Slight change of plans: the only reason I brought up the "objective" probabilities as early as I did was to help establish the idea of utilities. But with all the holes that seem to need to be patched to get from one to the other (continuity, etc), I decided to take a different route and define utilities more directly. So, for now, forget about "objective probabilities" and frequencies for a bit. I will get back to them a bit later on, but for now am leaving them aside.
So, we've got preference rankings, but not much of a sense of scale yet. We don't have much way of asking "how _much_ do you prefer this to that?" That's what I'm going to deal with in this post. There will be some slightly roundabout abstract bits here, but they'll let me establish utilities. And once I have that, more or less all I have to do is just use utilities as a currency to apply dutch book arguments to. (That will more or less be the shape of the rest of the sequence) The basic idea here is to work out a way of comparing the magnitudes of the differences of preferences. ie, How much you would have prefered some A2 to A1 vs how much you would have prefered some B2 to B1. But it seems difficult to define, no? "how much would you have wanted to switch reality to A2, if it was in state A1, vs how much would you have wanted to switch reality to B2, given that it was in B1?"
So far, the best solution I can think of is to ask "if you are equally uncertain about whether A1 or B1 is true, would you prefer to replace A1, if it would have been true, with A2, or similar for B1 to B2"? Specifically, supposing you're in a state of complete uncertainty with regards to two possible states/outcomes A1 or B1, so that you'd be equally surprised by both. Then consider that, instead of keeping that particular set of two possibilities, you have to choose between two substitutions: you can choose to either conditionally replace A1 with A2 (that is, if A1 would have been the outcome, you get A2 instead) _or_ you can choose to replace B1 with B2 in the same sense. So, you have to choose between (A2 or B1) and (A1 or B2) (where, again, your state of uncertainty is such that you'd be equally surprised by either outcome. That is, you can imagine that whatever it is that's giving rise to your uncertainty is effectively controlling both possibilities. You simply get to decide which of those are wired to the source of uncertainty) If you choose the first, then we will say that the amount of difference in your preference between A2 and A1 is bigger than between B2 and B1. And vice versa. And if you're indifferent, we'll say the preference difference of A2 vs A1 = the preference difference of B2 vs B1.
But wait! You might be saying "oh sure, that's all nice, but why the fluff should we consider this to obey any form of transitivity? Why should we consider this sort of comparison to actually correspond to a real ordered ranking of these things?" I'm glad you asked, because I'm about to tell you! Isn't that convinient? ;)
First, I'm going to introduce a slightly unusual notation which I don't expect to ever need use again. I need it now, however, because I haven't established epistemic probabilities, yet I need to be able to talk about "equivalent uncertainties" without assuming "uncertainty = probability" (which I'll basically be establishing over the next several posts.)
A v B v C v D ... will be defined to mean that you're in a state of uncertainty such that you'd be equally surprised by any of those outcomes. (Obviously, this is commutative. A v B v C is the same state as C v A v B, for instance.)
Next, I need to establish the following principle:
If you prefer Ai v Bi v Ci v... to Aj v Bj v Cj v..., then you should prefer Ai v Bi v Ci v... v Z to Aj v Bj v Cj v... v Z.
If this seems familiar, it should. However, this is a bit more abstract, since we don't yet have a way to measure uncertainty. I'm just assuming here that one can meaningfully say things like "I'd be equally surprised either way." We'll later revisit this argument once we start to apply a numerical measure to our uncertainty.
