The problem as stated is easy to misunderstand. I personally misunderstood (or "under-understood") it in at least three separate ways: 1. I considered the causal relation between Omega visiting me making that particular prediction and Alpha choosing me as potential receipant an unknown. 2. I considered what sort of predictions Omega would make in various counterfactuals an unknown. 3. I considered the truth value of "I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha's envelope." conditional on me always accepting the money if given a chance and the envelope being empty an unknown.
Even now that my current understanding seems to have have been indirectly confirmed by you my confidence that this understanding is correct is only about 0.95. Even if you were to confirm that I currently understand it correctly in a more direct way I doubt it would raise my confidence above 0.999. Unless the scenario was presented in a way that raised my confidence significantly higher (for example Omega stating: "this situation is in all relevant ways identical to how you eventually came to understand the "Omega's subcontracting to Alpha" scenario presented by Stuart_Armstrong) I'd still refuse the £10.
Alpha has sent me the envelope, and would do so whatever Omega decided to do. The causal decision as to why Omega visited me is irrelevant.
This is irrelevant.
"I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha's envelope." is true. To avoid ambiguity, recast is as: XNOR("I predicted you will refuse this £10", "there is £1000 000 in Alpha's envelope") is true.
As for the large ratio:
Omega snatches the £10 away from you, swallows his words, runs out and returns a bit later with a check fo...
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.