I assume you recast the problem this way: if the n-th bit of pi is 1, then Omega maybe gives you $10000, and if the bit is 0, then Omega asks for the $100.
If the bit is 1, Omega’s simulation of you can’t conclude that the bit is 0, because the bit is 1. Omega doesn’t compute what you’ll predict in reality, it computes what you would do if the bit was 0 (which it isn’t, in reality, in this case where it isn’t). And as you suggested, you decline to give away the $100 if the bit is 0, thus Omega’s simulation of counterfactual will say that you wouldn’t oblige, and you won’t get the $10000.
(I converted this comment to a top-level post. See Counterfactual Mugging and Logical Uncertainty. A little bit is left here in the original notation, as a reply.)
I assume you recast the problem this way: if the n-th bit of pi is 1, then Omega maybe gives you $10000, and if the bit is 0, then Omega asks for the $100.
If the bit is 1, Omega’s simulation of you can’t conclude that the bit is 0, because the bit is 1. Omega doesn’t compute what you’ll predict in reality, it computes what you would do if the bit was 0 (which it isn’t, in reality, in this case where it isn’t). And as you suggested, you decline to give away the $100 if the bit is 0, thus Omega’s simulation of counterfactual will say that you wouldn’t oblige, and you won’t get the $10000.