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Blueberry comments on Normal Cryonics - Less Wrong
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You may run into problems trying to create a utility function for some forms of deontology, at least if you're mapping into the real numbers. For instance, some deontologists would say that killing a person has infinite negative utility which can't be cancelled out by any number of positive utility outcomes.
That wouldn't be mapping into the real numbers, of course, since infinity isn't a real number.
As I understand it, utility functions are supposed to be equivalence classes of mappings into the real numbers, where two such mappings are said to be equivalent if they are related by a (positive) affine transformation (x -> ax + b where a>0).
Why do you think this restricts to positive affine transformations, rather than any strictly monotonic transformation?
Other monotonic transformations don't preserve preferences over gambles.
Ah, right, that's what I was missing. Thanks.
A strictly monotonic transformation will preserve your preference ordering of states but not your preference ordering for actions to achieve those states. That is, only affine transformations preserve the ordering of expected values of different actions.
Right, which is why I was saying that some ethical theories can't be expressed by a utility function. And there could be many such incomparable qualities: even adding in infinity and negative infinity may not be enough (though the transfinite ordinals, or the surreal numbers, might be).
I'm surprised at that +b, because that doesn't preserve utility ratios.
Ah, I see. But I'm still not actually sure that's true, though...see below.
Indeed not; utilities are measured on an interval scale, not a ratio scale. There's no "absolute zero". (I believe Eliezer made a youthful mistake along these lines, IIRC.) This expresses the fact that utility functions are just (scaled) preference orderings.