Probability of being the swing voter

There are 4 senate races that are still too close to call (I believe they are all in recount), and Missouri is too close to call in the presidential race. By the latest count, Al Franken is within 206 votes of winning the senate race in Minnesota, where the total number of votes was around 2.9 million. From Time:

But Jacobs says he does not expect a huge shift in recounting residual votes. "The bigger issue is how we handle these absentee ballots [which are the subject of the Franken lawsuit]," he says. Mark Jeranek, who voted for Franken, cast an absentee ballot in Beltrami County, located in northwestern Minnesota, that was rejected because he didn't sign the envelope in which he placed his ballot. The Franken campaign sent him an affidavit that he is considering signing. "I don't want to be a cause for revolution, but at the same time I want my vote to count," the 39 year-old environmental consultant says. "It's kind of neat — at least for a senatorial race — that it really does come down to every individual vote."

None of this squares with the typical public choice assumption, going back to Downs and Tullock, that the liklihood of being the swing voter is 1/n, where n is the number of voters. That is, the number of close elections indicates it is not a uniform distrubution, i.e. you need to take into account the median voter theorem. For this reason, I think a binomial distribution is more appropriate, such as that used by Barzel and Silberberg. The following graph illustrates how these two assumptions differ over the size of the electorate. I've assumed p=1/2 for the binomial distribution, and calculated P as the probability of being within 1 vote of a tie.




Clearly, in large elections such as those at the state and national level, both probabilities approach zero. Thus, voters in these elections aren't paying too much attention to this sort of calculus. Rather, expressive voting is probably a more important factor.

Addendum: To illustrate the main point here I've reproduced the same graph below but on a log-log scale. You can now see that especially in large elections the choice of binomial versus 1/n is critical. For instance, in Al Franken's race with 2.9 millions voters, if one assumes a binomial distribution, then there is a 1/711 chance of effecting the outcome. Therefore, if we assume the cost of voting is, say, $1, due to time lost, travel expense, etc., then one need only expect benefits greater than $711 for it to make sense to vote for one's preferred candidate. Compare that to the case of a uniform distribution, where benefits would need to exceed $2.9 million. You can see that the paradox of voting is not such a paradox if one makes realistic assumptions, even if we restict our analysis to non-expressive, i.e. instrumental, benefits.