I'm with incogn on this one: either there is predictability or there is choice; one cannot have both.

Incogn is right in saying that, from omega's point of view, the agent is purely deterministic, i.e. more or less equivalent to a computer program. Incogn is slightly off-the-mark in conflating determinism with predictability: a system can be deterministic, but still not predictable; this is the foundation of cryptography. Deterministic systems are either predictable or are not. Unless Newcombs problem explicitly allows the agent to be non-deterministic, but this is unclear.

The only way a deterministic system becomes unpredictable is if it incorporates a source of randomness that is stronger than the ability of a given intelligence to predict. There are good reasons to believe that there exist rather simple sources of entropy that are beyond the predictive power of any fixed super-intelligence -- this is not just the foundation of cryptography, but is generically studied under the rubric of 'chaotic dynamical systems'. I suppose you also have to believe that P is not NP. Or maybe I should just mutter 'Turing Halting Problem'. (unless omega is taken to be a mythical comp-sci "oracle", in which case you've pushed decision theory into that branch of set theory that deals with cardinal numbers larger than the continuum, and I'm pretty sure you are not ready for the dragons that lie there.)

If the agent incorporates such a source of non-determinism, then omega is unable to predict, and the whole paradox falls down. Either omega can predict, in which case EDT, else omega cannot predict, in which case CDT. Duhhh. I'm sort of flabbergasted, because these points seem obvious to me ... the Newcomb paradox, as given, seems poorly stated.