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.
The axiom of independence itself feels unassailable to me
Well, the axiom of independence is just that: an axiom. It doesn’t need to be assailed; we can take it as axiomatic, or not. If we do take it as axiomatic, certain interesting analyses become possible (depending on what other axioms we adopt). If we refuse to do so, then bad things happen—or so it’s claimed.
In any case, Dawes’ argument (and related ones) about the independence axiom fundamentally concerns the question of what properties of an outcome distribution we should concern ourselves with. (Here “outcome distribution” can refer to a probability distribution, or to some set of outcomes, distributed across time, space, individuals, etc., that is generated by some policy, which we may perhaps view as the output of a generator with some probability distribution.)
A VNM-compliant agent behaves as if it is maximizing the expectation of the utility of its outcome distribution. It is not concerned at all with other properties of that distribution, such as dispersion (i.e., standard deviation or some related measure) or skewness. (Or, to put it another way, a VNM-compliant agent is unconcerned with the form of the outcome distribution.)
What Dawes is saying is simply that, contra the assumptions of VNM-rationality, there seems to be ample reason to concern ourselves with, for instance, the skewness of the outcome distribution, and not just its expectation. But if we do prefer one outcome distribution to another, where the dis-preferred distribution has a higher expectation (but a “better” skewness), then we violate the independence axiom.
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.
Well, the axiom of independence is just that: an axiom. It doesn’t need to be assailed; we can take it as axiomatic, or not. If we do take it as axiomatic, certain interesting analyses become possible (depending on what other axioms we adopt). If we refuse to do so, then bad things happen—or so it’s claimed.
In any case, Dawes’ argument (and related ones) about the independence axiom fundamentally concerns the question of what properties of an outcome distribution we should concern ourselves with. (Here “outcome distribution” can refer to a probability distribution, or to some set of outcomes, distributed across time, space, individuals, etc., that is generated by some policy, which we may perhaps view as the output of a generator with some probability distribution.)
A VNM-compliant agent behaves as if it is maximizing the expectation of the utility of its outcome distribution. It is not concerned at all with other properties of that distribution, such as dispersion (i.e., standard deviation or some related measure) or skewness. (Or, to put it another way, a VNM-compliant agent is unconcerned with the form of the outcome distribution.)
What Dawes is saying is simply that, contra the assumptions of VNM-rationality, there seems to be ample reason to concern ourselves with, for instance, the skewness of the outcome distribution, and not just its expectation. But if we do prefer one outcome distribution to another, where the dis-preferred distribution has a higher expectation (but a “better” skewness), then we violate the independence axiom.
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.