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 think you are not allowed to refer explicitly to utility in the options.
I was going to answer that I can easily reword my example to not explicitly mention any utility values, but when I tried to that it very quickly led to something where it is obvious that u(A) = u(C). I guess my rewording was basically going through the steps of the proof of VNM theorem.
I am still not sure I am convinced by your objection, as I don’t think there’s anything self-referential in my example, but that did give me some pause.
The tricky bit is the question whether this also applies to one-shot problems or not.
This is the crux. It seems to me that the expected utility frame work means that if you prefer A to B in one time choice, then you must also prefer n repetitions of A to n repetitions of B, because the fact that you have larger variance for n=1 does not matter. This seems intuitively wrong to me.
Thanks, I looked at the discussion you linked with interest. I think I understand my confusion a little better, but I am still confused.
I can walk through the proof of the VNM theorem and see where the independence axiom comes in and how it leads to u(A)=u(B) in my example. The axiom of independence itself feels unassailable to me and I am not quite sure this is a strong enough argument against it. Maybe having a more direct argument from axiom of independence to unintuitive result would be more convincing.
Maybe the answer is to read Dawes book, thanks for the reference.
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?