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Squark comments on Anatomy of Multiversal Utility Functions: Tegmark Level IV - Less Wrong Discussion
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Thx for commenting!
Unbounded utility functions come with a world of trouble. For example, if your utility function is computable, its Solomonoff expectation value will almost always diverge since utilities will grow as BB(Kolmogorov complexity) whereas probabilities will only fall as 2^{-Kolmogorov complexity}. Essentially, it's Pascal mugging.
It is possible to consider utility functions that give a finite extra award for living forever. For example, say that the utility for T years of life is 1 - exp(-T / tau) whereas the utility for an infinite number of years of life is 2. Such a utility function is not lower semicontinuous, but as I explained in the post it seems that we only need upper semicontinuity.
"Finite extra reward" - sneaky, I like it! I'm still in doubt, mind you. Pascal's mugging might be part of a reason to abandon cardinal utility altogether, rather than restricting it to bounded forms.