The choice between qualitative probabilities and generalized quantitative probabilities is illusory

There are two approaches to generalizing probabilities beyond the classical real-valued numerical approach.

1. Switch the values of the probability function from real numbers to values in some other ordered algebraic entity (e.g., the hyperreals, the surreals, an arbitrary totally ordered monoid).

2. Switch to qualitative probabilities, where instead of assigning values to events, one just compares the events probabilistically (this one is at least as likely than that one).

In the literature (including stuff I wrote myself!), the qualitative probability approach is treated as more general. But in fact, it’s not, at least not if we assume that the “at least as likely as” relation is transitive and reflexive, i.e., is a partial preorder. For suppose that we have a collection F of events and a partial preorder ⪅ on them. Say that A ≈ B iff A ⪅ B and B ⪅ A. Let V be the set of equivalence classes of events under the relation ≈, and define the partial order ≤ on V by [A]≤[B] iff A ⪅ B, where [A] is the equivalence class of A. (All this makes sense if ⪅ is a partial preorder.) Define P(A)=[A].

And that’s it! You now have a probability function P on F whose values are a partially ordered set V. So, what you do with qualitative comparisons you can do with values. Now that I’ve said it, it’s obvious to me and I’m kicking myself for not noticing it earlier. Perhaps some people in the field have noticed it and found it so obvious that it’s not worth saying.

And rather than thinking of there being a debate between probability functions with values and qualitative probabilities, we can now just stick to the probability function approach, and say that there is a serious debate as to what kind of a structure the values have: are they a real closed field, a totally ordered field, a totally ordered monoid (and if so, does its operation correspond to addition or to multiplication in the classical case), a mere partially ordered set, etc.?

The point here applies in other contexts, such as qualitative utilities, moral value comparisons, etc. Using the apparatus of set theory, we can replace a comparison relation by a value-assignment, which makes it more convenient to apply all the apparatus of ordered algebraic entities of different sorts.

As an application, in a recent post I said that given two very plausible axioms on probability functions, one cannot have a regular rotationally-invariant probability function on the measurable subsets of the circle. Using the above remarks, that point immediately extends to qualitative probabilities that satisfy the following two axioms:

1. If A and B are disjoint, A and C are disjoint, and B and C are equally likely, then A ∪ B and A ∪ C are equally likely, and

2. Ω − A and Ω − B are equally likely iff A and B are equally likely,

with rotational invariance being understood as saying that A is at least as likely as B if and only if ρA is at least as likely as B if and only if A is at least as likely as ρB for every rotation ρ, and with regularity understood as saying that every non-empty event is more likely than the empty event.

The structure of the space of utilities

What kind of a structure do the utilities that egoists (on an individual level) and utilitarians (on a wider scale) want to maximize have? A standard approximation is that utilities are like real numbers. They have an order structure, so that we can compare utilities, an additive structure, so we can add utilities, and a multiplicative structure, so we can rescale them with probabilities. But that is insufficiently general. We want to allow for cases such as that any amount of value V₂ swamps any amount of value V₁. Thus, Socrates thought that any amount of virtue is better to have than any amount of pleasure. The structure of the real numbers won't allow that to happen.

A natural generalization is to note that the multiplicative structure of the space of utilities was overkill. We don't need to be able to multiply utilities by utilities. That operation need not make sense. We simply need to be able to multiply utilities by probabilities. Since probabilities are real numbers, a structure that will allow us to do that is that of a partially ordered vector space. However, we should not impose more structure on the utilities than there really is. It makes sense to multiply a utility by a probability in order to represent the value of such-and-such a chance at the utility. And since we have an additive structure on the utilities, we can make sense of multiplying a utility by a number greater than 1. E.g., 2.5U=U+U+(0.5)U. But it is not clear that it always makes conceptual sense to negate utilities. While it makes sense to think of a certain degree of pain as the negative of a certain degree of pleasure, it is not clear that such a negation operation is available in general.

Getting rid of the spurious structure of multiplying utilities by a negative number, and removing the unnecessary multiplication by numbers greater than 1, we get naturally get a structure as follows. Utilities are a partially ordered set with an operation + on them and there is an action of the commutative multiplicative monoid [0,1] on the utilities, with the order, addition and action all compatible.

A further generalization is that [0,1] may not be the best way to represent probabilities in general. So generalize that to a commutative monoid (with multiplicative notation). We now have this. A utility space is a pair (P,U) where P is a commutative monoid with multiplicatively written operation and an action on U, U is a commutative semigroup with an additively written operation + and a partial order ≤, where the operations, action and orders satisfy:

• (xy)a=x(ya) for x,y∈P and a∈U
• x(a+b)=xa+xb for x∈P and a,b∈U
• If a≤b, then xa≤xb for a,b∈U and x∈P
• If a≤b and c∈U, then a+c≤b+c.

I keep on going back and forth on whether U really should have an addition operation, though. I do not know if utilities can be sensibly added.