the problem still fails to compile if you entangle your decision with any property of the number that’s even a little bit related to primeness
That doesn’t seem completely right to me. For example, oddness is related to primeness. If I wanted to do the opposite of what Omega predicted, I might try to one-box on even numbers and two-box on odd numbers. But then Omega can just give me an odd number that isn’t prime. More generally, if we drop the lottery and simplify the problem to just transparent Newcomb’s with prime/composite, then for any player strategy that isn’t exactly “two-box if prime, one-box if composite”, Omega can find a way to be right.
Another problem is that Omega might have multiple ways to be right, e.g. if if your strategy is “one-box if prime, two-box if composite” or “one-box if odd, two-box if even”. But then it seems that regardless of how Omega chooses to break ties, as long as it predicts correctly, one-boxers cannot lose out to other strategies. That applies to the original problem as well, so I’m in favor of one-boxing there (see wedrifid’s and Carl’s comments for details).
Overall I agree that giving an underspecified problem and papering it over with “you don’t have a calculator” isn’t very nice, and it would be better to have well-specified problems in the future. For example, when Gary was describing the transparent Newcomb’s problem, he was careful to say that in the simulation both boxes are full. In our case the problem turned out to be kinda sorta solvable in the end, but I guess it was just luck.
Yep, this all seems correct; the player does not have enough degrees of freedom to prevent there from being a fixpoint, and it is possible to prove for all interpretations that no strategy does better than tying with the simple one-box strategy. But I feel, very strongly, that allowing this particular kind of ambiguity into decision theory problems is a reliably losing move. That road leads only to confusion, and that particular mistake is responsible for many (possibly most) previous failures to figure out decision theory.
That doesn’t seem completely right to me. For example, oddness is related to primeness. If I wanted to do the opposite of what Omega predicted, I might try to one-box on even numbers and two-box on odd numbers. But then Omega can just give me an odd number that isn’t prime. More generally, if we drop the lottery and simplify the problem to just transparent Newcomb’s with prime/composite, then for any player strategy that isn’t exactly “two-box if prime, one-box if composite”, Omega can find a way to be right.
Another problem is that Omega might have multiple ways to be right, e.g. if if your strategy is “one-box if prime, two-box if composite” or “one-box if odd, two-box if even”. But then it seems that regardless of how Omega chooses to break ties, as long as it predicts correctly, one-boxers cannot lose out to other strategies. That applies to the original problem as well, so I’m in favor of one-boxing there (see wedrifid’s and Carl’s comments for details).
Overall I agree that giving an underspecified problem and papering it over with “you don’t have a calculator” isn’t very nice, and it would be better to have well-specified problems in the future. For example, when Gary was describing the transparent Newcomb’s problem, he was careful to say that in the simulation both boxes are full. In our case the problem turned out to be kinda sorta solvable in the end, but I guess it was just luck.
Yep, this all seems correct; the player does not have enough degrees of freedom to prevent there from being a fixpoint, and it is possible to prove for all interpretations that no strategy does better than tying with the simple one-box strategy. But I feel, very strongly, that allowing this particular kind of ambiguity into decision theory problems is a reliably losing move. That road leads only to confusion, and that particular mistake is responsible for many (possibly most) previous failures to figure out decision theory.