If, instead of paying £50 to refusers, Omega doesn’t do that but takes away an additional £50 from payers (without giving them any choice), the problem seems to go away (at least, as far as I can tell—I need to check the maths later).
Yet we would expect this case to be identical to your case wouldn’t we?
If he did that, it would no longer be rational to accept to pay (expected return 0.5x(£260) + 0.5x(-£150 -£150) = -£20). The fact that this case is NOT identical to the first one is due to the whole Sleeping Beauty set-up.
i.e. a slightly clearer statement of my imagined setup is this:
Omega flips a coin. On tails, he asks you if you’ll pay £125 now, knowing that if this is day 1 he’ll wipe your memory and ask you again tomorrow.
On heads, he simulates you and sends you £260 if you pay.
There is never any money paid to non-payers.
(Basically, the only difference between this version and yours is that both paying and not paying have a return that is £25 lower than in your version. That surely shouldn’t make a difference, but it makes the problem go away.)
Not paying always gives £0.
Precommitting gives a return of 0.5 x £260 + 0.5 x (-£250) = £5
By your logic, at the time of the decision, return is 1/3(£260-£125-£125) = £3.33
If, instead of paying £50 to refusers, Omega doesn’t do that but takes away an additional £50 from payers (without giving them any choice), the problem seems to go away (at least, as far as I can tell—I need to check the maths later).
Yet we would expect this case to be identical to your case wouldn’t we?
If he did that, it would no longer be rational to accept to pay (expected return 0.5x(£260) + 0.5x(-£150 -£150) = -£20). The fact that this case is NOT identical to the first one is due to the whole Sleeping Beauty set-up.
No: I meant, if you pay, you pay a total of £250.
i.e. a slightly clearer statement of my imagined setup is this:
Omega flips a coin. On tails, he asks you if you’ll pay £125 now, knowing that if this is day 1 he’ll wipe your memory and ask you again tomorrow.
On heads, he simulates you and sends you £260 if you pay.
There is never any money paid to non-payers.
(Basically, the only difference between this version and yours is that both paying and not paying have a return that is £25 lower than in your version. That surely shouldn’t make a difference, but it makes the problem go away.)
Not paying always gives £0.
Precommitting gives a return of 0.5 x £260 + 0.5 x (-£250) = £5
By your logic, at the time of the decision, return is 1/3(£260-£125-£125) = £3.33
Isn’t that correct? I admit I suck at maths.