I’m not sure I’ve fully followed, but I’m suspicious that you seem to be getting something for nothing in your shift from a type of uncertainty that we don’t know how to handle to a type we do.
It seems to me like you must be making an implicit assumption somewhere. My guess is that this is where you used i to pair S with S′. If you’d instead chosen j=i∘ρ as the matching then you’d have uncertainty between whether m should be j or ρ−1∘j. My guess is that generically this gives different recommendations from your approach.
Nope! That gives the same recommendation (as does the same thing if you pre-compose with any other permutation of S). I thought about putting that fact in, but it took up space.
The recommendation given in both cases is just to normalise each utility function individually, using any of the methods that we know (which will always produce equivalent utility classes in this situation).
I’m not sure I’ve fully followed, but I’m suspicious that you seem to be getting something for nothing in your shift from a type of uncertainty that we don’t know how to handle to a type we do.
It seems to me like you must be making an implicit assumption somewhere. My guess is that this is where you used i to pair S with S′. If you’d instead chosen j=i∘ρ as the matching then you’d have uncertainty between whether m should be j or ρ−1∘j. My guess is that generically this gives different recommendations from your approach.
Nope! That gives the same recommendation (as does the same thing if you pre-compose with any other permutation of S). I thought about putting that fact in, but it took up space.
The recommendation given in both cases is just to normalise each utility function individually, using any of the methods that we know (which will always produce equivalent utility classes in this situation).