Probability against a random human is entirely irrelevant - what Omega must do is probability against the most uncooperative human being nontrivially >.5, as you can choose to be maximally uncooperative if you wish to.

The limit of how uncooperative you can be is determined by how much information can be stored in the quarks from which you are constituted. Omega can model these. Your recourse of uncooperativity is for your entire brain to be balanced such that your choice depends on quantum uncertainty. Omega then treats you the same way he treats any other jackass who tries to randomize with a quantum coin.

Free will is not imcompatible with Omega predicting your actions(Well, unless Omega delibrately manipulates your actions based on that predictive power, but it's outside the scope of this paradox), and Omega doesn't even need to see inside your head, just observe your behavior until it thinks it can predict your decisions with high accuracy only based on the input that you have received. Omega doesn't need to be 100% accurate anyway, only do better than 99,9%. Determinism is also not required, for the same reason.(Though yeah, you could toss a quantum coin to make the odds 50-50, but this seems a bit uninteresting case. Omega could predict that you select either at 50% chance, and thus you lose on average about 750 000$ every time Omega makes this offer to you.)

So basically, where we disagree is that Omega can choose first and you'd still have to one-box, since Omega knows the factors that correlate with your decision at high probability. Without breaking causality, without messing with free will and even without requiring determinism.

Thanks for the explanation. I think that if the right decision is always to 2-box (which it is if Omega is wrong 1/2-epsilon of the time), then all Omega has to do is flip a biased coin, and choose for the more likely alternative to believe that I 2-box. But I guess you disagree.

There's a true problem if you require Omega to make a deterministic decision; it's probably impossible to even postulate he's right with some specific probability. Maybe that's what you were getting at.

"Omega chooses first, so two box" trivial problems.

Yes. Omega chooses first. That's Newcomb's. The other one isn't.

It seems that the fact that both my decision and Omega's decision are determined (quantum acknowledged) by the earlier state of the universe utterly bamboozles your decision theory. Since that is in fact how this universe works your decision theory is broken. It is foolish to define a problem as 'ill-defined' simply because your decision theory can't handle it.

The current state of my brain influences both the decisions I will make in the future and the decisions other agents can make based on what they can infer of my from their observations. This means that intelligent agents will be able to predict my decisions better than a coin flip. In the case of superintelligences they can get a lot better than than 0.5.

Just how much money does Omega need to put in the box before you are willing to discard 'Serious' and take the cash?