Given that people can rationally have preferences that make essential reference to history and to the way events came about, why can't risk be one of those historical factors that matter? What's so "irrational" about that?
Nothing. Whoever said there was?
If your goal is to not be a thief, then expected utility theory recommends that you do not steal.
I suspect most of us do have 'do not steal' preferences on the scale of a few hundred pounds or more.
On the other hand, once you get to, say, a few hundred human lives, or the fate of the entire species, then I stop caring about the journey as much. It still matters, but the amount that it matters is too small to ever have an appreciable effect on the decision. This preference may be unique to me, but if so then I weep for humanity.
A desire to avoid arriving at an outcome via thievery does not violate the Axiom of Independence. A desire to avoid arriving via a risky procedure does. However, I'm not convinced that the latter is any more irrational than the former. And I take the point of this thread to be whether obeying the Axiom really is a requirement of rationality.
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?