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Jonathan_Lee comments on A problem with Timeless Decision Theory (TDT) - Less Wrong
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This ad-hoc fix breaks as soon as Omega makes a slightly messier game, wherein you receive a physical clue as to a computation output, and this computation and your decision determine your reward.
Suppose that for any output of the computation there is a a unique best decision, and that furthermore this set of (computation output, predicted decision) pairs are mapped to distinct physical clues. Then given the clue you can infer what decision to make and the logical computation, but this requires that you infer from a logical fact (the predictor of you) to the physical state to the clue to the logical fact of the computation.
The game is to pick a box numbered from 0 to 2; there is a hidden logical computation E yielding another value 0 to 2. Omega has a perfect predictor D of you. You choose C.
The payout is 10^((E+C)mod 3), and there is a display showing the value of F = (E-D)mod 3.
If F = 0, then:
And similarly for F = 1, F = 2 play C = F+1 as the only stable solution (which nets you 100 per play)
If you're not allowed to infer anything about E from F, then you're faced with a random pick from winning 1, 10 or 100, and can't do any better...
I'm not sure this game is well defined. What value of F does the predictor D see? (That is, it's predicting your choice after seeing what value of F?)
The same one that you're currently seeing; for all values of E there is a value of F such that this is consistent, ie that D has actually predicted you in the scenario you currently find yourself in.