Is it rational to vote?

Does it make sense to vote?

The short answer is "probably, since you'll feel better about having done your civic duty." But when people ask this question I think what they mean is "the cost of voting is about 10 minutes of your time, the benefit is a tiny probability multiplied by something huge--is that tiny multiplied by huge a big or small?

Gelman et al. estimate the probability your vote would have mattered in the 2008 election. The probability varies by state so they plot by state in this graph:

These probabilities are tiny. (And I think they are vast overestimates.) But look at the historic average growth rates for different income groups when Democrats are in office vs Republicans. The fact that Democrats have higher growth rates for all groups probably has little to do with policy. Democrats just happened to get voted into office at the right times of the business cycle. The fact that the growth is relatively even (and somewhat convergent) is possibly also an artifact of the Democrats being in office when there was a lot of growth in college and high school education, while the Republicans have presided during a period of more rapid technological change (or some such narrative).

But it's definitely possible that Democratic and Republican policies have something to do with the fact that income growth lags for the poorest under Republicans and is greatest for the poorest under Democrats. Suppose that we adjust the growth rates so the means are the same; then, by eye-balling, it looks like growth would be around 1% greater per year for the poorest income deciles. Using the numbers in this graph that means about $144 billion in value over 4 years for the bottom two income quintiles.

Suppose your time is worth $20 an hour. It takes ~10 minutes to vote. The value from voting is $144 billion worth of aid to the poor and working class. What would the probability need to be (at least) for the expected benefit of voting to exceed its cost? If you weight income to yourself and to others equally then it's just 1 in 43 billion.

Which means it rational (in the sense of expected value), for Democrats at least, to vote in almost every state in the union. That's definitely not a result most philosophers intuit (and is probably an artifact of greater overestimated probabilities).

But what if these numbers are right? Shouldn't the possibility be widely noted in the literature? And shouldn't estimates of the probability of your vote mattering note that there is a big difference between a 1 in 10²⁰ chance, which implies voting is irrational, and a 1 in 10⁹ or 1 in 10¹² chance, which may imply voting is rational?