Omega's subcontracting to Alpha

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.

I'll disregard my earlier comment and assume the latter interpretation for now.

So here are the things that can (and can't) happen:

1. Alpha puts £1 000 000 in the envelope.
   1. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope.
      1. You refuse the £10, and find £1 000 000 in the envelope.
      2. You accept the £10 — contradiction, won't happen.
   2. Omega has predicted you will accept the £10 and that the envelope is empty — contradiction, won't happen.
2. Alpha puts nothing in the envelope.
   1. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope — contradiction, won't happen.
   2. Omega has predicted you will accept the £10 and that the envelope is empty.
      1. You refuse the £10 — contradiction, won't happen.
      2. You accept the £10, and find nothing in the envelope.

So, starting with Alpha's coin flip, here are the only possible paths:

1. Alpha puts £1 000 000 in the envelope. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope. You refuse the £10, and find £1 000 000 in the envelope.
2. Alpha puts nothing in the envelope. Omega has predicted you will accept the £10 and that the envelope is empty. You accept the £10, and find nothing in the envelope.
3. Alpha puts £1 000 000 in the envelope. Omega predicts that if it were to offer you this puzzle (while honestly making the given prediction), you would accept the £10, so it has no choice but to leave you alone. You find £1 000 000 in the envelope.
4. Alpha puts nothing in the envelope. Omega predicts that if it were to offer you this puzzle (while honestly making the given prediction), you would reject the £10, so it has no choice but to leave you alone. You find nothing in the envelope.

Unlike in Newcomb's problem, in this case Omega's prediction is irrelevant to what Alpha actually did. The envelope either contains £1 000 000 or nothing, and you're going to receive it as-is, no matter what Omega says about your future actions. If the envelope has £1 000 000 in it, and you're the sort of person who would accept the £10, then Omega will not offer you this conundrum, because it couldn't honestly state the prediction as given — it would just leave you alone with your new riches. Same if the envelope is empty and you are the sort of person who would reject the £10. Your strategy affects nothing other than whether Omega will show up in the first place.

Conclusion: be the sort of person who would accept the £10. It won't affect whether you'll receive the envelope or what you find in it, and if it is empty, at least you'll get that £10 as a consolation prize.

(And now I'll read the rest of the thread to see if smarter people agree with me.)

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 of X given output of Y (but not Y).

[[A]] is the contents of the envelope (true/false, or 1/0), [[O(P,A)]] is Omega's decision to make the appearance, [[P(O)]] is player's decision.

Problem statement: Omega appeared ([[O(P,A)]] is true), it asserts that either envelope is full, or player takes its offer ([[A]] xor [[P(O)]] is true), P is money-maximizing, what is [[P(O)]]?

First, P(O) (the player that has observed Omega appearing, as distinguished from P that might or might not see Omega; note that Omega is parameterized with whole P, not just P(O)) constructs the expression for its payoff that depends on P(O). It's going to get the contents of the envelope if [[A]], and Omega's money if [[O(P,A)]], where P can be represented as {X,[[P(O)]]}, where X stands for the rest of P, while [[P(O)]] is specifically P's decision in this situation where P observed Omega.

The payoff is
V([[P(O)]])=10^6*[[A]]+10*[[P(O)]]*[[O({X,[[P(O)]]},A)]],
the payoff function is
V(t)=10^6*[[A]]+10*t*[[O({X,t},A)]].
This may be taken as part of problem statement, additionally specifying P(O) via a statement that P(O) has a property of maximizing V([[P(O)]]).

P(O) considers counterfactual actions for the role of [[P(O)]] (the counterfactuals are not assumed to be equal to [[P(O)]]; I use constants T=true and F=false to refer to them). That is, it's computing
[[P(O)]]:=arg max V(t)

First, consider T=true (take Omega's offer). The payoff is
V(T)=10^6*[[A]]+10*T*[[O({X,T},A)]]=10^6*[[A]]+10*[[O({X,T},A)]].
Second, consider F=false (refuse Omega's offer). The payoff is
V(F)=10^6*[[A]]+10*F*[[O({X,F},A)]]=10^6*[[A]].

Clearly, even given that we don't know what O(P,A) is, V(T)>=V(F). Therefore, [[P(O)]]=T (the player takes Omega's offer). Since [[A]] xor [[P(O)]], it follows that [[A]]=false, so there is no point in opening the envelope.

Refuse the 10 pounds.

The assumptions that you'll move Omega around or otherwise alter Omega's pattern of behavior seems speculative. Maybe Omega's going fishing for a few hundred years. Maybe she's feeling frisky and generous. Maybe I got the problem wrong.

It appears there's some chance that I'm improving my chance at a million pounds by some amount. Those "somes" may not be high, but my problem-uncertainty makes it an easy call. I see no reason to expect a lower or higher number of Omega appearances based on my decision. To the extent this might be true, it's dwarfed by the additional chance at a million pounds.

