How does this differ from the claim that any other preference pattern that does not violate a VNM axiom is modelled by expected utility?
Well if it doesn't violate an axiom - and specifically I'm worried about Independence - then the case is proven. So let me try to explain why I think it does violate independence. The Allais Paradox provides a case where the risk undertaken depends on so-called irrelevant alternatives. Take the version from Luke's Decision Theory FAQ. If I bet on the option (say "red or yellow") having 34 $24000-payoff balls, whether I take a big risk depends on how many other balls are in the lottery. If there are 66 zero-payoff green balls in the urn at the same time, then I do take a risk. If there are no other balls in the urn, then I don't. If I penalize the preferability of outcomes depending on the risk undertaken, then I will penalize the "red or yellow" bet if and only if the green balls are also involved. Say there are 33 yellow balls and 1 red one, and I get $27000 if I bet on "yellow" instead. I will penalize the outcome, bet on yellow and get $27000, in either scenario. If the penalty is not linear in the amount of risk, I could conceivably prefer to bet on yellow when the green balls are in the urn, and bet on [red or yellow] when there aren't.
I'm not sure quite what the best response to this is, but I think I wasn't understanding you up to this point. We seem to have a bit of a level mixing problem.
In VNM utility theory, we assign utility to outcomes, defined as a complete description of what happens, and expected utility to lotteries, defined as a probability distribution over outcomes. They are measured in the same units, but they are not the same thing and should not be compared directly.
VNM utility tells you nothing about how to calculate utility and everything about how to calculate expect...
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?