I think the most mathy (and thus, best :P) way to go about this is to think of the properties that these “utility” objects have, and just define them as objects with those properties.
For starters, you can compare them for size—The relationship is either bigger, or smaller, or the same. And you can do an operation to them that is a weighted sum—if you have two utilities that are different, you can do this operation to them and get a utility that’s in between them, with a third parameter (the probability of one versus the other) distinguishing between different applications of this operation.
Actually, I think this sort of thing is pretty much what Savage did.
I think the most mathy (and thus, best :P) way to go about this is to think of the properties that these “utility” objects have, and just define them as objects with those properties.
For starters, you can compare them for size—The relationship is either bigger, or smaller, or the same. And you can do an operation to them that is a weighted sum—if you have two utilities that are different, you can do this operation to them and get a utility that’s in between them, with a third parameter (the probability of one versus the other) distinguishing between different applications of this operation.
Actually, I think this sort of thing is pretty much what Savage did.