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-taker, then the probability that the envelope contains the £1000000 is zero. If I am a determined £10-refuser, then the probability that the envelope contains the £1000000 is one.
But I am neither of those things. I am some more complicated decision-making system which is capable of either taking or refusing the £10, depending on which act is to my advantage. And I can see that if I refuse the £10, then there must be £1000000 in the envelope, which I get to keep. So, I refuse the £10.
Now it might be argued that I just got lucky. If I was as rational as you and Vladimir, then Omega would only ever appear when there was no money in the envelope. But because I hadn't thought things through, it is possible for Omega to show up when there is money in the envelope, and in that case the right thing to do is what I did.
Basically, if you are already an entity which has reflectively optimized its decision procedure for Alpha-Omega situations, then you and Vladimir are making the right choice. But I was not such an entity, and so my choice was the right one for me.
If I am a determined £10-taker, then the probability that the envelope contains the £1000000 is zero. If I am a determined £10-refuser, then the probability that the envelope contains the £1000000 is one.
But I am neither of those things. I am some more complicated decision-making system which is capable of either taking or refusing the £10, depending on which act is to my advantage. And I can see that if I refuse the £10, then there must be £1000000 in the envelope, which I get to keep. So, I refuse the £10.
No. If you knowably refuse the £10 in this sit...
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.