I get what you are saying. You have convinced me that the following two statements are contradictory:
Axiom of Independence: preferring A to B implies preferring ApC to BpC for any p and C.
The variance and higher moments of utility matter, not just the expected value.
My confusion is that it intuitively it seems both must be true for a rational agent but I guess my intuition is just wrong.
Thanks for your comments, they were very illuminating.
I find it confusing that the only thing that matters to a rational agent is the expectation of utility, i.e., that the details of the probability distribution of utilities do not matter.
I understand that VNM theorem proves that from what seem reasonable axioms, but on the other hand it seems to me that there is nothing irrational about having different risk preferences. Consider the following two scenarios
A: you gain utility 1 with probability 1
B: you gain utility 0 with probability 1⁄2 or utility 2 with probability 1⁄2
According to expected utility, it is irrational to be anything but indifferent to between A and B. This seems wrong to me. I can even go a bit further, consider a third option:
C: you gain utility 0.9 with probability 1
Expected utility says it is irrational to prefer C to B, but this seems perfectly reasonable to me. It’s optimizing for the worst-case instead of the average case. Is there a direct way of showing that preferring B to C is irrational?