Definition:Lottery Induced by Preference Relation

Definition
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 $i$.