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Peter_de_Blanc comments on Expected utility without the independence axiom - Less Wrong
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Comments (65)
Your axiom talks about expected utility, but you have not defined that term yet.
The post assumes a knowledge of basic statistics throughout - in such a context, the meaning of "expected utility" is transparent.
Sorry, I meant the definition of utility.
[edit: this should have been a reply to Stuart Armstrong's comment below RobinZ's.]
Utility is the thing you want to maximize in your decision-making.
A decision-maker in general isn't necessarily maximizing anything. Von Neumann and Morgenstern showed that if you satisfy axioms 1 through 4, then you do in fact take actions which maximize expected utility for some utility function. But this post is ignoring axiom 4 and assuming only axioms 1 through 3. In that case, why should we expect there to be a utility function?
Thanks for bringing this up, and I've change my post to reflect your comments. Unfortunately, I have to decree a utility function ahead of time for this to make any sense, 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.
? This is just the standard definition. The mean of the random variable, when it is expressed in terms of utils.
Should this be specified in the post, or is it common knowledge on this list?
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.
You started out by assuming a preference relation on lotteries with various properties. The completeness, transitivity, and continuity axioms talk about this preference relation. Your "standard deviation bound" axiom, however, talks about a utility function. What utility function?