Refuse the 10 pounds.
The assumptions that you'll move Omega around or otherwise alter Omega's pattern of behavior seems speculative. Maybe Omega's going fishing for a few hundred years. Maybe she's feeling frisky and generous. Maybe I got the problem wrong.
It appears there's some chance that I'm improving my chance at a million pounds by some amount. Those "somes" may not be high, but my problem-uncertainty makes it an easy call. I see no reason to expect a lower or higher number of Omega appearances based on my decision. To the extent this might be true, it's dwarfed by the additional chance at a million pounds.
Once I have my million pounds, I'm getting a restraining order against Omega. He's always trying to screw with me.
--JRM
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.