Perhaps a stronger version is necessary. How about this: P(E|X) U(E|X) should tend to zero as U(E|X) tends to infinity. Or to put that with more mathematical clarity:
For any sequence of hypothetical events E_i, i=0, 1, …, if the sequence of utilities U(E_i|X) tends to infinity then the sequence of expectations P(E_i|X) U(E_i|X) must tend to zero.
Or perhaps an even stronger “uniform” version: For every e > 0 there exists a utility u such that for every event E with U(E|X) > u, its expected utility P(E|X) U(E|X) is less than e.
I called this an axiom, but it would be more accurate to call it a principle, something that any purported decision theory should satisfy as a theorem.
Hm, to be honest, I can’t quite wrap my head around the first version. Specifically, we’re choosing any sequence of events whatsoever, then if the utilities of the sequence tend to infinity (presumably equivalent to “increase without bound”, or maybe “increase monotonically without bound”?), then the expected utilities have to tend to zero? I feel like there’s not enough description of the early parts of the sequence. E.g. if it starts off as “going for a walk in nice weather, reading a mediocre book, kissing someone you like, inheriting a lot of money from a relative you don’t know or care about as you expected to do, accomplishing something really impressive...”, are we supposed to reduce probabilities on this part too? And if not, then do we start when we’re threatened with 3^^^3 disutilons, or only if it’s 3^^^^3 or more, or something?
I don’t think the second version works without setting further restrictions either, although I’m not entirely sure. E.g. choose u = (3^^^^3)^2/e, then clearly u is monotonically decreasing in e, so by the time we get to e = 3^^^^3, we get (approximately) that “an event with utility around 3^^^^3 can have utility at most 3^^^^3” with no further restrictions (since all previous e-u pairs have higher u’s, and therefore do not apply to this particular event), so that doesn’t actually help us any.
Anyway, it took me something like 20 minutes to decide on that, which mostly suggests that it’s been too long since I did actual math. I think the most reasonable and simple solution is to just have a bounded utility function (with the main question of interest being what sort of bound is best). There are definitely some alternative, more complicated, solutions, but we’d have to figure out in what (if any) ways they are actually superior.
Perhaps a stronger version is necessary. How about this: P(E|X) U(E|X) should tend to zero as U(E|X) tends to infinity. Or to put that with more mathematical clarity:
For any sequence of hypothetical events E_i, i=0, 1, …, if the sequence of utilities U(E_i|X) tends to infinity then the sequence of expectations P(E_i|X) U(E_i|X) must tend to zero.
Or perhaps an even stronger “uniform” version: For every e > 0 there exists a utility u such that for every event E with U(E|X) > u, its expected utility P(E|X) U(E|X) is less than e.
I called this an axiom, but it would be more accurate to call it a principle, something that any purported decision theory should satisfy as a theorem.
Hm, to be honest, I can’t quite wrap my head around the first version. Specifically, we’re choosing any sequence of events whatsoever, then if the utilities of the sequence tend to infinity (presumably equivalent to “increase without bound”, or maybe “increase monotonically without bound”?), then the expected utilities have to tend to zero? I feel like there’s not enough description of the early parts of the sequence. E.g. if it starts off as “going for a walk in nice weather, reading a mediocre book, kissing someone you like, inheriting a lot of money from a relative you don’t know or care about as you expected to do, accomplishing something really impressive...”, are we supposed to reduce probabilities on this part too? And if not, then do we start when we’re threatened with 3^^^3 disutilons, or only if it’s 3^^^^3 or more, or something?
I don’t think the second version works without setting further restrictions either, although I’m not entirely sure. E.g. choose u = (3^^^^3)^2/e, then clearly u is monotonically decreasing in e, so by the time we get to e = 3^^^^3, we get (approximately) that “an event with utility around 3^^^^3 can have utility at most 3^^^^3” with no further restrictions (since all previous e-u pairs have higher u’s, and therefore do not apply to this particular event), so that doesn’t actually help us any.
Anyway, it took me something like 20 minutes to decide on that, which mostly suggests that it’s been too long since I did actual math. I think the most reasonable and simple solution is to just have a bounded utility function (with the main question of interest being what sort of bound is best). There are definitely some alternative, more complicated, solutions, but we’d have to figure out in what (if any) ways they are actually superior.