I think there’s a specific clarification that might resolve the main concern.
The independence axiom is not about conditioning on observed information. If you observe C and then choose between A and B, conditioning your choice on what you observed is perfectly rational under EUT as well and nobody disputes this. The independence axiom constrains something different: your ex ante preference between two compound lotteries, before any uncertainty resolves.
Concretely: suppose you must choose right now between two lotteries, both of which contain the same common component C mixed in at the same probability:
Lottery 1: 90% chance of C, 10% chance of A Lottery 2: 90% chance of C, 10% chance of B
Independence says your preference between these two lotteries must be the same regardless of what C is. You don’t observe C before choosing. You choose between two complete packages.
The holistic objection is: the identity of C changes the overall risk profile of the package. Suppose A is risky (50/50 chance of doubling or halving your wealth) and B is safe ($5K certain gain). If C = “$1M for certain,” then both lotteries give you a 90% safety net, and you can afford to take the risky A in the remaining 10% branch. But if C = “$0,” then both lotteries give you a 90% chance of nothing, and in the remaining 10% branch, the safe B becomes much more valuable. Your ranking of the two packages flips, not because you observed C and updated, but because the overall distribution of the package changes with C even though C is common to both.
The conditional plan “if C, do B, otherwise do A” that you describe is a strategy for a sequential decision problem, which is indeed unproblematic. But the independence axiom applies to the static choice between compound lotteries, where C is not observed but mixed in. Your restaurant analogy of apple/blueberry/cherry pie is, as I see it, actually closer to “independence of irrelevant alternatives” (the Arrow axiom), which is a different axiom in a different framework. The vNM independence axiom is more like choosing between two fixed price menus that share the same dessert but differ in the main course: the axiom says which menu you prefer shouldn’t depend on the shared dessert, but a holistic diner evaluating the total meal might reasonably disagree.
Upon thinking about it, I agree that the resolute/sophisticated choice sections would benefit from more concrete examples, and I really agree with the point in your final paragraph.
then both lotteries give you a 90% safety net, and you can afford to take the risky A in the remaining 10% branch.
I’m having trouble seeing how this works. Regardless of whether C is in the pool, I run a 5% risk of halving my wealth by taking the first gamble. I think the safety net metaphor only makes sense if the outcome can’t be worse than C, but in this example it seems like there’s a hole in the net I can fall through.
I think there’s a specific clarification that might resolve the main concern.
The independence axiom is not about conditioning on observed information. If you observe C and then choose between A and B, conditioning your choice on what you observed is perfectly rational under EUT as well and nobody disputes this. The independence axiom constrains something different: your ex ante preference between two compound lotteries, before any uncertainty resolves.
Concretely: suppose you must choose right now between two lotteries, both of which contain the same common component C mixed in at the same probability:
Lottery 1: 90% chance of C, 10% chance of A Lottery 2: 90% chance of C, 10% chance of B
Independence says your preference between these two lotteries must be the same regardless of what C is. You don’t observe C before choosing. You choose between two complete packages.
The holistic objection is: the identity of C changes the overall risk profile of the package. Suppose A is risky (50/50 chance of doubling or halving your wealth) and B is safe ($5K certain gain). If C = “$1M for certain,” then both lotteries give you a 90% safety net, and you can afford to take the risky A in the remaining 10% branch. But if C = “$0,” then both lotteries give you a 90% chance of nothing, and in the remaining 10% branch, the safe B becomes much more valuable. Your ranking of the two packages flips, not because you observed C and updated, but because the overall distribution of the package changes with C even though C is common to both.
The conditional plan “if C, do B, otherwise do A” that you describe is a strategy for a sequential decision problem, which is indeed unproblematic. But the independence axiom applies to the static choice between compound lotteries, where C is not observed but mixed in. Your restaurant analogy of apple/blueberry/cherry pie is, as I see it, actually closer to “independence of irrelevant alternatives” (the Arrow axiom), which is a different axiom in a different framework. The vNM independence axiom is more like choosing between two fixed price menus that share the same dessert but differ in the main course: the axiom says which menu you prefer shouldn’t depend on the shared dessert, but a holistic diner evaluating the total meal might reasonably disagree.
Upon thinking about it, I agree that the resolute/sophisticated choice sections would benefit from more concrete examples, and I really agree with the point in your final paragraph.
I’m having trouble seeing how this works. Regardless of whether C is in the pool, I run a 5% risk of halving my wealth by taking the first gamble. I think the safety net metaphor only makes sense if the outcome can’t be worse than C, but in this example it seems like there’s a hole in the net I can fall through.