I don't think you can always get out of the Pascal's mugging by just assuming that the probability of the mugger telling the truth shrinks in proportion to the stakes. It might work for finite stakes, but what if the bugger tells you that infinitely many people will be tortured if you refuse? You would need a credence of exactly zero to …
I don't think you can always get out of the Pascal's mugging by just assuming that the probability of the mugger telling the truth shrinks in proportion to the stakes. It might work for finite stakes, but what if the bugger tells you that infinitely many people will be tortured if you refuse? You would need a credence of exactly zero to cancel out the stakes, and that seems absurd: Although the claim is very implausible, you can't rule it out with certainty.
To avoid the mugging without modifying the way you perform EV calculations to ignore small probabilities, you must either find some way that it's equally plausible that giving in to the buggers demand could lead to an infinite loss, or perhaps use a decision theory that explicitly takes into account the fact that you're being exploited and acts in a way that prevents the possibility of exploitation.
Zero probability events can still happen. (Suppose God picks a random integer. For any given value i, P(i) = 0. Still, there is some such i that he ends up choosing.) So the mere fact that you "can't rule something out with certainty" doesn't suffice to show that you should give it credence greater than zero.
(Intuitively, some kind of "infinitesimal" value would be nice to work with here, but I gather mathematicians tend to be suspicious of that notion.)
I know zero probability events can still happen - I'm just using Cromwell's Rule here. You still have to be infinitely certain that the mugger is lying to assign a probability of exactly zero that he's telling the truth, and there's no plausible way you could have acquired such certainty. You haven't collected an infinite amount of evidence that he's lying, nor would it be rational to be a priori so confident that no person could actually do what the mugger claims to be doing that you could assign it zero probability.
And for all practical purposes, a probability of 1 is certainty, even though mathematicians technically call it "almost certainty". The only cases where it actually makes sense to talk about zero probability events happening is when the zero-probability event is one option out of uncountably infinitely many mutually exclusive options, which isn't the case with Pascal's mugging.
I don't think you can always get out of the Pascal's mugging by just assuming that the probability of the mugger telling the truth shrinks in proportion to the stakes. It might work for finite stakes, but what if the bugger tells you that infinitely many people will be tortured if you refuse? You would need a credence of exactly zero to cancel out the stakes, and that seems absurd: Although the claim is very implausible, you can't rule it out with certainty.
To avoid the mugging without modifying the way you perform EV calculations to ignore small probabilities, you must either find some way that it's equally plausible that giving in to the buggers demand could lead to an infinite loss, or perhaps use a decision theory that explicitly takes into account the fact that you're being exploited and acts in a way that prevents the possibility of exploitation.
Zero probability events can still happen. (Suppose God picks a random integer. For any given value i, P(i) = 0. Still, there is some such i that he ends up choosing.) So the mere fact that you "can't rule something out with certainty" doesn't suffice to show that you should give it credence greater than zero.
(Intuitively, some kind of "infinitesimal" value would be nice to work with here, but I gather mathematicians tend to be suspicious of that notion.)
I know zero probability events can still happen - I'm just using Cromwell's Rule here. You still have to be infinitely certain that the mugger is lying to assign a probability of exactly zero that he's telling the truth, and there's no plausible way you could have acquired such certainty. You haven't collected an infinite amount of evidence that he's lying, nor would it be rational to be a priori so confident that no person could actually do what the mugger claims to be doing that you could assign it zero probability.
And for all practical purposes, a probability of 1 is certainty, even though mathematicians technically call it "almost certainty". The only cases where it actually makes sense to talk about zero probability events happening is when the zero-probability event is one option out of uncountably infinitely many mutually exclusive options, which isn't the case with Pascal's mugging.