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RobinZ comments on Expected utility without the independence axiom - Less Wrong
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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.