If this seems counterintuitive, it’s because you can’t really just go ahead and imagine something being 10x better. It’s not because of the infinities involved. Substitute “infinite” for Graham’s number of years and it’s basically the same.
I know there are weird paradoxes involving infinite value, but none of them seem to be showing up in this problem, and I’ve got to decide somehow, so I might as well multiply. It’s not like I have a better backup algorithm for when multiplication fails.
You can multiply by Graham’s number and get a meaningful result. Try finding the expected return of four possible distributions: One in which you have epsilon chance of infinite negative utility and a 1-epsilon chance of doubleplusgood utility, one in which you have 95% chance of -infinity and 5% chance of plusgood, one in which you have epsilon chance of Graham’s number negative utility and a 1-epsilon chance of doubleplusgood utility, and one where you have 100% chance of doubleplusungood utility.
Consider the case where epsilon is BB(Graham’s number).
The first has an expected utility of -infinity, the second has the same value, but the third has an expected value of roughly doubleplusgood, despite having identical outcomes to the first one.
If this seems counterintuitive, it’s because you can’t really just go ahead and imagine something being 10x better. It’s not because of the infinities involved. Substitute “infinite” for Graham’s number of years and it’s basically the same.
I know there are weird paradoxes involving infinite value, but none of them seem to be showing up in this problem, and I’ve got to decide somehow, so I might as well multiply. It’s not like I have a better backup algorithm for when multiplication fails.
You can multiply by Graham’s number and get a meaningful result. Try finding the expected return of four possible distributions: One in which you have epsilon chance of infinite negative utility and a 1-epsilon chance of doubleplusgood utility, one in which you have 95% chance of -infinity and 5% chance of plusgood, one in which you have epsilon chance of Graham’s number negative utility and a 1-epsilon chance of doubleplusgood utility, and one where you have 100% chance of doubleplusungood utility.
Consider the case where epsilon is BB(Graham’s number).
The first has an expected utility of -infinity, the second has the same value, but the third has an expected value of roughly doubleplusgood, despite having identical outcomes to the first one.