The Interrupted Ultimate Newcomb's Problem

When you consider expected utility in a broader context, expected utility in a smaller event (a particular scenario) only contributes proportionally to the probability of that event. The usual heuristic of locally (independently) maximizing expected utility of particular scenarios stops working if your decisions within the scenario are able to control the probability of the scenario. And this happens naturally when the decision as to whether to set up the scenario is based on predictions of your hypothetical decisions within the scenario.

One way of influencing whether a hypothetical scenario is implemented is by making it impossible. When faced with the hypothetical scenario, make a decision that contradicts the assumptions of the hypothetical (such as a prediction of your decision, or something based on that prediction). That would make the hypothetical self-contradictory, impossible to implement.

If you find yourself in an "independent hypothetical", the prior probability of this event is not apparent, but it matters if it can be controlled, and it can be controlled by for example making decisions that contradict the assumptions of the hypothetical. If a certain decision increases expected utility of a game conditional on the game being played, but decreases the probability that the game gets played, this can well make the outcome worse, if the alternative to the game being played is less lucrative.

In our case, if you make a decision that implies that a prime number is composite, given the assumption that the prediction is accurate, this makes the scenario (in the case where the number is prime) impossible, its measure zero, and its conditional expected utility irrelevant (it doesn't contribute to the overall expected utility). Alternatively, if we permit some probability of error in the prediction, this significantly reduces the probability of the scenario, as it now requires the prediction to be in error. (By eliminating the case where the Lottery number is prime, you increase the conditional probability of the number being composite, given the assumption that the game gets played, but you don't change the absolute probability of the number being composite, outside the assumption that the game gets played.)

What's to prevent omega from performing the simulation of you wherein a sign appears reading "INDEPENDENT SCENARIO BEGIN", and he tells you "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."?

Suppose you play this interrupted newcomb + lottery problem 1 jillion times, and the lottery outputs a suitably wide range of numbers. Which strategy wins, 1-boxing or 1-boxing except for when the numbers are the same, then 2-boxing?

Or suppose that you only get to play one game of this problem in your life, so that people only get one shot. So we take 1 jillion people and have them play the game - who wins more money during their single try, the 1-boxers or the 2-box-if-same-ers?

How can 1-boxers win more over 1 jillion games when they win less when the numbers are the same? Because seeing identical numbers is a lot less common for 1-boxers than 2-boxers.

That is, not only are you controlling arithmetic, you're also controlling the probability that the thought experiment happens at all, and yes, you do have to keep track of that.

Gary Drescher fleshes this out in Good and Real (referring to Newcomb's problem where both boxes are transparent, emphasis is mine):

Another way to justify the one-box choice is to note that for all you know, you might be the simulated you; hence you should act in part for your (causal) influence on the simulation outcome....Say the real you assumes that it is the real you. Nothing false can logically follow from that true (even if unjustified) assumption; in particular, nothing false follows as to which choice is in fact more lucrative for you to make. (The simulated you might, however, infer false conclusions from its false assumption that it is the real you.)

And I paraphrase the last sentence of that footnote: Therefore, the one-box choice is more lucrative for you only if you cannot safely assume that you are the real you.

(The original sentence was, "So the one-box choice is more lucrative for you only if you cannot infer otherwise by assuming that you are indeed the real you," which, to me, is most clearly read as expressing the opposite of what Drescher meant to express.)

I still one-box. If Omega believes I will one-box, then I get a million dollars. I want Omega to think I will one-box, because that gives me the maximum payoff. Therefore, I one-box.