Which strategy wins, 1-boxing or 1-boxing except for when the numbers are the same, then 2-boxing?
The 2-boxers, because you've misunderstood the problem.
The thought experiment only ever occurs when the numbers coincide. Equivalently, this experiment is run such that Omega will always output the same number as the lottery, in addition to its other restrictions. That's why it's called the Interrupted Newcomb's Problem: it begins in medias res, and you don't have to worry about the low probability of the coincidence itself - you don't have to decide your algorithm to optimize for the "more likely" case.
Or at least, that's my argument. It seems fairly obvious, but it also says "two-box" on a Newcomb-ish problem, so I'd like to have my work checked :p.
I guess I'm not totally clear on how you're setting up the problem, then - I thought it was the same as in Eliezer's post.
Consider this extreme version though: let's call it "perverse Newcomb's problem with transparent boxes."
The way it works is that the boxes are transparent, so that you can see whether the million dollars is there or not (and as usual you can see $1000 in the other box). And the reason it's perverse is that Omega will only put the million dollars there if you will not take the box with the thousand dollars in it no matter what...
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.)