It seems to me there that utility functions are not only equivalent up to affine transformations. Both utility functions and subjective probability distributions seem to take some relevant real world factor into account. And it seems you can move these representations between your utility function and your probability distribution while still giving exactly the same choice over all possible decisions.

In the case of discounting, you could for example represent uncertainty in a time-discounted utility function, or you do it with your probability distribution. You could even throw away your probability distribution and have your utility function take into account all subjective uncertainty.

At least I think thats possible. Have there been any formal analyses of this idea?