Well, sure, by mangling enough the events you can re-establish the axioms...Of course, by splitting the events like that, you'll reestablish independence - but by showing the need to mangle choices to make fit the axioms, you in fact have shown the axioms don't work in the general case, when the choices you're given are not independent, as it often is in real life.
'If I write out my arithmetic like "one plus two", your calculator can't handle it! Proving that arithmetic doesn't work in the general case. Sure, you can mangle these words into these things you call numbers and symbols like "1+2", but often in real life we don't use them.'
Hrm, could you try to steelman instead of strawmaning my position ?
It's not just a matter of formulation or translating words to symbols. Having to split the choices offered in the real world into an undefined number of virtual choices is not just a switch of notation. Real world choices have far-fetched consequences, and having to split apart all possible interactions between those choices can easily lead to combinatorial explosion of possible choices. benelliott split my choices into 2, but he could have split them into much more : different level of pre...
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?