Reduction of compound lotteries ($\mathrm{Pr}({\omega }_j)={\sum }_k{p}_k{q}_{kj}$)

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 ${\sum }_k{p}_k{q}_{kj}$. The law of total expectation gives the same expected utility in either description: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.