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bsterrett comments on The Ultimate Newcomb's Problem - Less Wrong
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If I one-box then (ignoring throughout the tiny probabilities of Omega being wrong) the number is prime. I receive $1M from Omega and $0 from the Lottery.
If I two-box then the number is composite, Omega pays me $1K, and the Lottery pays me $2M.
Therefore I two-box.
I think this line of reasoning relies on the Number Lottery's choice of number being conditional on Omega's evaluation of you as a one-boxer or two-boxer. The problem description (at the time of this writing) states that the Number Lottery's number is randomly chosen, so it seems like more of a distraction than something you should try to manipulate for a better payoff.
Edit: Distraction is definitely the wrong word. As ShardPhoenix indicated, you might be able to get a better payoff by making your one-box / two-box decision depend on the outcome of the Number Lottery.
Those are not the only precommitments one can make for this type of situation.
Of course! I meant to say that Richard's line of thought was mistaken because it didn't take into account the (default) independence of Omega's choice of number and the Number Lottery's choice of number. Suggesting that there are only two possible strategies for approaching this problem was a consequence of my poor wording.
What? I can't even parse that.
There IS a number in the box which is the same as the one at the Lottery Bank. The number either is prime or it is composite.
According to the hypothetical, if I two-box, there is a 99.9% correlation with Omega putting a composite number in his box, in which case my payooff is $2,001,000. There is a 0.1% correlation with Omega putting a prime number in the box in which case my payoffis $1,001,000. If the correlation is a good estimate of probability, then my expected payoff from two-boxing is $2million more or less. If I one-box, blah blah blah expected payoff is $1million.
Sorry for my poor phrasing. The Number Lottery's number is randomly chosen and has nothing to do with Omega's prediction of you as a two-boxer or one-boxer. It is only Omega's choice of number that depends on whether it believes you are a one-boxer or two-boxer. Does this clear it up?
Note that there is a caveat: if your strategy for deciding to one-box or two-box depends on the outcome of the Number Lottery, then Omega's choice of number and the Lottery's choice of number are no longer independent.