Vladimir (and you!) get £1000000 zero percent of the time on those occasions when Omega appears
Exactly. Which is our purpose here. We want Omega to give £10 when we can accept it, not when we have to reject it. Which brings us back to my earlier statement:
So to maximize the expected value, you should accept the £10. That way, you get 50% time £1 000 000 and 50% £10. Otherwise you get 50% time £1 000 000 and 50% time £0
If you accept the £10, you get £10, and envelope will be empty. However, just as often(I'm assuming for simplicity that Omega appears always when possible) you receive envelope with £1 000 000 in it.
If you refuse £10, you find that the envelope holds £1 000 000. However, just as often you receive empty envelopes. Your expected value here is £500 000, whereas by accepting your expected value would be £500 005.
Your choice doesn't affect what the envelope holds. It will just as often hold £1 000 000 and be empty. Only thing you can affect here is when does the Omega appear. This is very much unlike the Newcombs problem, where your choice actually affects what the boxes contain.
So effectively, only thing we do here is shift Omega-appearances to the times when we can accept the £10. Like I noted earlier, your choice has already caused Omega to appear, but it has not, and cannot, affect what the envelope contains.
Edit: I should clarify that Omega appearing is a double conditional, if you won, you won regardless. If you lost, you lost regardless. For Omega to appear, your choice, given Omega appearing, has to be the right kind, and result of Alpha coin toss has to be the right kind. If you're the kinda guy to turn down the £10, for Omega to appear envelope has to contain the £1 000 000. Regardless of what you choose, you won anyway. This way however, if you didn't win, Omega wouldn't appear, offering you £10.
Like I noted earlier, your choice has already caused Omega to appear, but it has not, and cannot, affect what the envelope contains.
The nature of my decision procedure affects the conditions under which Omega can appear.
When I first confront this problem, I have not thought it through, but I know that Omega has appeared. So I ask: given that fact, what is the probability that the envelope contains the £1000000?
Without any knowledge of what my decision procedure is, the probability that the envelope contains the £1000000 is .5.
If I am a determined £10...
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.