> So if a few hundred thousand people decided to not vote because their one vote doesn't matter, it's conceivable that the result in a state could be swayed.
Yes, but it's extremely unlikely that your vote would sway the result, and I'm operating under the assumption that you can only control your own decision to vote.
> So why try to persuade people their vote doesn't matter? If enough of them (an admittedly large number) listen, it will matter.
You're confusing two things. I would never try to persuade people that the sum of everyone's votes doesn't matter, because it obviously does (discounting for the moment the possibility of election fraud). But I would try to persuade individuals that their vote will not matter with a very high probability.
Even though this is often phrased in such a way that it almost sounds like a contradiction or paradox, it's not at all. It's analogous to the lottery: the more people that play, the higher the chance that someone will win (assuming random picks), but one person buying a single lottery ticket has such a small probability of winning that I would recommend against relying on the lottery to change one's financial situation.
