What deal would you have made if you didn’t know that the first digit of pi was odd, or if you didn’t know that 1+1=2?
It seems, the important factor is how does Omega makes its choice of which digit of pi (or other logical fact) to check. If Omega uses a quantum coin to select a number N between 1 and 84, and then checks the N-th digit of pi, then you should cooperate. If Omega searches through a roundish number N such that N-th digit of pi is odd, then the answer appears to further depend on Omega’s motivation. If Omega made the choice of “seeking for ‘odd’ roundish numbers” vs. “‘even’ roundish numbers” by tossing a quantum coin, then you should cooperate. Otherwise… etc.
It seems, the important factor is how does Omega makes its choice of which digit of pi (or other logical fact) to check. If Omega uses a quantum coin to select a number N between 1 and 84, and then checks the N-th digit of pi, then you should cooperate. If Omega searches through a roundish number N such that N-th digit of pi is odd, then the answer appears to further depend on Omega’s motivation. If Omega made the choice of “seeking for ‘odd’ roundish numbers” vs. “‘even’ roundish numbers” by tossing a quantum coin, then you should cooperate. Otherwise… etc.