Omega's subcontracting to Alpha

Stuart_Armstrong

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.

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.

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?

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

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

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.
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:

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.
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.
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.
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.)

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Omega's subcontracting to Alpha

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Omega's subcontracting to Alpha

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