The Von-Neumann Morgenstern axioms talk just about preference over lotteries, which are simply probability distributions over outcomes. That is you have an unstructured set O of outcomes, and you have a total preordering over Dist(O) the set of probability distributions over O. They do not talk about a utility function. This is quite elegant, because to make decisions you must have preferences over distributions over outcomes, but you don’t need to assume that O has a certain structure, e.g. that of the reals.
The expected utility theorem says that preferences which satisfy the first four axioms are exactly those which can be represented by:
A ⇐ B iff E[U;A] ⇐ E[U;B]
for some utility function U: O → R, where
E[U;A] = \sum{o} A(o) U(o)
However, U is only defined up to positive affine transformation i.e. aU+b will work equally well for any a>0. In particular, you can amplify the standard deviation as much as you like by redefining U.
Your axioms require you to pick a particular representation of U for them to make sense. How do you choose this U? Even with a mechanism for choosing U, e.g. assume bounded nontrivial preferences and pick the unique U such that \sup{x} U(x) = 1 and \inf{x} U(x) = 0, this is still less elegant than talking directly about lotteries.
Can you redefine your axioms to talk only about lotteries over outcomes?
The Von-Neumann Morgenstern axioms talk just about preference over lotteries, which are simply probability distributions over outcomes. That is you have an unstructured set O of outcomes, and you have a total preordering over Dist(O) the set of probability distributions over O. They do not talk about a utility function. This is quite elegant, because to make decisions you must have preferences over distributions over outcomes, but you don’t need to assume that O has a certain structure, e.g. that of the reals.
The expected utility theorem says that preferences which satisfy the first four axioms are exactly those which can be represented by:
A ⇐ B iff E[U;A] ⇐ E[U;B]
for some utility function U: O → R, where
E[U;A] = \sum{o} A(o) U(o)
However, U is only defined up to positive affine transformation i.e. aU+b will work equally well for any a>0. In particular, you can amplify the standard deviation as much as you like by redefining U.
Your axioms require you to pick a particular representation of U for them to make sense. How do you choose this U? Even with a mechanism for choosing U, e.g. assume bounded nontrivial preferences and pick the unique U such that \sup{x} U(x) = 1 and \inf{x} U(x) = 0, this is still less elegant than talking directly about lotteries.
Can you redefine your axioms to talk only about lotteries over outcomes?
Alas no. I’ve changed my post to explain the difficulties as I can change the mean and SD of any distribution by just changing my utility function.
I have a new post up that argues that where small sums are concerned, you have to have a utility function linear in cash.