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AndySimpson comments on Counterfactual Mugging - Less Wrong
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Yes, then, following the utility function you specified, I would gladly risk $100 for an even chance at $10000. Since Omega's omniscient, I'd be honest about it, too, and cough up the money if I lost.
So after you observe the coin toss, and find yourself in a position where you've lost, you'll give Omega your money? Why would you? It won't ever reciprocate, and it won't enforce the deal, its only enforcement are those $10000 that you know got away anyway, because you didn't win the coin toss.
Yes, I'll give Omega the money, because if I'm going to refuse to give Omega the money after the coin toss occurs, Omega knows ahead of time on account of his omniscience. If I had won, Omega could look at me and say, "You get no money, because I know you wouldn't have really given me the $100 if you'd lost. Your pre-commitment wasn't genuine."