Counterfactual mugging is the typical problem used to motivate updateless decision theory:
Omega flips a coin. If Heads, it asks you for $1. If Tails, it offers you $5 only if it predicts you would have given it the $1 had the coin come up Heads.
Counterfactuals are confusing though, so let us re-phrase it this way:
Omega flips a coin. If Heads, it offers you $ (i.e. -$1 if the coin came up heads; +$5 if it came up tails). If Tails, it simulates a version of you in which the coin came up Heads, and lets that simulated version make the decision as to whether to take the offer.
[Take a moment to convince yourself that these are indeed equivalent.]
This is already arguably more convincing than the original problem: if Omega comes to you and says the coin came up Heads, then you might be in the simulated world, and might actually be choosing for the outside version of you. Of course, the simulated version of you needn't care about the outside version of you, but a rational agent would self-modify to have identical versions of itself care about each other.
Omega flips a coin. If Heads, it has a 90% chance to LetYouChoose. If Tails, it has a 10% chance to LetYouChoose. If it LetsYouChoose, it offers you $, and your decision in this case will be used in all circumstances.
This "all circumstances" needs to be made precise: we must ensure the simulated cases add up to the correct 90% 10% 10% 90%. E.g.
There are 10 identical copies of you, with a shared bank account. Omega flips a coin. If Heads, it gives 9 of you a GreenMarble. If Tails, it gives 1 of you a GreenMarble. For those with a GreenMarble, it offers a single payment of $ to the bank account.
Now it is apparent that Psy-kosh's problem is in fact exactly the same.
There are 10 identical copies of you, with a shared bank account. Omega flips a coin. If Heads, it gives 9 of you a GreenMarble. If Tails, it gives 1 of you a GreenMarble. For those with a GreenMarble, it offers a single payment of $ to the bank account.
[Actually, Psy-kosh's problem is negated, but this is OK: you still have two choices; saying YES in the first case is equivalent to saying NO in Psy-kosh's case. Basically in the 90% -> 100% limit of Psy-kosh's problem, Omega says: I'll give you $1, but if you accept it then I would have murdered you if the coin had come up tails. So accepting the $1 in Psy-kosh is equivalent to not paying the $1 in Counterfactual Mugging.]
Counterfactual mugging is the typical problem used to motivate updateless decision theory:
Counterfactuals are confusing though, so let us re-phrase it this way:
[Take a moment to convince yourself that these are indeed equivalent.]
This is already arguably more convincing than the original problem: if Omega comes to you and says the coin came up Heads, then you might be in the simulated world, and might actually be choosing for the outside version of you. Of course, the simulated version of you needn't care about the outside version of you, but a rational agent would self-modify to have identical versions of itself care about each other.
This "you only choose in some worlds" logic may remind you of Psy-kosh's non-anthropic problem or Conitzer 2017's Dutch book for EDT Sleeping Beauties (see also my simplified version here). Indeed, soften the problem as follows:
This "all circumstances" needs to be made precise: we must ensure the simulated cases add up to the correct 90% 10% 10% 90%. E.g.
Now it is apparent that Psy-kosh's problem is in fact exactly the same.
[Actually, Psy-kosh's problem is negated, but this is OK: you still have two choices; saying YES in the first case is equivalent to saying NO in Psy-kosh's case. Basically in the 90% -> 100% limit of Psy-kosh's problem, Omega says: I'll give you $1, but if you accept it then I would have murdered you if the coin had come up tails. So accepting the $1 in Psy-kosh is equivalent to not paying the $1 in Counterfactual Mugging.]