= Solution
\b[No: these two strict comparisons violate the <independence axiom for lottery preferences>.] Let $z$ denote the certain zero-money <lottery>. <Reduction of compound lotteries> gives
$$
\ell_1=\frac14\ell_3+\frac34z,\qquad
\ell_2=\frac14\ell_4+\frac34z.
$$
For 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 imply
$$
\frac14\ell_3+\frac34z\succ\frac14\ell_4+\frac34z,
$$
which 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.
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