The cholera example was definitely a bit silly - after all, "cholera" and "apple vs orange" are usually really independent in the real world, you've to make very far-fetched circumstances for them to be dependent. But an axiom is supposed to be valid everywhere - even in far-fetched circumstances ;)
But overall, I understand the thing much better now: in fact, the independence principle doesn't strictly hold in the real world, like there are no strictly right angle in the real world. But yet, like we do use the Pythagoras theorem in the real world, assuming an angle to be right when it's "close enough" to be right, we apply the VNM axioms and the related expected utility theory when we consider the independence principle to have enough validity?
But do we have any way to measure the degree of error introduced by this approximation? Do we have ways to recognize the cases where we shouldn't apply the expected utility theory, because we are too far from the ideal model?
My point never was to fully reject VNM and expected utility theory - I know they are useful, they work in many cases, ... My point was to draw attention on a potential problem (making it an approximation, making it not always valid) that I don't usually see being addressed (actually, I don't remember ever having seen it that explicitly).
I think we have almost reached agreement, just a few more nitpicks I seem to have with your current post.
the independence principle doesn't strictly hold in the real world, like there are no strictly right angle in the real world
Its pedantic, but these two statements aren't analogous. A better analogy would be
"the independence principle doesn't strictly hold in the real world, like the axiom that all right angles are equal doesn't hold in the real world"
"there are no strictly identical outcomes in the real world, like there are no strict...
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?