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.