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Mitchell_Porter comments on Omega's subcontracting to Alpha - Less Wrong
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Comments (90)
Huh? Omega is there and says that if and only if you refuse will there be £1000 000 in the envelope. Aren't you turning down £1000 000 for £10?
Nope. I find my explanation pretty clear, can you point to what in particular you don't follow?
I haven't worked through your formalization, but I do know that if I refuse, I get the £1000000! So I think something must be wrong with your implementation of the concept "money-maximizing".
This doesn't clarify the problem you are having.
But you're the one having the problem! :-) ... I think. Omega, always right, says: "I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha's envelope." So refusing the £10 is my only chance at the £1000000, and I actually have the envelope where the £1000000 may be. Unless it spontaneously combusts, or someone snatches it away, the larger sum should be mine.
Your choice doesn't change what's inside the envelope. Not even a-causally. Your choice only affects whether or not Omega comes and offers you £10 or not, and you maximize your expected value there by being the kinda guy who takes £10 that's offered. That way the 50% of time Alpha doesn't send you £1 000 000, you get £10. Otherwise those 50% time you wouldn't get anything.
But with Vladimir's assumptions, he gets the £1000000 zero percent of the time! I quote:
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,
correct
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 the envelope don't have any, direct or indirect, entanglement with what does the envelope contain.
Your actions however are entangled with whether or not Omega comes along to offer you £10. If you're the kinda guy to accept the £10, Omega makes this deal only when Alpha didn't sent you £1 000 000. If you're the kinda guy that refuses £10, Omega comes only when Alpha sent you £1 000 000.
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
Vladimir (and you!) get £1000000 zero percent of the time on those occasions when Omega appears, and by hypothesis this is one of those occasions! You are committing a higher-order version of the two-box mistake.
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:
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.
Omega only appears conditionally on at least the statement it asserts being correct. By taking/not taking its offer, you are only controlling the conditions under which Omega appears, and not contents of the envelope. By refusing the £10, you make sure that Omega appears only when the envelope is full (but you don't make the envelope full, though it's going to be full given that you've made this decision), and by accepting the £10, you make sure that Omega appears only when the envelope is empty.
It's admittedly confusing that you can (acausally) control the conditions under which Omega appears (when the envelope is full/empty), when Omega remains right in front of you during the decision-making (this is analogous to controlling the contents of the big box in Newcomb's problem) but at the same time, you don't control the contents of the envelope.
And by assuming you are a certain sort of agent (which you incorrectly call money-maximizing), you set those conditions to your own disadvantage! An agent which just flips a coin to decide whether to accept or refuse the £10 will have a bigger expected payoff than you. So surely a rational entity can do better.
You are setting the conditions for appearance of Omega. The best conditions for Omega to appear are those where you take its money, since it's good for nothing else.
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.
(Added this paragraph to the top-level comment.)
I'm not 100% sure but it seems like you and Jonii are calculating correctly. It's just ironic that if the situation as described happens to you, it means you were unlucky and there's no money in Alpha's envelope, whereas if it happens to someone like me, it means I was lucky and the £1000000 is there.
So you refuse the £10?
No, I don't. Why?
I'm sorry - I was confused when I wrote that comment.