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In expected-utility portfolio choice, an increasing utility function prefers greater wealth, while concavity models aversion to risk.

A utility function assigns numerical values to outcomes consistently with preference comparisons. It represents utility; the representation and the preference concept are distinct.

For a finite outcome space, complete lottery preferences, a transitive relation, continuity of lottery preferences and the independence axiom for lottery preferences are equivalent to an expected utility representation of the displayed form. The representing outcome utility function is unique up to a positive affine function. The theorem is associated with John von Neumann and Oskar Morgenstern.

A useful proof construction selects a best outcome $a$ and a worst outcome $b$. If all outcomes are indifferent, a constant utility function suffices. Otherwise assign $u(a)=1$, $u(b)=0$. Continuity of lottery preferences makes each outcome indifferent to a mixture $u(\omega )a+(1-u(\omega ))b$, and mixture monotonicity makes the coefficient unique. Substitution of indifferent lotteries and reduction of compound lotteries turn any lottery into a best–worst mixture with best-outcome probability ${\sum }_{\omega }L(\omega )u(\omega )$. Mixture monotonicity then ranks all such mixtures. The reverse direction follows by taking expected values; positive affine functions preserve every comparison.^[ref]

A finite rule base pairs propositional formulas with weights and sums the weights of the formulas satisfied by a Boolean valuation. Over a finite set $S$, associate one Boolean variable to membership of each element. Any utility function $u:2^S\to \mathbb{R}$ is represented by giving each subset's minterm weight $u(T)$. Exactly one rule fires at each valuation, so at most $2^{|S|}$ rules suffice. This bound counts rules, not the symbols required to write all formulas.

For computational questions, the weight encoding matters. Binary rational numbers permit exact addition and comparison in polynomial time; unrestricted real constants do not specify an ordinary finite input encoding.