FAWS comments on Omega's subcontracting to Alpha - Less Wrong
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Comments (90)
I assume the problem is to be interpreted as Omega saying, "Either (1) (I have predicted you will refuse the $10, and there is $1000,000 in the envelope) xor (2) (I have predicted you will take the $10, and there is $0 in the envelope)", rather than asserting some sort of entanglement above and beyond this.
If so, I take the $10 and formulate the counterfactual, "If I were the sort of person who rejected the $10, Omega would have told me something else to begin with, like 'if you refuse the $10 then the envelope will be empty', but the digit of pi would have been the same".
As previously noted, though, I can't quite say how to compute this formally.
I assume you would consider "You will take this $10 if and only if Barack Obama is president of the United States." true even if you were completely certain you would take the $10 if John McCain was President. If and only if this was the intended meaning I would agree with your conclusion.