- This is irrelevant.
No, it's not. If, conditional on me always rejecting the £10 when Omega makes this specific prediction, Omega would visit when the envelope was empty, offer £10 and make the different prediction that I'd take it (the assumption being that I wouldn't refuse it without reason so Omega can't make the true prediction that I'd do so), or if, conditional on me always taking the £10 when Omega makes this specific prediction, Omega would visit when the envelope was full, offer £10 and make the different prediction that I'd take it that would change the payoff. If only the first was true that would make the scenarios equivalent.
Omega snatches the £10 away from you, swallows his words, runs out and returns a bit later with a check for £100 000. "Out of deference to your uncertainties", he says, sighing, "I've decided to renew the experiment with a lesser ratio. But just this once!"
I take it of course.
This is a variant built on Gary Drescher's xor problem for timeless decision theory.
You get an envelope from your good friend Alpha, and are about to open it, when Omega appears in a puff of logic.
Being completely trustworthy as usual (don't you just hate that?), he explains that Alpha flipped a coin (or looked at the parity of a sufficiently high digit of pi), to decide whether to put £1000 000 in your envelope, or put nothing.
He, Omega, knows what Alpha decided, has also predicted your own actions, and you know these facts. He hands you a £10 note and says:
"(I predicted that you will refuse this £10) if and only if (there is £1000 000 in Alpha's envelope)."
What to do?
EDIT: to clarify, Alpha will send you the envelope anyway, and Omega may choose to appear or not appear as he and his logic deem fit. Nor is Omega stating a mathematical theorem: that one can deduce from the first premise the truth of the second. He is using XNOR, but using 'if and only if' seems a more understandable formulation. You get to keep the envelope whatever happens, in case that wasn't clear.