This essentially resolves Pascal’s mugging by fixing some large number X and assigning probability 0 to claims about more than X people.
I understand why this is from a theoretical perspective: if you define X as a finite number, then an “infinite” gamble with low probability can have lower expected value than a finite gamble. It also seems pretty clear that increasing X if the probability of an X-achieving event gets too low is not great.
But from a practical perspective, why do we have to define X in greater detail than just “it’s a very large finite number but I don’t know what it is” and then compare analytically? That is to say
comparing infinite-gambles to finite gambles by analytically showing that, for large enough X, one of them is higher value than the other
comparing infinite-gambles to finite gambles by analytically showing that, for large enough X, the infinite-gamble is higher value than the finite gamble
compare finite gambles to finite gambles as normal
Another way to think about this is that, when we decide to take an action, we shouldn’t use the function
If limX→∞EVAction A(X)>limx→∞EVAction B(X) then return Action A, Otherwise Action B
because we know X is a finite number and taking the limit washes out the important of any terms that don’t scale with X. Instead, we should put the decision output inside the limit, in keeping with the definition that X is just an arbitrarily large finite number:
To the query, "should I Action A over Action B", outputlimx→∞(EVAction A(X)>EVAction B(X))
If we analogize Action A and Action B to wager A and wager B, we see that the “>” evaluator returns FALSE for all X larger than some value of X. Per the epsilon-delta definition of a limit, this concludes that we should not take wager A over wager B and gives us the appropriate decision.
However, if we analogize Action A to “take Pacal’s Mugging” and Action B to “Don’t do that”, we see that at some finite X, the “EV(Pascal’s Mugging) > EV(No Pascal’s Mugging)” function will return TRUE and always return TRUE for larger values of X. Thus we conclude that we should be Pascally mugged.
And obviously, for all finite gambles, the Expected Values of the finite gambles become independent of X for large enough X so we can just evaluate them without the limit.
I understand why this is from a theoretical perspective: if you define X as a finite number, then an “infinite” gamble with low probability can have lower expected value than a finite gamble. It also seems pretty clear that increasing X if the probability of an X-achieving event gets too low is not great.
But from a practical perspective, why do we have to define X in greater detail than just “it’s a very large finite number but I don’t know what it is” and then compare analytically? That is to say
comparing infinite-gambles to finite gambles by analytically showing that, for large enough X, one of them is higher value than the other
comparing infinite-gambles to finite gambles by analytically showing that, for large enough X, the infinite-gamble is higher value than the finite gamble
compare finite gambles to finite gambles as normal
Another way to think about this is that, when we decide to take an action, we shouldn’t use the function
If limX→∞EVAction A(X)>limx→∞EVAction B(X) then return Action A, Otherwise Action B
because we know X is a finite number and taking the limit washes out the important of any terms that don’t scale with X. Instead, we should put the decision output inside the limit, in keeping with the definition that X is just an arbitrarily large finite number:
To the query, "should I Action A over Action B", outputlimx→∞(EVAction A(X)>EVAction B(X))
If we analogize Action A and Action B to wager A and wager B, we see that the “>” evaluator returns FALSE for all X larger than some value of X. Per the epsilon-delta definition of a limit, this concludes that we should not take wager A over wager B and gives us the appropriate decision.
However, if we analogize Action A to “take Pacal’s Mugging” and Action B to “Don’t do that”, we see that at some finite X, the “EV(Pascal’s Mugging) > EV(No Pascal’s Mugging)” function will return TRUE and always return TRUE for larger values of X. Thus we conclude that we should be Pascally mugged.
And obviously, for all finite gambles, the Expected Values of the finite gambles become independent of X for large enough X so we can just evaluate them without the limit.