There's a potential problem with the "don't vote" reasoning that I haven't quite managed to nail down. It's roughly the "But if everyone thought like that...", but not quite. And it's not so much "if everyone" as "if forty or fifty people".
Let's get mathematical:
Let a be the result of the election ( candidate X votes - candidate Y votes, to be simple) if I don't vote
Let b be the result of the election if I do vote (say for candidate X).
Now, b = a + 1, so the actual outcome of the election will only be different if a=0 or a=-1 (whatever the rules are for tied elections).
But what is the real justification for saying b = a + 1?
We can assume that my vote doesn't affect anyone else's vote. After all, they're not supposed to know.
But that's not sufficent. For b to equal a + 1, the votes of other people have to be statistically independent from mine. Can I assume that?
Now we get philosophical. The common view of me as a mind with "free will" seems to imply the independence assumption. But it isn't backed up by sociology or neurobiology. On the basis of either observation or a reductionist, mechanistic view of the human brain, my vote is likely to be significantly correlated with other peoples' votes. That, after all, is the assumption behind opinion polling.
And based on that correlation, b - a cannot be assumed to be 1. It might be 5, or 100, or 10000.
Now, if I vote for X, the chance that X will win could be substantially higher than if I stay at home.
The above argument doesn't feel right — it seems like there's a flaw somewhere. I can't find it though.
These questions of probability and independence are notoriously tricky and counter-intuitive, but using the result of your own decision-making process as a model for the behaviour of other people, and feeding that model back into your original decision, all makes the Monty Hall problem look easy.
This has been nagging at me for years, but I think it was originally triggered by something I read by Douglas Hofstadter.
Anomaly UK