vote even when the probability of decisive vote is 1 in 10 million.

P.S. Typo in Figure 1 caption above fixed.

P.P.S. See here for more discussion of why we can compute the probability of a decisive vote, even though the election might be decided by a recount

I spoke on the book at NYU on Tuesday and used the line about religion as the “opiate of the elites,” showing this graph:

Pat Egan was my discussant and presented some data casting doubt on this story. He took poll data from 2004, I think it was, and fit logistic regressions to vote preference, given attitudes on three issues: taxes, abortion, and Iraq. He fit separate models to high, middle, and low-income voters. His hypothesis was, if social issues are truly the opiate of the elites, that the coefficient for attitude on abortion would be higher among the rich than the poor. But this didn’t happen: the coefficient for abortion was slightly higher among the poor–that is, social issues, by this measure, seemed to be more important among lower income voters.

What do I make of this? My quick response was that, in their Purple America paper, Ansolabehere, Rodden, and Snyder put together pretty comprehensive measures of social and economic attitudes, and found social attitudes to be more important among the rich than the poor in predicting vote.

I was thinking more about this and have a few other thoughts. Perhaps Pat or someone else will be interested in looking in to this.

- Measurement is key. Responses to different issues have low correlations, so it can help to use multiple issue questions for each issue domain, rather than simply one question per domain.

- My impression is that, for richer voters, both economic and social issues are more closely tied to party preference and also more closely tied to each other. I’m not sure how this should come out in Pat’s multiple regression analysis. I’m still struggling to see how the “opiate of the elites” story fits in to this.

- Religious attendance, and even religious belief, isn’t the same as social attitudes. Not by a long shot. (Similarly, income is not so strongly correlated with economic ideology.) See the graphs on page 90. A challenge is to fit together our findings on income and religion with what we know about issue attitudes.

Katy Gartside pointed me to this site. It’s fun to see the new states gradually being added as the years go on. Here was the exciting 1796 election. (At that time most people weren’t allowed to vote.)

And here’s the Twentieth Century Reversal. First, the McKinley-Bryan matchup of 1896:

Red states in the northeast, upper midwest, and west coast; blue in the south, plains, and mountain west. That sounds right . . . hey, wait a minute!

Let’s go to the Bush-Gore map from 2000. The red states have switched to blue and vice-versa:

See here for more on the Twentieth Century Reversal, including some scatterplots. It turns out that, at the local level, it was not a simple switching of red to blue or a simple switching of party positions.

P.S. The historical maps on the above-linked site are great, but don’t take the probabilities on the site seriously.

This is a long one because it has a lot of information (collected and analyzed by others, not ourselves). To start with, some data from a study by Baldassare of Californians:

But political scientists generally hold that voters and nonvoters aren’t really so different

This 1999 article by Highton and Wolfinger summarizes the basic political-science view of nonvoters and election outcomes:

Analyses of survey data show that no objectively achieved increase in turnout–including compulsory voting–would be a boon to progressive causes or Democratic candidates. Simply put, voters differ minimally from all citizens; outcomes would not change if everyone voted. . . . The absence of a consequential link between outcomes and turnout can be explained by answering a hitherto neglected question: who does not vote? . . .

If everyone had voted in 1992, Bill Clinton’s share of the vote would have shrunk by 1.2 percentage points, compared to a loss of 2.5 points by President Bush. Clinton’s margin over Bush would have risen from 13.7 to an even 15 points. Ross Perot would have picked up a few more votes and marginal candidates would have done marginally worse. . . . Both Republicans and Democrats were just barely more numerous among voters than in the total population. . . . By and large, voters were representative of the entire sample on most policy questions. Voters were, by five percentage points, more conservative than the whole sample about whether “it is important for the government to provide many more services [in areas such as health and education] even if it means an increase in spending.” Employment preferences for blacks were more popular among the entire sample, 21 percent of whom supported this policy, compared to 17 percent of voters. Voters were less favorable about the federal government guaranteeing a job and a good standard of living to everyone but more sympathetic to abortion rights. On all other issues, differences in either direction did not exceed two percentage points.

