I’m no decision theorist, but I see no reason utilities can’t function on the basis of a partial ordering rather than a totally ordered numerical function.
The total ordering is really nice because it means we can move from the messy world of outcomes to the neat world of real numbers, whose values are probabilistically relevant. If we move from total ordering to partial ordering, then we are no longer able to make probabilistic judgments based only on the utilities.
If you have some multidimensional utility function, and a way to determine your probabilistic preferences between any uncertain gamble between outcomes x and y and a certain outcome z, then I believe you should be able to find the real function that expresses those probabilistic preferences, and that’s your unidimensional utility function. If you don’t have that way to determine your preferences, then you’ll be indecisive, which is not something we like to build in to our decision theories.
The total ordering is really nice because it means we can move from the messy world of outcomes to the neat world of real numbers, whose values are probabilistically relevant. If we move from total ordering to partial ordering, then we are no longer able to make probabilistic judgments based only on the utilities.
If you have some multidimensional utility function, and a way to determine your probabilistic preferences between any uncertain gamble between outcomes x and y and a certain outcome z, then I believe you should be able to find the real function that expresses those probabilistic preferences, and that’s your unidimensional utility function. If you don’t have that way to determine your preferences, then you’ll be indecisive, which is not something we like to build in to our decision theories.