Is there a particular reason to express utility frameworks with representation theorems, such as the one by Bolker? I assume one motivation for “representing” probabilities and utilities via preferences is the assumption, particularly in economics, that preferences are more basic than beliefs and desires. However, representation arguments can be given in various directions, and no implication is made on which is more basic (which explains or “grounds” the others).
See the overview table of representation theorems here, and the remark beneath:
Notice that it is often possible to follow the arrows in circles—from preference to ordinal probability, from ordinal probability to cardinal probability, from cardinal probability and preference to expected utility, and from expected utility back to preference. Thus, although the arrows represent a mathematical relationship of representation, they do not represent a metaphysical relationship of grounding.
So rather than bothering with Bolker’s numerous assumptions for his representation theorem, we could just take Jeffrey’s desirability axiom:
If and then
Paired with the usual three probability axioms, the desirability axiom directly axiomatizes Jeffrey’s utility theory, without going the path (detour?) of Bolker’s representation theorem. We can also add as an axiom the plausible assumption (frequently used by Jeffrey) that
This lets us prove interesting formulas for operations like the utility of a negation (as derived by Jeffrey in his book) or the utility of an arbitrary non-exclusive disjunction (as I did it a while ago), analogous to the familiar formulas for probability, as well as providing a definition of conditional utility .
Note also that the tautology having 0 utility provides a zero point that makes utility a ratio scale, which means a utility function is not invariant under addition of arbitrary constants, which is stronger than what the usual representation theorems can enforce.
cc @Annapurna