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MBlume comments on Counterfactual Mugging - Less Wrong
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There are various intuition pumps to explain the answer.
The simplest is to imagine that a moment from now, Omega walks up to you and says "I'm sorry, I would have given you $10000, except I simulated what would happen if I asked you for $100 and you refused". In that case, you would certainly wish you had been the sort of person to give up the $100.
Which means that right now, with both scenarios equally probable, you should want to be the sort of person who will give up the $100, since if you are that sort of person, there's half a chance you'll get $10000.
If you want to be the sort of person who'll do X given Y, then when Y turns up, you'd better bloody well do X.
That's not the situation in question. The scenario laid out by Vladimir_Nesov does not allow for an equal probability of getting $10000 and paying $100. Omega has already flipped the coin, and it's already been decided that I'm on the "losing" side. Join that with the fact that me giving $100 now does not increase the chance of me getting $10000 in the future because there is no repetition.
Perhaps there's something fundamental I'm missing here, but the linearity of events seems pretty clear. If Omega really did calculate that I would give him the $100 then either he miscalculated, or this situation cannot actually occur.
-- EDIT --
There is a third possibility after reading Cameron's reply... If Omega is correct and honest, then I am indeed going to give up the money.
But it's a bit of a trick question, isn't it? I'm going to give up the money because Omega says I'm going to give up the money and everything Omega says is gospel truth. However, if Omega hadn't said that I would give up the money, then I wouldn't of given up the money. Which makes this a bit of an impossible situation.
Assuming the existence of Omega, his intelligence, and his honesty, this scenario is an impossibility.
Omega hasn't told you his predictions in the given scenario.