To deal with a couple possible ambiguities, first imagine you use the same source of uncertainty no matter which outcome you choose. So the only thing you get to choose is which outcomes are plugged into the "consequence slots". Then, imagine that you switch the source of uncertainty with an equivalent one. Unless you place some inherent value in something about whatever it is that is leading to your state of uncertainty or you have additional information (in which case it's not an equivalent amount of uncertainty, so doesn't even apply here), you should value it the same either way, right? Basically an "it's the same, unless it's different" principle. :) But for now, if it helps, imagine it's the same source of uncertainty, just different consequences plugged in. Suppose you prefer the Aj v Bj ... v Z to Ai v Bi ... v Z. You have equal amount of expectation (in the informal sense) of Z in either case, by assumption, so makes no difference which of the two you select as far as Z is concerned. And if Z doesn't happen, you're left with the rest. (Assuming appropriate mutual exclusiveness, etc...) So that leaves you back at the "i"s vs the "j"s, but by assumption you already prefered, overall, the set of "i"s to the set of "j"s. So the result of prefering the second set with Z simply means that either Z is the outcome, which could have happened the same either way or if not Z, then what you have left is equivalent to the Aj v Bj v... option, which, by assumption, you prefer less than the "i"s. So, effectively, you either gain nothing, or end up with a set of possibilities that you prefer less overall. By the power vested in me by Don't Be Stupid, I say that therefore if you prefer Ai v Bi v ... to Aj v Bj v ..., then you must have the same preference ordering when the Z is tacked on.
There is, however, a possibility that we haven't quite eliminated via the above construction: being indifferent to Ai v Bi v... vs Aj v Bj v... while actually prefering one of the versions with the Z tacked on. All I can say to that is "unless you, explicitly, in your preference structure have some term for certain types of sources of uncertainty set up in certain ways leading to certain preferences" here, I don't see any reasonable way that should be happening. ie, where would the latter preference be arising from, if it's not arising from prefereces relating to the individual possibilities?
I admit, this is a weak point. In fact, it may be the weakest part, so if anyone has any actual concrete objections to this bit, I'd be interested in hearing it. But the "reasonableness" criteria seems reasonable here. So, for now at least, I'm going to go with it as "sufficiently established to move on."
So let's get to building up utilities: Suppose the preference difference of A2 vs A1 is larger than preference difference B2-B1, which is larger than preference difference of C2 vs C1.
Is preference difference A2 vs A1 larger than C2 vs C1, in terms of the above way for comparing the magnitudes of preference differences?
Let's find out (Where >, <, and = are being used to represent preference relations)
A2 v B1 > A1 v B2
We also have
B2 v C1 > B1 v C2
Let's now use our above theorem of being able to tack on a "Z" without changing preference ordering.
The first one we will transform into (by tacking an extra C1 onto both sides):
A2 v B1 v C1 > A1 v B2 v C1
The second comparison will be transfomed into (by tacking an extra A1 onto both sides):
A1 v B2 v C1 > A1 v B1 v C2
aha! now we've got an expression that shows up in both the top and the bottom. Specifically A1 v B2 v C1
By earlier postings, we've already established that prefence rankings are transitive, so we must therefore derive:
A2 v B1 v C1 > A1 v B1 v C2
And, again, by the above rule, we can chop off a term that shows up on both sides, specifically B1:
A2 v C1 > A1 v C2
Which is our definition for saying the preference difference between A2 and A1 is larger than that between C2 and C1. (given equal expectation (in the informal sense) of A1 or C1, you'd prefere to replace the possibility A1 with A2 then to replace the possibility C1 with C2). And a similar argument applies for equality. So there, we've got transitivity for our comparisons of differences of preferences. Woooo!
Well then, let us call W a utility function if it has the property that W(B2) - W(B1) =/>/< W(A2) - W(A1) implies the appropriate relation applies to the preference differences. For example, if we have this:
W(B2) - W(B1) > W(A2) - W(A1), then we have this:
B2 v A1 > B1 v A2.
(and similar for equality.)
In other other words, differences of utility act as an internal currency. Gaining X points of utility corresponds to a climb up your preference ranking that's worth the same no matter what the starting point is. This gives us something to work with.
Also, note the relations will hold for arbitrary shifting of everything by an equal amount, and by multiplying everything by some positive number. So, basically, you can do a (positive) affine transform on the whole thing and still have all the important properties retained, since all we care about are the relationships between differences, rather than the absolute values.
And there it is, that's utilities. An indexing of preference rankings with the special property that differences between those indices actually corresponds in a meaningful way to _how much_ you prefer one thing to another.