I'm way late to this party, but aren't we ignoring something obvious? Such as imperfect knowledge of how likely Omega is to be right about its prediction of what you would do? If you live in a universe where Omega is a known fact and nobody thinks themselves insane when they meet him, well, then it's the degenerate case where you are 100% certain that Omega predicts correctly. If you lived in such a universe presumably you would know it, and everyone in that world would pre-commit to giving Omega $100, just like in ours pizza-deliverers pre-commit to not carrying more than a small amount of cash with them.
There may be other universes where Omega is known to be right and do what he says he will do 80% of the time. Or ones where there are rumors of an omniscient Omega that always makes good on his word, but you assign them 80% probability of being true. And so on.
Given the $5000 expected payoff and the $50 expected cost for pre committing, you should do it if the probability of Omega being both right and trustworthy is greater than or equal to 0.01.
But, if you, knowing what you know about THIS universe, suddenly found yourself in the presence of some alien entity making the claim Omega makes in the above scenario, what kind of evidence would you demand for this claim before assigning a probability greater than 0.01?
It occurs to me that the dude in the robe and mask pretending to be Omega could up the ante to $1000000, and if I wouldn't believe him more than 0.01% given a $10000 payoff, it probably wouldn't matter to me what he offered as a payoff, because if he has enough delusions and/or chutzpah to make this claim in this universe, there's no reason for him to balk at adding on a few extra decimal places. I'm not sure how to formalize that mathematically, though.
Related to: Can Counterfactuals Be True?, Newcomb's Problem and Regret of Rationality.
Imagine that one day, Omega comes to you and says that it has just tossed a fair coin, and given that the coin came up tails, it decided to ask you to give it $100. Whatever you do in this situation, nothing else will happen differently in reality as a result. Naturally you don't want to give up your $100. But see, Omega tells you that if the coin came up heads instead of tails, it'd give you $10000, but only if you'd agree to give it $100 if the coin came up tails.
Omega can predict your decision in case it asked you to give it $100, even if that hasn't actually happened, it can compute the counterfactual truth. Omega is also known to be absolutely honest and trustworthy, no word-twisting, so the facts are really as it says, it really tossed a coin and really would've given you $10000.
From your current position, it seems absurd to give up your $100. Nothing good happens if you do that, the coin has already landed tails up, you'll never see the counterfactual $10000. But look at this situation from your point of view before Omega tossed the coin. There, you have two possible branches ahead of you, of equal probability. On one branch, you are asked to part with $100, and on the other branch, you are conditionally given $10000. If you decide to keep $100, the expected gain from this decision is $0: there is no exchange of money, you don't give Omega anything on the first branch, and as a result Omega doesn't give you anything on the second branch. If you decide to give $100 on the first branch, then Omega gives you $10000 on the second branch, so the expected gain from this decision is
-$100 * 0.5 + $10000 * 0.5 = $4950
So, this straightforward calculation tells that you ought to give up your $100. It looks like a good idea before the coin toss, but it starts to look like a bad idea after the coin came up tails. Had you known about the deal in advance, one possible course of action would be to set up a precommitment. You contract a third party, agreeing that you'll lose $1000 if you don't give $100 to Omega, in case it asks for that. In this case, you leave yourself no other choice.
But in this game, explicit precommitment is not an option: you didn't know about Omega's little game until the coin was already tossed and the outcome of the toss was given to you. The only thing that stands between Omega and your 100$ is your ritual of cognition. And so I ask you all: is the decision to give up $100 when you have no real benefit from it, only counterfactual benefit, an example of winning?
P.S. Let's assume that the coin is deterministic, that in the overwhelming measure of the MWI worlds it gives the same outcome. You don't care about a fraction that sees a different result, in all reality the result is that Omega won't even consider giving you $10000, it only asks for your $100. Also, the deal is unique, you won't see Omega ever again.