For voters with ‘social’ preferences, the expected utility of voting is
approximately independent of the size of the electorate, suggesting that
rational voter turnouts can be substantial even in large elections.
That’s from a paper by Edlin, Gelman (who talks about it here) and Kaplan. Their logic goes like this:
The probability that a voter’s vote is decisive is
p = K / n
Where K is a constant (on the order of 10 for close elections) and n is the number of voters. The potential benefits of voting are
B = Bself +α N B social
Where Bself are the selfish benefits of voting, α is a constant to correct for the fact that benefits are altruistic (they don’t accrue to the voter), N is the number of people in the population, and Bsocial is the benefits to society as percieved by the voter
Bexpect = (K / n) (Bself +α N B)
When n is large (pretty much in any realistic case) this simplifies to
Bexpect = α Bsocial K N / n
If α = .1 (I discount benefits to others by 10 relative to benefits to me) and K = 10, which the authors state is reasonable and Bsocial = B$25 which seams plausible, and N / n = 3 meaning 1/3rd of the population votes, which is again, plausible, then
Bexpect = $75
Which would certainly make voting rational.
My main qualm is that the authors do not discuss people’s tendency to discount low probability events even when those events have large consequences (see risk framing). It is very difficult to have an intuitive understanding of low probability events. Often this effect is unimportant because low probabilities mean that the expected effect is small, but when the expected effect is large this effect can be important. I very much doubt that voters have an intuitive grasp of the calculus involved in determining the expected value of voting.