Utility functions are invariant under ordering-preserving transformations.
Utility functions in the sense of VNM, Savage, de Finetti, Jeffrey-Bolker, etc. are not invariant under all ordering-preserving transformations, only affine ones. Exponentiation is not affine.
What sort of utility function do you have in mind?
Oops, you’re right. I clearly took too many mental shortcuts when formulating that response.
What sort of utility function do you have in mind?
The reason this still works is because in the actual formulation I had in mind, we then plug the utility-function-transformed-into-a-probability-distribution into a logarithm function, canceling out the exponentiation. Indeed, that was the actual core statement in my post: that maximizing expected utility is equivalent to minimizing the cross-entropy between some target distribution and the real distribution.
But evidently I decided to skip some steps and claim that the utility function is directly equivalent to the target distribution. That was, indeed, unambiguously incorrect.
Utility functions in the sense of VNM, Savage, de Finetti, Jeffrey-Bolker, etc. are not invariant under all ordering-preserving transformations, only affine ones. Exponentiation is not affine.
What sort of utility function do you have in mind?
Oops, you’re right. I clearly took too many mental shortcuts when formulating that response.
The reason this still works is because in the actual formulation I had in mind, we then plug the utility-function-transformed-into-a-probability-distribution into a logarithm function, canceling out the exponentiation. Indeed, that was the actual core statement in my post: that maximizing expected utility is equivalent to minimizing the cross-entropy between some target distribution and the real distribution.
But evidently I decided to skip some steps and claim that the utility function is directly equivalent to the target distribution. That was, indeed, unambiguously incorrect.