In any event, you didn't answer the question I asked, which was at what point in time does the two-boxer label the decision "irrational". Is it still "irrational" in their estimation to two-box, in the case where Omega decides after they do?

Time is irrelevant to the two-boxer except as a proof of causal independence so there's no interesting answer to this question. The two-boxer is concerned with causal independence. If a decision cannot help but causally influence the brain scan then the two-boxer would one-box.

Notice that in both cases, the decision arises from information already available: the state of the chooser's brain. So even in the original Newcomb's problem, there is a causal connection between the chooser's brain state and the boxes' contents. That's why I and other people are asking what role time plays: if you are using the correct causal model, where your current brain state has causal influence over your future decision, then the only distinction two-boxers can base their "irrational" label on is time, not causality.

Two-boxers use a causal model where your current brain state has causal influence on your future decisions. They are interested in the causal effects of the decision not the brain state and hence the causal independence criterion does distinguish the cases in their view and they need not appeal to time.

If a two-boxer argues that their decision cannot cause a past event, they have the causal model wrong. The correct model is one of a past brain state influencing both Omega's decision and your own future decision.

They have the right causal model. They just disagree about which downstream causal effects we should be considering.

For me, the simulation argument made it obvious that one-boxing is the rational choice, because it makes clear that your decision is algorithmic. "Then I'll just decide differently!" is, you see, still a fixed algorithm. There is no such thing as submitting one program to Omega and then running a different one, because you are the same program in both cases -- and it's that program that is causal over both Omega's behavior and the "choice you would make in that situation". Separating the decision from the deciding algorithm is incoherent.

No-one denies this. Everyone agrees about what the best program is. They just disagree about what this means about the best decision. The two-boxer says that unfortunately the best program leads us to make a non-optimal decision which is a shame (but worth it because the benefits outweigh the cost). But, they say, this doesn't change the fact that two-boxing is the optimal decision (while acknowledging that the optimal program one-boxes).

How does your hypothetical two-boxer respond to simulation or copy arguments? If you have no way of knowing whether you're the simulated version of you, or the real version of you, which decision is rational then?

I suspect that different two-boxers would respond differently as anthropic style puzzles tend to elicit disagreement.

To put it another way, a two-boxer is arguing that they ought to two-box while simultaneously not being the sort of person who would two-box -- an obvious contradiction. The two-boxer is either arguing for this contradiction, or arguing about the definitions of words by saying "yes, but that's not what 'rational' means".

Well, they're saying that the optimal algorithm is a one-boxing algorithm while the optimal decision is two-boxing. They can explain why as well (algorithms have different causal effects to decisions). There is no immediate contradiction here (it would take serious argument to show a contradiction like, for example, an argument showing that decisions and algorithms are the same thing). For example, imagine a game where I choose a colour and then later choose a number between 1 and 4. With regards to the number, if you pick n, you get $n. With regards to the colour, if you pick red, you get $0, if you pick blue you get $5 but then don't get a choice about the number (you are presumed to have picked 1). It is not contradictory to say that the optimal number to pick is 1 but the optimal colour to pick is blue. The two-boxer is saying something pretty similar here.

What "ought" you do, according to the two-boxer. Well that depends what decision you're facing. If you're facing a decision about what algorithm to adopt, then adopt the optimal algorithm (which one-boxers on all future versions of NP though not ones where the prediction has occurred). If you are not able to choose between algorithms but are just choosing a decision for this occasion then choose two-boxing. They do not give contradictory advice.