You control the posterior probability of 1033 being composite, conditional on the event of playing the game where both numbers are 1033. Seen from outside that event (i.e. without conditioning on it), what you are controlling is the probability of Omega's number being composite, but not the probability of Lottery number being composite. The expected value of composite 1033 comes from the event of Lottery number being composite, but you don't control the probability of this event. Instead, you control the conditional probability of this event given another event (the game with both 1033). This conditional probability is therefore misleading for the purposes of overall expected utility maximization, where outcomes are weighed by their absolute (prior) probability, not conditional probability on arbitrary sub-events.
Ah, I see. (Thanks! Some rigour helps a lot.)
So, I can definitely see why this applies to the Ultimate Newcomb's Problem. As a contrast to help me understand it, I've adjusted this problem so that P(playing the game, both numbers 1033|playing the game) = ~1. See my response to Manfred here.
It is, of course, possible that your algorithm results in you not playing the game at all - but if Omega does this every year, say, then the winners will be the ones who make the most when the numbers are the same, since no other option exists.
While figuring out my error in my solution to the Ultimate Newcomb's Problem, I ran across this (distinct) reformulation that helped me distinguish between what I was doing and what the problem was actually asking.
... but that being said, I'm not sure if my answer to the reformulation is correct either.
The question, cleaned for Discussion, looks like this:
You approach the boxes and lottery, which are exactly as in the UNP. Before reaching it, you come to sign with a flashing red light. The sign reads: "INDEPENDENT SCENARIO BEGIN."
Omega, who has predicted that you will be confused, shows up to explain: "This is considered an artificially independent experiment. Your algorithm for solving this problem will not be used in my simulations of your algorithm for my various other problems. In other words, you are allowed to two-box here but one-box Newcomb's problem, or vice versa."
This is motivated by the realization that I've been making the same mistake as in the original Newcomb's Problem, though this justification does not (I believe) apply to the original. The mistake is simply this: that I assumed that I simply appear in medias res. When solving the UNP, it is (seems to be) important to remember that you may be in some very rare edge case of the main problem, and that you are choosing your algorithm for the problem as a whole.
But if that's not true - if you're allowed to appear in the middle of the problem, and no counterfactual-yous are at risk - it sure seems like two-boxing is justified - as khafra put it, "trying to ambiently control basic arithmetic".
(Speaking of which, is there a write up of ambient decision theory anywhere? For that matter, is there any compilation of decision theories?)
EDIT: (Yes to the first, though not under that name: Controlling Constant Programs.)