The trouble is, any utility function where 1 doesn’t hold is vulnerable to intuition pumps. If you can’t say which of A, B and C is better (e.g. A > B, B > C, C > A), then I can charge you a penny to switch from C → B, then B → A, then A → C, and you’re three pennies poorer.
I really, really hope my utility function’s “set B” can be mapped to the reals. If not, I’m screwed. (It’s fine if what I want varies with time, so long as it’s not circular at a given point in time.)
The trouble is, any utility function where 1 doesn’t hold is vulnerable to intuition pumps. If you can’t say which of A, B and C is better (e.g. A > B, B > C, C > A), then I can charge you a penny to switch from C → B, then B → A, then A → C, and you’re three pennies poorer.
I really, really hope my utility function’s “set B” can be mapped to the reals. If not, I’m screwed. (It’s fine if what I want varies with time, so long as it’s not circular at a given point in time.)