Gwern said pretty much everything I wanted to say to this, but there's an extra distinction I want to make
What you're doing is saying you can't use A, B, and C when there is dependency, but have to create subevents like C1="C when you are you sure you'll have either A or C".
The distinction I made was things like A2="A when you prepare" not A2="A when you are sure of getting A or C". This looks like a nitpick, but is in fact incredibly important. The difference between my A1 and A2, is important, they are fundamentally different outcomes which may have completely different utilities, they have no more in common than B1 and C2. They are events in their own right, there is no 'sub-' to it. Distinguishing between them is not 'mangling', putting them together in the first place was always an error.
you in fact have shown the axioms don't work in the general case
It is easily possible to imagine three tennis players A, B and C, such that A beats B, B beats C and C beats A (perhaps A has a rather odd technique, which C has worked hard at learning to deal with despite being otherwise mediocre). Then we have A > B and B > C but not A > C, I have just shown that the axiom of transitivity is not true in the general case!
Well, no, I haven't.
I've shown that the axiom of transitivity does not hold for tennis players. This may be an interesting fact about tennis, but it has not 'disproven' anything, nobody ever claimed that transitivity applied to tennis players.
What the VNM axioms are meant to refer to, are outcomes, meaning a complete description of what will happen to you. "Trip to Ecuador" is not an outcome, because it does not describe exactly what will happen to you, and in particular leaves open whether or not you will prepare for the trip.
This sort of thing is why I think everyone with the intellectual capacity to do so should study mathematical logic. It really helps you learn to keep things cleanly separated in your mind and avoid mistakes like this.
First, I did study mathematical logic, and please avoid such kind of ad hominem.
That said, if what you're referring to is the whole world state, the outcomes are, in fact, awlays different. Even if only because there is somewhere in your brain the knowledge that the choice is different.
To take the formulation in the FAQ : « The independence axiom states that, for example, if an agent prefers an apple to an orange, then she must also prefer the lottery [55% chance she gets an apple, otherwise she gets cholera] over the lottery [55% chance she gets an orang...
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?