He uses the axioms required to derive the useful results he aimed for, given his approach to formalizing decision problems, and no more than that.
Can you give me two examples of useful results he derives from the axioms? That'll help me target my response. (I should note that the commentary in the grandparent is targeted at the 2004 paper in the context of the other things you've quoted on this page; if there's relevant material in one of the other links I probably missed it.)
I don't think Peterson denies the usefulness of traditional axiomatic decision theory for checking the consistency of one's preferences, he's just saying that it would also be nice to have a decision theory that can tell you what you should choose given what you believe and what you value.
Agreed. In this comment I want to differentiate between "decision theory" and a component of it, "expected utility theory" (I didn't differentiate between them in the grandparent). The first studies how to make decisions, and the second studies a particular mathematical technique to isolate the highest scoring of a set of alternative actions. My claim is that expected utility theory is and should be silent on the design of human-appropriate utility functions, but that decision theory should include a component focused on the design of human-appropriate utility functions. That component will be primarily researched by psychologists- what makes humans happy, what do humans want, how do we align those, what common mistakes do humans make, what intuitions do humans have and when are those useful, and so on.
Peterson's axioms look to me like trying to shoehorn human-appropriate utility functions into expected utility theory, which doesn't seem to augment the math of calculating expected utilities or augment the actual design of human-appropriate utility functions. As far as I can tell, that field is too young to profit from an axiomatic approach.
But I said "profit" from axioms and you said "justified" with axioms, and those are different things. It's not clear to me that Peterson's axioms are useful at justifying the use of expected utility theory, and my hesitance hinges on the phrase "given what you believe and what you value" from the parent. That means that's Peterson's decision theory takes your beliefs and values as inputs and outputs decisions- which is exactly what traditional decision theory does, and so they look the same to me (and if they're different, I think it's because Peterson made his worse, not better). The underlying problem as I see it is that beliefs and values are not given, they have to be extracted- and traditional decision theory underestimated the difficulty of that extraction.
(Side note: decision theory underestimating the difficulty and decision theorists underestimating the difficulty are very different things. Indeed, it's likely that decision theorists realized the problem was very hard, and so left it to the reader so they wouldn't have to do it!)
Then the question is how much Peterson 2004 helps its readers extract their beliefs and values. As far as I can tell, there's very little normative or prescriptive content.
My claim is that expected utility theory is and should be silent on the design of human-appropriate utility functions, but that decision theory should include a component focused on the design of human-appropriate utility functions.
What do you mean by "the design of human-appropriate utility functions"?
Can you give me two examples of useful results he derives from the axioms? That'll help me target my response.
Actually, let me show you a section of Peterson (2009), which is an updated and (I think) clearer presentation of his axiomatic ex ...
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.