Could someone please explain to me exactly, precisely, what a utility function is? I have seen it called a perfectly well-defined mathematical object as well as not-vague, but as far as I can tell, no one has ever explained what one is, ever.
The words "positive affine transformation" have been used, but they fly over my head. So the For Dummies version, please.
Given an agent with some set X of choices, a utility function u maps from the set X to the real numbers R. The mapping is such that the agent prefers x1 to x2 if and only if u(x1) > u(x2). This completes the definition of an ordinal utility function.
A cardinal utility function satisfies additional conditions which allow easy consideration of probabilities. One way to state these conditions is that probabilities defined on X are required to be linear over u. This means that we can now consider probabilistic mixes of choices from X (with probabilities sum...
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