To show that my utility for Frank is infinite you have to establish that I wouldn’t trade an arbitrarily small probability of his death for the nanofab. I would make the trade at sufficiently small probabilities.
Also, the surreal numbers are almost always unnecessarily large. Try the hyperreals first.
Not at all. I wouldn’t trade any secular value for Frank’s life, but if I got a deal saying that Frank might die (or live) at a probability of 1/3^^^3, I’d be more curious about how on earth even Omega can get that level of precision than actually worried about Frank.
Not at all. I wouldn’t trade any secular value for Frank’s life
Eh? Do you mean you wouldn’t make the trade at any probability? That would be weird; everyone makes decisions every day that put other people in small probabilities of danger.
Well of course. That’s why I put this in a white room.
(Also, just because I should choose something doesn’t mean I’m actually rational enough to choose it.)
Assuming I am perfectly rational (*cough* *cough*) in the real world, the decision I’m actually making is “some fraction of myself living” versus “small probability of someone else dying.”
What’s wrong with the surreals? It’s not like we have reason to keep our sets small here. The surreals are prettier, don’t require an arbitrary nonconstructive ultrafilter, are more likely to fall out of an axiomatic approach, and can’t accidently end up being too small (up to some quibbles about Grothendieck universes).
I agree with all of that, but I think we should work out what decision theory actually needs and then use that. Surreals will definitely work, but if hyperreals also worked then that would be a really interesting fact worth knowing, because the hyperreals are so much smaller. (Ditto for any totally ordered affine set).
On second thoughts, I think the surreal numbers are what you want to use for utilities. If you choose any subset of the surreals then you can construct a hypothetical agent who assigns those numbers as utilities to some set of choices. So you sometimes need the surreal numbers to express a utility function. And on the other hand the surreal numbers are the universally embedding total order, so they also suffice to express any utility function.
To show that my utility for Frank is infinite you have to establish that I wouldn’t trade an arbitrarily small probability of his death for the nanofab. I would make the trade at sufficiently small probabilities.
Also, the surreal numbers are almost always unnecessarily large. Try the hyperreals first.
Affirm this reply as well.
Not at all. I wouldn’t trade any secular value for Frank’s life, but if I got a deal saying that Frank might die (or live) at a probability of 1/3^^^3, I’d be more curious about how on earth even Omega can get that level of precision than actually worried about Frank.
Eh? Do you mean you wouldn’t make the trade at any probability? That would be weird; everyone makes decisions every day that put other people in small probabilities of danger.
Well of course. That’s why I put this in a white room.
(Also, just because I should choose something doesn’t mean I’m actually rational enough to choose it.)
Assuming I am perfectly rational (*cough* *cough*) in the real world, the decision I’m actually making is “some fraction of myself living” versus “small probability of someone else dying.”
What’s wrong with the surreals? It’s not like we have reason to keep our sets small here. The surreals are prettier, don’t require an arbitrary nonconstructive ultrafilter, are more likely to fall out of an axiomatic approach, and can’t accidently end up being too small (up to some quibbles about Grothendieck universes).
I agree with all of that, but I think we should work out what decision theory actually needs and then use that. Surreals will definitely work, but if hyperreals also worked then that would be a really interesting fact worth knowing, because the hyperreals are so much smaller. (Ditto for any totally ordered affine set).
On second thoughts, I think the surreal numbers are what you want to use for utilities. If you choose any subset of the surreals then you can construct a hypothetical agent who assigns those numbers as utilities to some set of choices. So you sometimes need the surreal numbers to express a utility function. And on the other hand the surreal numbers are the universally embedding total order, so they also suffice to express any utility function.