But you're the one having the problem! :-) ... I think. Omega, always right, says: "I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha's envelope." So refusing the £10 is my only chance at the £1000000, and I actually have the envelope where the £1000000 may be. Unless it spontaneously combusts, or someone snatches it away, the larger sum should be mine.
Your choice doesn't change what's inside the envelope. Not even a-causally. Your choice only affects whether or not Omega comes and offers you £10 or not, and you maximize your expected value there by being the kinda guy who takes £10 that's offered. That way the 50% of time Alpha doesn't send you £1 000 000, you get £10. Otherwise those 50% time you wouldn't get anything.
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.