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 orange, otherwise she gets cholera]. More generally, this axiom holds that a preference must hold independently of the possibility of another outcome (e.g. cholera). »
That has no meaning if you consider whole world states, not just specific outcomes. Because in the lottery it's not "apple or orange" then but "apple with the knowledge I almost got cholera" vs "orange with the knowledge I almost got cholera". And if there is an interaction between the two, then you have different ranking between them. Maybe you had a friend who died of cholera and loved apple, and that'll change how much you appreciate apples knowing you almost had cholera. Maybe not. But anyway, if what you consider are whole world states, then by definition the whole world state is always different when you're offered even a slightly different choice. How can you define an independence principle in that case ?
First, I did study mathematical logic, and please avoid such kind of ad hominem.
Fair enough
That said, if what you're referring to is the whole world state, the outcomes are, in fact, always different. Even if only because there is somewhere in your brain the knowledge that the choice is different.
I thought this would be your reply, but didn't want to address it because the comment was too long already.
Firstly, this is completely correct. (Well, technically we could imagine situations where the outcomes removed your memory of there ever having been a...
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?