= Solution
\b[$\ell_2\succ\ell_1$.] 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 give
$$
u(\omega_3)=\frac35,\qquad u(\omega_2)=\frac34+\frac14\frac35=\frac9{10}.
$$
Taking the <expected utility> of each <lottery> yields
$$
U(\ell_1)=\frac25+\frac15\frac9{10}+\frac15\frac35=\frac7{10},\qquad
U(\ell_2)=\frac25\frac9{10}+\frac35\frac35=\frac{18}{25}.
$$
Their 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>, gives
$$
\omega_2\sim\frac9{10}\omega_1+\frac1{10}\omega_4,
$$
and consequently
$$
\ell_1\sim\frac7{10}\omega_1+\frac3{10}\omega_4,
\qquad
\ell_2\sim\frac{18}{25}\omega_1+\frac7{25}\omega_4.
$$
The <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.<Expected utility theory by EconPort>[“The Expected Utility Property” and the preceding preference axioms]{ref}
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