But with Vladimir's assumptions, he gets the £1000000 zero percent of the time! I quote:
the money-maximizing agents will only be visited by Omega when the envelope is empty
The description "money-maximizing" is wrong, but he is talking about a type of agent which does indeed make it impossible for Omega to ever show up while the £1000000 is there.
To return to your own comment,
our choice only affects whether or not Omega comes and offers you £10 or not,
correct
and you maximize your expected value there by being the kinda guy who takes £10 that's offered.
wrong!
You're missing the fact that Alpha sending a letter happened regardless of Omega, and thus regardless of what you choose, you'd get £1 000 000 from Alpha 50% of time. You can't choose so that you'd get £1 000 000 zero percent of the time simply because your choice doesn't affect that.
I repeat that, since that seems to be the key problem here. Alpha flipped a coin to decide whether or not to send you £1 000 000. Your past or future actions don't have any control over Alpha doing this, and sending you £1 000 000. In particular, your actions, upon receiving t...
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.