Someone asked me to formalize an argument I made some time ago about how voting is an irrational act. The essence of the argument is as follows. Given that hundreds of thousands of people vote, the chance of a tie is minuscule. So the chances that an individual’s vote decides the election is equally minuscule. Hence it is almost certain that the act of voting has no consequences. So why vote?
At first glance, it seems like an infantile argument to make, most people would reply: if everybody thought/did that no one would vote. And truth be told, the first time I heard that reply I was fooled. But of course, that reply is wholly irrelevant to the matter. It’s an example of argumentum ad consequentiam, and on top of that, something being true does not imply people believe it. So the bottom line stands, and you can readily formulate it in decision-theoretic terms. As follows:
Expected Utility of Voting = (Probability of Deciding * Utility of Deciding) – Cost of voting
Because the probability of deciding is so small the term in parentheses vanishes, and the utility of voting is negative, the cost of voting, which includes explicit costs of going to vote plus opportunity costs of what you could have done otherwise.
I decided to briefly google the subject and it turns out it’s actually a controversial issue in rational choice theory. It was initially formulated exactly as above by Anthony Downs in 1957. Nope, it wasn’t just something of an anecdote anymore, but an open “problem” as of today. Here are some example calculations from  including probability estimates
Consider an election in which 5 million voters are expected to cast ballots and candidate 1’s expected vote share is 50.1 percent, while candidate 2 is expected to receive 49.9 percent ofthe votes cast. Myerson (2000) develops a formula in which the number of people who vote is a random number drawn from a Poisson distribution with mean n. According to Myerson’s formula, the probability a vote is pivotal for candidate 2 is 8.1079 x 10^-9. Thus, the benefit to a voter who prefers candidate 2 must be more than 8 billion times greater than the cost to vote. For example, if voting costs $.01, then the expected benefit of electing one’s favored candidate must be greater than $80 million dollars. Expected benefits at such levels seem unreasonable.
Because rational choice theory has the pretense of describing actual human behavior, and because in fact millions of people do vote, there is an apparent contradiction. It’s called the Paradox of Voting, and has been described as “the paradox that ate rational choice theory” .
I haven’t looked extensively into the literature at how attempts are made to resolve the “paradox”. I’m pretty sure one can invoke all sorts of technical wizardry to get the desired empirical results (game theory and Nash equilibrium come to mind). But to be frank, I presume it’s just ad-hockery to arrive where you were trying to get to initially.
It’s much simpler to just accept that people are either not rational (surprise surprise) or that there are motivations besides those related to the act of deciding itself (surprise surprise). People may go to vote because they have nothing else to do, out of a sense of duty, because it’s amusing, because it’s what everyone else does , or simply because they are outright irrational in estimating cost/benefit. What’s that? Do I hear anybody crying heresy at this scandalous violation of democracy’s sanctitude?
As I said previously, this matter has been debated extensively since 1957. But it seems to me that it’s just as simple as I’m making it out to be; there is no Paradox of Voting just as there is no Paradox of Buying Lottery.
 Downs, Anthony. 1957. An Economic Theory of Democracy.
 Riker, William and Peter Ordeshook. 1968. “A Theory of the Calculus of Voting.”
 Fiorina, Morris (1990). Information and rationality in elections.
 These motivations for voting can be modeled as a consumption benefit in Riker and Ordeshook (1968):
Riker and Ordeshook (1968) modify the calculus of voting by assuming that, in addition to a cost to vote, voters get a consumption benefit D > 0 from the act of voting.