So, when people say 'risk aversion', they can mean one of three different things:
I) I have a utility function that penalises world-histories in which I take risks.
II) I have a utility function which offers diminishing returns in some resource, so I am risk averse in that resource
III) I am risk averse in utility
Out of the three (III) is irrational and violates VNM. (II) is not irrational, and is an extremely common preference among humans wrt some things, but not others (money vs lives being the classic one). (I) is not irrational, but is pretty weird, I'm really not sure I have preferences like this, and when other people claim they do I become a bit suspicious that it is actually a case of (II) or (III).
Since we agree that (I) is not irrational, it remains to show that someone with that preference pattern (and not pattern III) still must have a VNM utility function - then my objection will be answered. Indeed, before we can even attribute "utility" to this person and thus go to case III, we must show that their preferences obey certain rules (or maybe just that their rational ones would).
I don't think preference (I) is weird at all, though I don't share it. Also not rare: a utility function that rewards world histories in which one takes risk...
Followup to : Is risk aversion really irrational?
After reading the decision theory FAQ and re-reading The Allais Paradox I realized I still don't accept the VNM axioms, especially the independence one, and I started thinking about what my true rejection could be. And then I realized I already somewhat explained it here, in my Is risk aversion really irrational? article, but it didn't make it obvious in the article how it relates to VNM - it wasn't obvious to me at that time.
Here is the core idea: information has value. Uncertainty therefore has a cost. And that cost is not linear to uncertainty.
Let's take a first example: A is being offered a trip to Ecuador, B is being offered a great new laptop and C is being offered a trip to Iceland. My own preference is: A > B > C. I love Ecuador - it's a fantastic country. But I prefer a laptop over a trip to Iceland, because I'm not fond of cold weather (well, actually Iceland is pretty cool too, but let's assume for the sake of the article that A > B > C is my preference).
But now, I'm offered D = (50% chance of A, 50% chance of B) or E = (50% chance of A, 50% chance of C). The VNM independence principle says I should prefer D > E. But doing so, it forgets the cost of information/uncertainty. By choosing E, I'm sure I'll be offered a trip - I don't know where, but I know I'll be offered a trip, not a laptop. By choosing D, I'm no idea on the nature of the present. I've much less information on my future - and that lack of information has a cost. If I know I'll be offered a trip, I can already ask for days off at work, I can go buy a backpack, I can start doing the paperwork to get my passport. And if I know I won't be offered a laptop, I may decide to buy one, maybe not as great as one I would have been offered, but I can still buy one. But if I chose D, I've much less information about my future, and I can't optimize it as much.
The same goes for the Allais paradox: having certitude of receiving a significant amount of money ($24 000) has a value, which is present in choice 1A, but not in all others (1B, 2A, 2B).
And I don't see why a "rational agent" should neglect the value of this information, as the VNM axioms imply. Any thought about that?