Take the £10, and don't bother opening the envelope. You are not (acausally) controlling whether £1'000'000 are in the envelope, but are controlling whether to take the £10, so you'll take the £10 (since you are money-maximizing), and if Omega is correct, the envelope is going to be empty.

The agents that refuse the £10 in this situation will only be visited by Omega when the envelope contains the £1'000'000, while the money-maximizing agents will only be visited by Omega when the envelope is empty. By your decision, you don't control whether the envelop contains money, but you do control whether Omega appears (since the statement asserted by Omega is about you). Thus, by deciding to take the money in this situation, you add expected £5 (or however often Omega appears) to your balance, by acausally summoning Omega.

By refusing the £10, you maximize the amount of money that the agents who see Omega get, by moving Omega around. It's similar to trying to become a lottery winner by selling to existing lottery winners the same dietary supplement you take, since this makes the takers of this dietary supplement more likely to be lottery winners.

I give formalization of this solution in another comment.

This is really cool puzzle. By accepting the £10, you're in a conditional "Alpha never sent you the money", but by refusing you're in conditional "Alpha sent you the money". However, that choice doesn't actually affect Alpha sending or not sending you the money. This is unlike the Newcomb's problem, where you can truly choose, acausally, what the opaque box will contain.

I assume the problem is to be interpreted as Omega saying, "Either (1) (I have predicted you will refuse the $10, and there is $1000,000 in the envelope) xor (2) (I have predicted you will take the $10, and there is $0 in the envelope)", rather than asserting some sort of entanglement above and beyond this.

If so, I take the $10 and formulate the counterfactual, "If I were the sort of person who rejected the $10, Omega would have told me something else to begin with, like 'if you refuse the $10 then the envelope will be empty', but the digit of pi would have been the same".

As previously noted, though, I can't quite say how to compute this formally.

I think this problem would be clearer with a smaller ratio between the two payments. As it is the risk that you might have misunderstood the problem or made an unwarranted assumption dominates and you should not take the £10 just to be safe you aren't making a big mistake, even if you think that's a losing move.

This is formalization of the decision procedure corresponding to the informal solution I gave in another comment (obviously, it includes a lot of detail unnecessary for this problem, but for the purpose of demonstrating the method, details are not omitted):

Programs for the participants:
P - player
O - Omega deciding whether to make the offer
A - Alpha

Notation: [[X]] is the output of program X, X(Y) is a program that is composition of X and Y, where X expects program Y as argument. Thus, [[X(Y)]] is the output of X given argument Y, and X([[Y]]) is the output ... (read more)

What Eliezer and Vladimir said (though if anyone's counting, I decided this before looking at the comments). My choice controls whether or not Omega made its prediction, not the contents of the envelope. (How would one express this using a world-program?)

I humbly request that future thought experiments not be done in £, since there is no "£" key on my keyboard.

I don't understand the theory, but the one-boxing solution seems obvious: given that Omega is correct, if I am such that I would refuse the £10, I would not be offered the choice unless the £1 000 000 is in the envelope, therefore I should refuse the £10 ...

... unless I believe Omega is over u(£1 000 000)/u(£10) times more likely to offer the deal to agents who take the £10 than to agents who refuse. In that case, being willing to take the £10 is expected to pay off.

Edit (after timtyler's reply): Vladimir Nesov's analysis has caused me to reconsider - I would now take the £10.

Isn't this just a reformulation of Newcomb's problem ?

Mechanically, "Omega + alpha + the random generator" is equivalent to Newcomb's Omega.

[Edit: OK, it isn't :)]

The answer is dependong on what Omega would have done if he had predicted that you will refuse the 10 iff there is nothing in Alpha's envelope. Two possibilities :

• Omega1 would have brought you the envelope anyway, but said nothing else

• Omega2 wouldn't have bothered to come, since there's no paradox involved.

When dealing with Omega1, take the £10, yay, free money ! (there wasn't anything in the envelope anyway, otherwise Omega wouldn't have visited you, the taker-of-free-money - see Vladimir's explanation)

The post as stated doesn't tell us which Omega... (read more)

Hmm. Some commentators appear to be assuming that you don't get to keep the contents of the envelope which Alpha sent you. The problem is not 100% clear on this issue - and it makes a difference to the answer!

I one-box on Newcomb's. I two-envelope on this. This situation, however, is absurd. [ETA: Now that I think about it more, I'm now inclined to one-envelope and also more irritated by the hidden assumptions in this whole hypothetical.]

Omega's prediction is bizarre, because there's no apparent way that the contents of the envelope are entangled with my decision to accept the money - whether I am the kind of person who two-boxes or one-boxes, the contents of the envelope were decided by a coin toss. It seems the only way for Omega to make a reliable prediction... (read more)

How are we to read Omega's statement?

if and only if there is £1000 000 in Alpha's envelope.

Or:

I predicted that <you will refuse this £10 if and only if there is £1000 000 in Alpha's envelope>.

The former interpretation leaves open the possibility that, if there is £1000 000 in the envelope, Omega made no prediction one way or the other.

I would translate this scenario into the following world-program:

U(S) =
{
    envelopeIsFilled = coinflip()
    acceptNote = S()

    if (acceptNote == envelopeIsFilled)
        CONTRADICTION
    else
        return (envelopeIsFilled ? 1e6 : 0) + (acceptNote ? 10 : 0)
}

Based on this world-program, it is obvious that you should refuse the note.