\begin{definition}[Definition:Lottery]
A '''lottery''' is a game in which the consequence of each move is determined by the realization of a random variable.
Let $X$ denote the set of prizes which a player may receive.
Let $\Omega$ denote the set of possible states.
A '''lottery''' is a mapping $f: X \times \Omega \to \R_{\ge 0}$ such that:
:$\forall t \in \Omega: \ds \sum_{x \mathop \in X} \map f {x, t} = 1$
\end{definition}