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A preference relation on lotteries is complete when every pair is comparable, allowing an indifference relation. Completeness does not require strict ranking of distinct lotteries. Together with a transitive relation, it gives the ordering assumption in the Von Neumann-Morgenstern utility theorem.

One standard mixture formulation says that if $A⪰B⪰C$, some $p\in [0,1]$ satisfies $B\sim pA+(1-p)C$. Under the other Von Neumann-Morgenstern utility theorem assumptions, this excludes lexicographic comparisons and permits numerical utility function scales. It concerns mixtures of lotteries, rather than continuity of a monetary utility function on a prespecified consumption domain.

An economist who developed foundations of game theory and the Von Neumann-Morgenstern utility theorem with John von Neumann.

${\ell }_2\succ {\ell }_1$. 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]