I spent quite a few hours going through Peterson's 2008 book (online copy available here) to see if there were any interesting ideas, and found the time largely wasted. (This was my initial intuition, but I thought I'd take a closer look since Luke emailed me directly to ask me to comment.) It would take even more time to write up a good critique, so I'll just point out the most glaring problem: Peterson's proposal for how to derive a utility function from one's subjective uncertainty about one's own choices, as illustrated in this example:
This means that if the probability that you choose salmon is 2/3, and the probability that you choose tuna is 1/3, then your utility of salmon is twice as high as that of tuna.
What if we apply this idea to the choice between $20 and $30?
Let us now return to the problem of perfect discrimination mentioned above. As explained by Luce, the problem is that ‘the [utility] scale is defined only over a set having no pairwise perfect discriminations, which is probably only a small portion of any dimension we might wish to scale That is, the problem lies in the assumption that p(x > y) != 0,1 for all x,y in B. After all, this condition is rather unlikely to be satisfied, because most agents know for sure that they prefer $40 to $20, and $30 to $20, etc.
Peterson tries to solve this problem in section 5.3, but his solution makes no sense. From page 90:
Suppose, for example, that I wish to determine my utility of $20, $30, and $40, respectively. In this case, the non-perfect object can be a photo of my beloved cat Carla, who died when I was fourteen. If offered a choice between $20 and the photo, the probability is 1/4 that I would choose the money; if offered a choice between $30 and the photo, the probability is 2/4 that I would choose the money; and if offered a choice between $40 and the photo, the probability is 3/4 that I would choose the money. This information is sufficient for constructing a single ratio scale for all four objects. Here is how to do it: The point of departure is the three local scales, which have one common element, the photo of Carla. The utility of the photo is the same in all three pairwise choices. Let u(photo) = 1. Then the utility of money is calculated by calibrating the three local scales such that u(photo) = 1 in all of them.
So we end up with u($20)=1/3, u($30)=u(photo)=1, u($40)=3. But this utility function now implies that given a choice between $20 and $30, you'd choose $20 with probability 1/4, and $30 with probability 3/4, contradicting the initial assumption that you'd choose $30 with certainty. I have no idea how Peterson failed to notice this.
I've only read your comment, not anything by Peterson, so I'm just asking for clarification on what he claims to do:
In your first quote of him, he claims to derive utilities from a certain kind of subjective probability. But does he also claim to make the converse derivation? That is, does he also claim to derive those same subjective probabilities from utilities as you do in your final paragraph? It's not clear to me that your first quote of him commits him to doing this.
In the standard approach to axiomatic Bayesian decision theory, an agent (a decision maker) doesn't prefer Act #1 to Act #2 because the expected utility of Act #1 exceeds that of Act #2. Instead, the agent states its preferences over a set of risky acts, and if these stated preferences are consistent with a certain set of axioms (e.g. the VNM axioms, or the Savage axioms), it can be proven that the agent's decisions can be described as if the agent were assigning probabilities and utilities to outcomes and then maximizing expected utility. (Let's call this the ex post approach.)
Peterson (2004) introduces a different approach, which he calls the ex ante approach. In many ways, this approach is more intuitive. The agent assigns probabilities and utilities directly to outcomes (not acts), and these assignments are used to generate preferences over acts. Using this approach, Peterson claims to have shown that the principle of expected utility maximization can be derived from just four axioms.
As Peterson (2009:75,77) explains:
Jensen (2012:428) calls the ex ante approach "controversial," but I can't find any actual published rebuttals to Peterson (2004), so maybe Jensen just means that Peterson's result is "new and not yet percolated to the broad community."
Peterson (2008) explores the ex ante approach in more detail, under the unfortunate title of "non-Bayesian decision theory." (No, Peterson doesn't reject Bayesianism.) Cozic (2011) is a review of Peterson (2008) that may offer the quickest entry point into the subject of ex ante axiomatic decision theory.
Peterson (2009:210) illustrates the controversy nicely:
I'm not a decision theory expert, so I'd be very curious to hear what LW's decision theorists think of the axiomatization in Peterson (2004) — whether it works, and how significant it is.