= Solution
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 give
$$
u(\omega_2)=\frac35,\qquad u(\omega_3)=\frac15.
$$
The two <lotteries> have <expected utilities>
$$
\begin{aligned}
U(\ell_1)&=\frac14\left(1+\frac35+\frac15+0\right)=\frac9{20}=0.45,\\
U(\ell_2)&=\frac3{20}+\frac12\frac35+\frac3{20}\frac15=\frac{12}{25}=0.48.
\end{aligned}
$$
Hence \b[$\ell_2\succ\ell_1$], 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>.
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