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A probability distribution over possible outcomes. A finite lottery assigning probabilities $p_j$ to outcomes ${\omega }_j$ has expected utility ${\sum }_j{p}_{j}u({\omega }_j)$. Mixing lotteries mixes their outcome probabilities; the independence axiom for lottery preferences describes how this operation preserves preference comparisons.

Assume $0\le q<p\le 1$ and write $L_q=qA+(1-q)B$. The independence axiom for lottery preferences gives $A\succ L_q$ because $1-q>0$. Set $\lambda =(p-q)/(1-q)>0$; then $L_p=\lambda A+(1-\lambda )L_q$. A second use of the same axiom gives $L_p\succ L_q$. Thus a higher probability of the strictly preferred outcome strictly improves a two-outcome lottery.

An affine function preserves mixtures, soThereforeThe positive mixture weight preserves the strict sign; the common component cancels. This is the independence axiom for lottery preferences for the affine utility on probability measures.

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

No: these two strict comparisons violate the independence axiom for lottery preferences. Let $z$ denote the certain zero-money lottery. Reduction of compound lotteries givesFor example, the second expression assigns probability $(1/4)(4/5)=1/5$ to the nonzero reward and $4/5$ to zero. The claimed ${\ell }_3\succ {\ell }_4$ must therefore implywhich is ${\ell }_1\succ {\ell }_2$, contradicting the other claimed strict preference.

The same obstruction is visible in any proposed expected utility representation. Subtract $u(0)$ from the utility function, and write $a=u(3000)-u(0)$ and $b=u(4000)-u(0)$. The first comparison requires $(1/5)b>(1/4)a$, hence $(4/5)b>a$, whereas the second requires $a>(4/5)b$. No real $a,b$ satisfy both. This argument requires neither risk neutrality nor an assumption that the utility function is increasing in money.

Write $A$ and $B$ for the two deterministic outcomes. Under the expected utility hypothesis, indifference between them implies $u(A)=u(B)=c$. The expected utility of the fair lottery is consequentlyIt must therefore be indifferent to each deterministic outcome, contradicting the asserted strict preference. These preferences do not admit an expected utility representation over the stated outcomes. Equivalently, affine utility on probability measures preserves this indifference under mixing, as required by the independence axiom for lottery preferences.

A concern for the fairness of the procedure can be perfectly intelligible. It would require enlarging the outcome description to include the procedure; it cannot be represented by an expected utility over only the two final gift recipients while retaining the stated strict preference.

Under the expected utility hypothesis, use positive affine uniqueness of affine preference representations to normalize the extreme outcomes to $u({\omega }_1)=1$ and $u({\omega }_4)=0$. The stated indifference relations then giveThe two lotteries have expected utilitiesHence ${\ell }_2\succ {\ell }_1$, with normalized expected utility difference $3/100$. Equivalently, substitution reduces them to lotteries with probabilities $0.45$ and $0.48$ of the better extreme outcome. The inference uses the independence axiom for lottery preferences; the three displayed outcome comparisons alone would not determine preferences over all compound lotteries without the expected utility hypothesis.

Apply the independence axiom for lottery preferences in both weak directions of the indifference relation. At $p=0$ the two mixtures are identical. Repeating this substitution and using transitivity lets one replace any finite collection of indifferent component lotteries, after which reduction of compound lotteries combines their outcome probabilities.

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]