In 1996, universal turnout would have expanded Clinton’s share of the vote from 53.1 to 59.5 percent, chopped Bob Dole’s vote more than 11 points, and doubled Clinton’s winning margin. Changes among other candidates were trivial, and four percent of the sample had no candidate preference. In contrast to 1992, Democrats were slightly less numerous among voters and Republicans somewhat less so. As always, Independents were scarcer among voters than in the general public.

The pattern of differences on issues was somewhat more pronounced in 1996. This was particularly the case on redistributive economic questions. By anywhere from five to nine percentage points, voters were more conservative than the whole sample. . . .

These findings are incontestable evidence that on some major issues voters were more conservative than the entire adult population in 1996. We defer for the moment trying to appraise the importance of this tilt to the right, other than to note that the differences between voters and the entire electorate, while indisputable, are relatively modest; none is as high as 10 percentage points. . . .

How can one estimate what nonvoters would do if they were to vote? . . . we attributed to nonvoters the perspectives of voters in their respective income quintile. We assigned to poor nonvoters the political preferences of poor voters, rich nonvoters the characteristics of rich voters, and so on. . . . If everyone had voted in 1992, Clinton would have gained two and a half percent more of the vote and Bush would have lost 1.5 percent . . . The results would have been about the same in 1996, except for a slightly larger rise in Clinton’s vote share. In the 1990s, universal turnout would have been a slight benefit to Democratic presidential candidates. These results confirm the conventional wisdom, although the modest size of the advantage might disappoint some readers.

Why is it surprising that with universal turnout voters would differ so little from those who actually go to the polls? . . . No single characteristic is shared by a majority of those who did not vote in 1992 or 1996; the “party of nonvoters” is rather diverse. The two most common demographic features of nonvoters are residential mobility and youth, two characteristics that do not suggest political distinctiveness, let alone a Mother Lode of votes for Democratic candidates or pressure for liberal causes. In both 1992 and 1996, fully 43 percent of nonvoters had moved within two years of the election and one third were under the age of thirty.

Would election outcomes and the substance of public policy in the United States dramatically change if more people voted? Contrary to the expectations of many others, we have found that universal turnout would bring modest changes. . . . Taken as a whole, nonvoters appear well represented by those who vote.

I’d say that Highton and Wolfinger’s view is accepted by most political scientists and well-informed journalists (just for example, see this 1988 article by E. J. Dionne, ” If Nonvoters Had Voted: Same Winner, but Bigger”).

A new synthesis?

But Jan Leighley and Jonathan Nagler are writing a book, Who Votes Now? And Does It Matter?, about the importance of differences between voters and nonvoters. In their words:

Who votes does matter in terms of policy benefits. Hill and Leighley (1992), for example, find that state electorates in which the poor vote as much as the wealthy provide significantly higher
welfare benefits. Similarly, Martin (2003) finds that members of Congress allocate federal
grant awards to areas where turnout is highest. Thus, those who vote more are rewarded with more substantive policy benefits. . . .

Consistent with the results for partisanship . . . the ideological distribution of voters and non-voters in 2004 is fairly similar to that in 1972: moderates are most under-represented, while conservatives are over-represented. Importantly, the magnitude of these differences increases between 1972 and 2004. . . We therefore draw a very different inference than did Wolfinger and Rosenstone regarding the policy representativeness of voters. Our data, too, suggest that the differences in 1972 were modest. However, our results for each of these three issues after 1972 suggests though that these differences on class-based issues are enduring and increasing. . . .
the conventional wisdom suggesting that “who votes” does not matter due to the relatively representative policy preferences expressed by voters seemed to conflict with a broader appreciation of the notable demographic, economic, and political changes that have occurred in the U.S. since Wolfinger and Rosenstone’s classic statement. . . . we take issue with the assumption that voters are indeed representative of non-voters.

This is important! It would be good to resolve the disagreement between the different experts in this area.

P.S. Here is John Sides’s summary of the situation.