= Reduction of compound lotteries
{title2=$\Pr(\omega_j)=\sum_k p_kq_{kj}$}
A compound <lottery> first selects branch $k$ with <probability> $p_k$ and then outcome $\omega_j$ with conditional <probability> $q_{kj}$. Its reduced <lottery> assigns $\omega_j$ the total <probability> $\sum_kp_kq_{kj}$. The <law of total expectation> gives the same <expected utility> in either description:
$$
\sum_kp_k\sum_jq_{kj}u(\omega_j)=\sum_j\left(\sum_kp_kq_{kj}\right)u(\omega_j).
$$
This rule assumes that only the final outcome matters. A preference for a particular randomization procedure requires including that procedure in the outcome description; it cannot be captured by changing the arithmetic of <expected utility> on a fixed outcome set.
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