A Decision Problem

DragonGod

The idea for this problem is gotten from dmytryl.

Omega makes a simulation of you. One of you is presented with an offer Omega offers them $1000.
1. If the simulation is offered the $1000 dollars and rejects it, the real you gets a $10,000.
2. If the simulation is offered the $1,000 dollars and accepts it, the real you gets $100.
3. If the real you is offered $1000 and accepts it, the real you gets $1000.
4. If the real you is offered $1000 and rejects it, the real you gets $0.

Immeidately after completion of the decision problem, the simulation is terminated.
The probability of selecting simulation or real you by Omega is not known. (OMega may always select one option, select both options with equal probability, or select options with any valid probabilities).

You find yourself in the game, with the rules explained as such to you. You don't know if you're the simulation or real, do you accept the $1000 or reject it?

The payoffs only need be of the form: 1. $k*X (k: 1 < k) (X: 1 < X)
2. $X/k
3. $X
4. $0

If $1000 is irrelevant to you, then substitute for any enticing value of X, and replace $X with X utils. There is no diminishing returns on the utility you gain from the reward Omega gives you.

Do you have a strategy for a general form of this problem?

The idea for this problem is gotten from dmytryl.

Omega makes a simulation of you. One of you is presented with an offer Omega offers them $1000.
1. If the simulation is offered the $1000 dollars and rejects it, the real you gets a $10,000.
2. If the simulation is offered the $1,000 dollars and accepts it, the real you gets $100.
3. If the real you is offered $1000 and accepts it, the real you gets $1000.
4. If the real you is offered $1000 and rejects it, the real you gets $0.

Immeidately after completion of the decision problem, the simulation is terminated.
The probability of selecting simulation or real you by Omega is not known. (OMega may always select one option, select both options with equal probability, or select options with any valid probabilities).

You find yourself in the game, with the rules explained as such to you. You don't know if you're the simulation or real, do you accept the $1000 or reject it?

The payoffs only need be of the form: 1. $k*X (k: 1 < k) (X: 1 < X)
2. $X/k
3. $X
4. $0

I also agree with Dagon's first paragraph. Then, since I don't know which game Omega is playing except that either is possible, I will assign 0.5 probability to each game, calculate expected utilities (reject -> $5000, accept -> $550) and reject.

For general form I will reject if k > 1/k + 1, which is the same as k*k - k - 1 > 0 or k > (1+sqrt(5))/2. Otherwise i will accept.

It seems like I'm missing something, though, because it's not clear why you chose these payoffs and not the ones that give some kind of nice answer.

Either is possible and no mention is made of how it's chosen (in fact, it's explicitly stated that the probability is not known), so why would you assign 50% rather than 0% to the chance of game A? If Omega mentioned a few irrelevant options (games C through K) which favored reject, but which it NEVER used (but you don't know that), would you change your acceptance?

0kvas_it9y

You could also make a version where you don't know what X is. In this case always reject strategy doesn't work since you would reject k*X in real life after the simulation rejected X. It seems like if you must precommit to one choice, you would have to accept (and get (X+X/k)/2 on average) but if you have a source of randomness, you could try to reject your cake and eat it too. If you accept with probability p and reject with probability 1 - p, your expected utility would be (p*X + (1-p)*p*k*X + p*p*X/k)/2. If you know the value of k, you can calculate the best p and see if random strategy is better than always-accept. I'm still not sure where this is going though.