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.
The problem as stated is easy to misunderstand. I personally misunderstood (or “under-understood”) it in at least three separate ways: 1. I considered the causal relation between Omega visiting me making that particular prediction and Alpha choosing me as potential receipant an unknown. 2. I considered what sort of predictions Omega would make in various counterfactuals an unknown. 3. I considered the truth value of “I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha’s envelope.” conditional on me always accepting the money if given a chance and the envelope being empty an unknown.
Even now that my current understanding seems to have have been indirectly confirmed by you my confidence that this understanding is correct is only about 0.95. Even if you were to confirm that I currently understand it correctly in a more direct way I doubt it would raise my confidence above 0.999. Unless the scenario was presented in a way that raised my confidence significantly higher (for example Omega stating: “this situation is in all relevant ways identical to how you eventually came to understand the “Omega’s subcontracting to Alpha” scenario presented by Stuart_Armstrong) I’d still refuse the £10.
Alpha has sent me the envelope, and would do so whatever Omega decided to do. The causal decision as to why Omega visited me is irrelevant.
This is irrelevant.
“I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha’s envelope.” is true. To avoid ambiguity, recast is as:
XNOR(“I predicted you will refuse this £10″, “there is £1000 000 in Alpha’s envelope”) is true.
As for the large ratio:
Omega snatches the £10 away from you, swallows his words, runs out and returns a bit later with a check for £100 000. “Out of deference to your uncertainties”, he says, sighing, “I’ve decided to renew the experiment with a lesser ratio. But just this once!”
No, it’s not. If, conditional on me always rejecting the £10 when Omega makes this specific prediction, Omega would visit when the envelope was empty, offer £10 and make the different prediction that I’d take it (the assumption being that I wouldn’t refuse it without reason so Omega can’t make the true prediction that I’d do so), or if, conditional on me always taking the £10 when Omega makes this specific prediction, Omega would visit when the envelope was full, offer £10 and make the different prediction that I’d take it that would change the payoff. If only the first was true that would make the scenarios equivalent.
Omega snatches the £10 away from you, swallows his words, runs out and returns a bit later with a check for £100 000. “Out of deference to your uncertainties”, he says, sighing, “I’ve decided to renew the experiment with a lesser ratio. But just this once!”
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.
The large ratio is deliberate (and it’s not so huge that ‘all my theories are wrong!’ is going to dominate).
The problem as stated is easy to misunderstand. I personally misunderstood (or “under-understood”) it in at least three separate ways: 1. I considered the causal relation between Omega visiting me making that particular prediction and Alpha choosing me as potential receipant an unknown. 2. I considered what sort of predictions Omega would make in various counterfactuals an unknown. 3. I considered the truth value of “I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha’s envelope.” conditional on me always accepting the money if given a chance and the envelope being empty an unknown.
Even now that my current understanding seems to have have been indirectly confirmed by you my confidence that this understanding is correct is only about 0.95. Even if you were to confirm that I currently understand it correctly in a more direct way I doubt it would raise my confidence above 0.999. Unless the scenario was presented in a way that raised my confidence significantly higher (for example Omega stating: “this situation is in all relevant ways identical to how you eventually came to understand the “Omega’s subcontracting to Alpha” scenario presented by Stuart_Armstrong) I’d still refuse the £10.
Alpha has sent me the envelope, and would do so whatever Omega decided to do. The causal decision as to why Omega visited me is irrelevant.
This is irrelevant.
“I predicted that you will refuse this £10 if and only if there is £1000 000 in Alpha’s envelope.” is true. To avoid ambiguity, recast is as: XNOR(“I predicted you will refuse this £10″, “there is £1000 000 in Alpha’s envelope”) is true.
As for the large ratio:
Omega snatches the £10 away from you, swallows his words, runs out and returns a bit later with a check for £100 000. “Out of deference to your uncertainties”, he says, sighing, “I’ve decided to renew the experiment with a lesser ratio. But just this once!”
No, it’s not. If, conditional on me always rejecting the £10 when Omega makes this specific prediction, Omega would visit when the envelope was empty, offer £10 and make the different prediction that I’d take it (the assumption being that I wouldn’t refuse it without reason so Omega can’t make the true prediction that I’d do so), or if, conditional on me always taking the £10 when Omega makes this specific prediction, Omega would visit when the envelope was full, offer £10 and make the different prediction that I’d take it that would change the payoff. If only the first was true that would make the scenarios equivalent.
I take it of course.