“Of course, it is unlikely that one’s vote decides the outcome of the election. One’s vote has an impact on the outcome only when (1) the votes of all other voters are evenly split between the two candidates, or (2) one’s preferred candidate would lose by one vote if one did not vote.
P [the probability that your vote is decisive] has been calculated in several ways. Under one approach, each voter can be viewed as picking a ball out of a bag in which p fraction of the balls are labeled candidate 1 and (1 – p) are labeled candidate. Each voter is assumed to have a prior as to what p is. If there are N voters and N is odd, then P1 for any voter is simply the probability that exactly one half of the remaining (N – 1) voters would pick a ball labeled candidate 1 and the remaining one half would pick a ball labeled candidate 2, given this voter’s prior p. P then becomes:
P = 3e^[(-2)(N-1)(p-½)^2]/[2*(2PI(N-1))^(½)]
P declines as N increases, and as p diverts from 1/2.” […]
“Voters do not decide how to vote by picking balls out of hats. On election day, it is more reasonable to assume that all voters are committed to voting for either candidate 1 or candidate 2. Each voter has some prior, p, of the fraction of the population of potential voters who are committed to candidate 1, based perhaps on preelection polls. The rational voter knows, however, that this p is measured with error. Thus, in deciding whether to vote, a rational voter must calculate the probability that her vote will make or break a tie, given p, and the inaccuracy with which it is estimated. This probability is inversely related to (Np (1-p))^(½), the standard deviation of the estimated number of people voting for candidate 1, and thus also becomes infinitesimal as N becomes large.(3)
Several people have noted that the probability of being run over by a car going to or returning from the polls is similar to the probability of casting the decisive vote.(4) If being run over is worse than having one’s preferred candidate lose, then this potential cost of voting alone would exceed the potential gain, and no rational self-interested individual would ever vote. But millions do, and thus the paradox.
There are essentially three ways around the paradox: (1) redefine the rational voter’s calculus so that the rational action is now to vote; (2) relax the rationality assumption; (3) relax the self-interest assumption. All three routes have been pursued.” […and the rest of the chapter deals with these]
From Mueller, chapter 14: The paradox of voting. Note that if you relax the also somewhat unrealistic assumption that everybody know who they’ll vote for beforehand, voting becomes more risky and thus less attractive given risk averse voters.
There’s an election coming up and it’s likely that I’ll post a bit more on related matters in the time to come. Before people start to claim in the comments section that Danes are people who care a lot about the poor and stuff and that’s why we usually have a relatively high voter turnout from an international perspective, actually some of the numbers are telling a quite different story. Dealing with the economic aspects of voting, we’re a bunch of selfish bastards compared to other countries:
“More direct comparisons with Hudson and Jones’s test of the ethical voter hypothesis are obtained in studies of economic voting, which estimate the relative weights placed on egotropic and sociotropic variables. Egotropic variables measure voter expectations regarding the effect of the government’s policies on the voter’s own income, employment status, and so on. Sociotropic variables measure voter expectations regarding the effect of the government’s policies on the economy at large, that is, on the welfare of all citizens. By linking voters’ support for the government to their answers to these sorts of questions, researchers have been able to estimate equivalents to θ in (14.4), where θ = 1 implies full weight on sociotropic variables, and θ = 0 implies full weight on the egotropic variables. Estimates of θ falling between 0.5 and 1.0 have been made for the United States, the United Kingdom, France and Germany.(20) Only Danish voters seem to conform largely to the egotropic economic man assumption in studies by Nannestad and Paldam (1996, 1997). They estimate a θ for Denmark of about 0.15.(21)”
When it comes to voting, just like in the case of lotteries it’s easy to argue the math but hard to argue the preferences. If you derive pleasure from voting, by all means vote. At least as long as the pleasure you derive from voting is relatively unrelated to the impact your vote will have on the election outcome.