\begin{definition}[Definition:Lottery Induced by Preference Relation]
Let $G$ be a game.
Let $N$ be the set of players of $G$.
Let $A$ be the set of moves available to player $i \in N$.
Let $C$ be the set of consequences of those moves.
Let the consequences of those moves be affected by a random variable on a probability space $\Omega$ whose realization is not known to the players before they make their moves.
Let $g: A \times \Omega \to C$ be the consequence function for player $i$.
Then the '''lottery on $C$ induced by the profile of preference relations over $C$''' is defined by:
:$\forall a, b \in A: \left({a \succsim_i b}\right) \iff \left({g \left({a, \omega_a}\right) \succsim_i^* g \left({b, \omega_b}\right)}\right)$
where $\succsim_i$ is the preference relation for player $i$.
\end{definition}