I have one minor question about this problem, would I be allowed to say, offer omega $50 instead of the $100 he asked for in exchange for $5000 and the promise that, if it had occured that the coin landed head, it would give me $5000 and ask me for $50, which he (going to refer to all sentinents as he, that way I don't have to waste time typing figuring out whether the person I'm talking about is he,she, or it.) would know to do since Omega would simulate the me when the tail landed tails, and thus the simulated me would offer him this proposition. Which should not be too difficult to accept, given that the cost to Omega is basically zero across all possibilities, unless part of the point of the exercise was to mess with me.
In the event that he rejects this offer, I'm going to give him $100, and then mug him. (Assuming of course that the probability is not 0 that I succeed in the mugging, if he were to say kill me in response to the mugging, I'd simply have the copy that succeeded in mugging him force him to use some of his powers to resurrect my less fortunate copies. Assuming of course that that power is part of his omnipotence, else I wouldn't mug him, no point in betting my life in something that does not generate more of my life (given that if I succeeded I can have him make all copies of me immortal. Of course, if he had this power, there's a might be a chance that his retaliation might have been to simply wipe out all copies of me in all existances, in that case, the probability of success should be computed as a negative value, given I CAN fail more times then I try.) In the case I succeed in the mugging, I'd get at least my $100 back, and the cases I fail, I doubt I'd care about $100.
In the case that neither of the above are possible, I would not give him the $100, given that the diminishing returns of increasing amounts of money might well make the $10000 less utility than 2x instances of $100. (The 2x instances of $100 scale linearly, where each increasing $100 in the $10000 diminishes in value. As in, each instance of $100 would be worth just as much as a prior instance of $100 since it's being distributed among different copies of me, so diminishing returns does not kick in, whereas the $10000 all goes to one instance. It should be obvious that I prefer $50 to 50% chance of $100)
Of course, due to the above, there's a fourth possiblity, one where the iteration of me being offered the choice is being very much affected by the diminishing returns on the value of money. In that case, I would give $100 to omega, since this action would partially smooth out the differing amounts of wealth among multiple copies of me across worlds. Or rather, diminish the number of me who are "poorer," since the copies that are in need of money do not give up $100, but will recieve some regardless, unless that doesn't work out because Omega simulates the exact version, including current finiancial assets, which rather nullifies his capabilities as an interdimentional arbitrageur among copies of me. But at that point, the diminishing returns on money should be such that each additional $100 should be roughly equal in value, since diminishing returns ALSO suffer from diminishing returns, with increasing amounts of diminishing returns diminishing less returns.
In short, the options are, I offer him $50 for a constant $5000 across all outcomes of the coin flip, and he accepts, I give him $100 then I mug him, I do not give him $100 if diminishing returns is not yet itself affected by much by diminishing returns, or I give him $100 if it is.
We have to presume you can't just mug Omega. (He is omniscient, may as well make him omnipotent too.) Otherwise the problem is totally different.
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.