In Richard Jeffrey’s utility theory there is actually a very natural distinction between positive and negative motivations/desires. A plausible axiom is U(⊤)=0 (the tautology has zero desirability: you already know it’s true). Which implies with the main axiom[1] that the negation of any proposition with positive utility has negative utility, and vice versa. Which is intuitive: If something is good, its negation is bad, and the other way round. In particular, if U(X)=U(¬X) (indifference between X and ¬X), then U(X)=U(¬X)=0.
More generally, U(¬X)=−(P(X)/P(¬X))U(X). Which means that positive and negative utility of a proposition and it’s negation are scaled according to their relative odds. For example, while your lottery ticket winning the jackpot is obviously very good (large positive utility), having a losing ticket is clearly not very bad (small negative utility). Why? Because losing the lottery is very likely, far more likely than winning. Which means losing was already “priced in” to a large degree. If you learned that you indeed lost, that wouldn’t be a big update, so the “news value” is negative but not large in magnitude.
Which means this utility theory has a zero point. Utility functions are therefore not invariant under adding an arbitrary constant. So the theory actually allows you to say X is “twice as good” as Y, “three times as bad”, “much better” etc. It’s a ratio scale.
In Richard Jeffrey’s utility theory there is actually a very natural distinction between positive and negative motivations/desires. A plausible axiom is U(⊤)=0 (the tautology has zero desirability: you already know it’s true). Which implies with the main axiom[1] that the negation of any proposition with positive utility has negative utility, and vice versa. Which is intuitive: If something is good, its negation is bad, and the other way round. In particular, if U(X)=U(¬X) (indifference between X and ¬X), then U(X)=U(¬X)=0.
More generally, U(¬X)=−(P(X)/P(¬X))U(X). Which means that positive and negative utility of a proposition and it’s negation are scaled according to their relative odds. For example, while your lottery ticket winning the jackpot is obviously very good (large positive utility), having a losing ticket is clearly not very bad (small negative utility). Why? Because losing the lottery is very likely, far more likely than winning. Which means losing was already “priced in” to a large degree. If you learned that you indeed lost, that wouldn’t be a big update, so the “news value” is negative but not large in magnitude.
Which means this utility theory has a zero point. Utility functions are therefore not invariant under adding an arbitrary constant. So the theory actually allows you to say X is “twice as good” as Y, “three times as bad”, “much better” etc. It’s a ratio scale.
If P(X∧Y)=0 and P(X∨Y)≠0 then U(X∨Y)=P(X)U(X)+P(Y)U(Y)P(X)+P(Y).