Yesterday, I took part in a panel on the question of the significance of voting.
Early in my portion, I said that when we ask about the significance of voting, we should be careful to distinguish among the following three things: 1) The significance and value of democracy as a whole. 2) The significance and value of the aggregate of all votes, of all the votes as a whole. 3) The significance and value of individual votes. If we had more time, I would have added 4) the significance and value of having the right to vote.
Obviously, you can't make an argument of the form, "Democracy is very valuable, therefore, every individual vote is valuable." That's kind of like saying "Harvard University is valuable, therefore this particular speck of dirt under the third chair from the left in the second row of Emerson 104 is valuable." Oddly, one of the other panelists ignored this distinction entirely and proceeded to make fallacious arguments of the form, "Democracy is good, therefore individual votes are good".
Instead, I pointed out that while 1 might be very valuable, 2 is of questionable value (as Achen and Bartels say, elections are essentially random events), and 3 is generally of no value. Voting is a very high status form of ineffective altruism. It's perhaps important that enough people vote, and it's arguably important that the right kind of cross-section of people vote (e.g., this is one reason why we should probably use voting lotteries rather than voluntary voting), but it's almost never important that any one of us vote.
That's not even a criticism of democracy, by the way. In general, if your individual vote makes a big difference, something has gone wrong.