I think what this argument typically indicates is an individual who is using the following logic, more or less consciously:
- My vote isn't going to be the vote that changes the outcome.
- Therefore, my vote is of exactly zero value.
Ultimately, this is a very selfish line of reasoning, because it implicitly is claiming that only their vote should count. Well, I got news. There are about three hundred million other people in the good ole' US of A, many of them able to vote, and their votes count too.
Your vote does count; it is just one of many.
Moreover, even speaking purely rationally, vote ratios do matter. A Senator who wins every election by a 90-10 margin will be much more powerful in Congress then one that squeaks by; they will be more able to take a particular stand that may be somewhat unpopular, and Senators who last longer tend to rise to the top of various powerful committees. By voting for or against someone, even if their win is assured, you are still having a small, but real, effect on the power they will have. This holds true for all elected offices, though we usually only speak of a "mandate" for the Presidency.
In other words, voting outcomes are not really a digital thing, as the logic implicitly assumes. There is certainly a big discontinuity as you cross the threshold to win the vote, whatever it may be, but the results of a 50-50 vote will almost always be sigificantly different than a 90-10 vote, even when it comes to voting on laws or resolutions. (A law just barely passed may be more likely to go unenforced, or opposition to it will more quickly form to repeal it then for a 90-10.)
I will concede that in game theoretic terms, the likely payoff of voting does most likely exceed the expected value to any given individual. But we can't always think like this, for the obvious reasons. Plus, the intangibles should be counted as well, a sense of ownership, involvement, the ability to affect outcomes beyond just changing the winner. I honestly can't tell you with 100% confidence that this is a wrong way of thinking, but I would encourage everyone who thinks this way to at least consider the larger issues beyond just "who won".
I'm not going to end this with an exhortation to vote. If you don't care enough to vote, then don't. I'd much rather only people who have taken time to make up their minds and care enough to vote actually do so. But I would encourage you to research and vote.
(This is #4 in the Government Myth series.)