Once I have my million pounds, I'm getting a restraining order against Omega. He's always trying to screw with me.

--JRM

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?)

def P():
    envelope_full = pi_parity(10^6)
    refuse = Omega_Predict(S, "I predicted that you will refuse this ￡10 if and only if there is ￡1000 000 in Alpha's envelope.")
    if envelope_full:
        u += 1000000
    if refuse == envelope_full:
        if not S("I predicted that you will refuse this ￡10 if and only if there is ￡1000 000 in Alpha's envelope."):
            u += 10

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

I'd refuse the £10 unless I was extremely confident (>99.999%) that if I took the £10 I couldn't actually exist because the scenario as given was inconsistent and that the real me would end up with £10 more if this was true.

(i. e. the offer was independent of my decision but the prediction not, Omega would take exactly the same action regardless of whether the envelope was filled, the prediction would be false if I took the £10 in either case, and I would take the £10 in either case)

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 :)]

Take the £10, my reasoning goes as follows: if I precommit to refuse it, either I get the £1,000,000 and refuse £10, or I get £0 and omega doesn't even show up; if I precommit to accept it, either I get the £1,000,000 and omega doesn't even show up, or I get £10 from omega showing up and me accepting (the respective expected utilities being £500,000 and £500,005). I do better by precommitting to take it, so to be reflectively consistent (and win), I must now take it.

I like this problem because it seems to operate on the same intuitions that lead to one-boxing and two-boxing for those who don't do any actual analysis, but the one-boxing intuition leads you astray (though not by much).

Personally, I'd take the £10 on reflection but would have refused the £10 based on my intuitions. I'm pretty sure Omega wouldn't be giving me £10, since if confronted with the situation I would be forced to think, "If I say 'no' now, there's lots of money in that envelope."

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 we're dealing with, so I would have to guess. I'd say Omega2 (so I wouldn't take the coin), but any information about Omega may switch that the other way.

When dealing with Omega2, don't take the £10, 'cause Omega2 doesn't visit takers-of-free-money when the envelope is full, and you want omega to be visiting you!

I'll disregard my earlier comment and assume the latter interpretation for now.

So here are the things that can (and can't) happen:

1. Alpha puts £1 000 000 in the envelope.
   1. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope.
      1. You refuse the £10, and find £1 000 000 in the envelope.
      2. You accept the £10 — contradiction, won't happen.
   2. Omega has predicted you will accept the £10 and that the envelope is empty — contradiction, won't happen.
2. Alpha puts nothing in the envelope.
   1. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope — contradiction, won't happen.
   2. Omega has predicted you will accept the £10 and that the envelope is empty.
      1. You refuse the £10 — contradiction, won't happen.
      2. You accept the £10, and find nothing in the envelope.

So, starting with Alpha's coin flip, here is what can happen:

1. Alpha puts £1 000 000 in the envelope. Omega has predicted you will refuse the £10 and that there is £1 000 000 in the envelope. You refuse the £10, and find £1 000 000 in the envelope.
2. Alpha puts nothing in the envelope. Omega has predicted you will accept the £10 and that the envelope is empty. You accept the £10, and find nothing in the envelope.
3. Alpha puts £1 000 000 in the envelope. Omega predicts that if it were to offer you this puzzle (while remaining completely honest about its own prediction), you would accept the £10, so it has no choice but to leave you alone. You find £1 000 000 in the envelope.
4. Alpha puts nothing in the envelope. Omega predicts that if it were to offer you this puzzle (while remaining completely honest about its own prediction), you would reject the £10, so it has no choice but to leave you alone. You find nothing in the envelope.

If you happen to get as far as having Omega present you with this conundrum as stated, and you find yourself able to convince yourself to refuse the £10, then do so. If the envelope were empty and you were the sort of person who would make this choice, then Omega would not have made the prediction in the first place, as it would have been wrong. It would have left you alone with your empty envelope. Same if the envelope contained the £1 000 000 and you could not convince yourself to refuse the £10, as Omega would have known that in advance.

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 would be to predict my response to Omega's deal, and then phrase the deal such that my predicted response is tied to the actual contents of the envelope. That is, Omega knows the envelope is empty, and he knows I will accept the offer, so he says "reject offer iff envelope is full."

In other words, I don't actually believe Omega can reliably make this same prediction, as he could theoretically do in Newcomb's; this hypothetical is absurd. If you had a thousand people, roughly half of them would have opened envelopes full of money (or whatever percentage would emerge from Alpha's random generator). It seems inconceivable that those half must also have rejected the ten pound note, and that the other half must have accepted it. If I saw a large enough sample illustrating this effect occurring consistently, I'd have to throw up my hands in confusion and reject the ten-pound note, but this outcome is outlandishly unlikely.

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.

U2(S) =
{
    try {
        return U(S)
    } catch(contradiction) {
        return 0
    }
}

This is just Newcomb's problem with a coin flipped on how the boxes are labeled, so of course I onebox.