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${\ell }_2\succ {\ell }_1$. The distinction is small but strict: their normalized expected utilities are $18/25$ and $7/10$, respectively.

By the Von Neumann-Morgenstern utility theorem, use a representing utility function and apply a positive affine function to set $u({\omega }_1)=1$ and $u({\omega }_4)=0$. This is possible because the given strict preference implies $u({\omega }_1)>u({\omega }_4)$. The two specified indifference relations then giveTaking the expected utility of each lottery yieldsTheir difference is $1/50>0$, establishing the stated strict preference. These numbers measure utility, not monetary reward or an assumed ordering of the subscripts.

Here is the underlying argument directly in terms of the axioms. The independence axiom for lottery preferences permits substitution of indifferent lotteries inside a common mixture. Replacing ${\omega }_3$ by its given mixture of ${\omega }_1,{\omega }_4$, and then replacing ${\omega }_2$, followed by reduction of compound lotteries, givesand consequentlyThe transitive relation axiom combines the successive substitutions. Finally, mixture monotonicity says that among mixtures of a strictly better and a strictly worse outcome, the greater probability of the better outcome is strictly preferred; this follows from the independence axiom for lottery preferences and transitivity. Since $18/25>7/10$ and ${\omega }_1\succ {\omega }_4$, the comparison follows. Complete lottery preferences and continuity of lottery preferences are part of the representation theorem's hypotheses; the direct substitution calculation uses the already supplied indifferences and does not need to invoke continuity of lottery preferences again.^[ref]

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]