Lottery

A probability distribution over possible outcomes. A finite lottery assigning probabilities $p_j$ to outcomes ${\omega }_j$ has expected utility ${\sum }_j{p}_{j}u({\omega }_j)$. Mixing lotteries mixes their outcome probabilities; the independence axiom for lottery preferences describes how this operation preserves preference comparisons.

One standard mixture formulation says that if $A⪰B⪰C$, some $p\in [0,1]$ satisfies $B\sim pA+(1-p)C$. Under the other Von Neumann-Morgenstern utility theorem assumptions, this excludes lexicographic comparisons and permits numerical utility function scales. It concerns mixtures of lotteries, rather than continuity of a monetary utility function on a prespecified consumption domain.

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.