The line isn't arbitrary - if you're told "you'll receive a gift", you're given much less information than if you're told "you'll receive a trip as a gift". The same goes here : in option D (a coin is tossed, heads you're given a trip to Ecuador, tails you're given a laptop) you are given much less information than in option E (a coin is tossed, heads you're given a trip to Ecuador, tails you're given a trip to Iceland). In option E you know you'll be given a trip - and you can prepare for a trip. In option D, if you prepare for a trip, you've 50% chance of the preparation being wasted.
Now take your grandma dialogue. It needs to be added an additional option to see where it contradicts the VNM independence hypothesis: consider grandma could either offer one of us a trip around the world (T), or offer one of us a driving license (L). I value a trip around the world $2000 (in subjective dollars), and a driving license $1850 (in subjective dollars).
To make the trip around the world, as you said, I've to spend $100 in paperwork, but there is no such cost for driving lessons. So the total the gain $1900 if I'm offered a trip around the world, and $1850 if offered a driving lesson, just for me. I prefer T over L.
But if we are in the situation of your dialogue, I'm offered (50% chance of T) but I need to make the paperwork and spend the $100 anyway. So in fact, the total value of that offer is $2000/2-$100 = $900. While if I'm offered (50% chance of L) then the total value of that offer is $1850/2 = $925.
So I prefer T over L, but I prefer 50% chance of L to 50% chance of T. Which violates the independence principle.
...It needs to be added an additional option to see where it contradicts the VNM independence hypothesis: consider grandma could either offer one of us a trip around the world (T), or offer one of us a driving license (L). I value a trip around the world $2000 (in subjective dollars), and a driving license $1850 (in subjective dollars). To make the trip around the world, as you said, I've to spend $100 in paperwork, but there is no such cost for driving lessons. So the total the gain $1900 if I'm offered a trip around the world, and $1850 if offered a drivin
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?