= Von Neumann-Morgenstern utility theorem
{c}
{title2=$U(L)=\sum_\omega L(\omega)u(\omega)$}
{wiki=Von_Neumann–Morgenstern_utility_theorem}
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.<Expected utility theory by EconPort>[“The Expected Utility Property”; preceding axioms]{ref}
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