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:

$\forall a, b \in A: \paren {a \succsim_i b} \iff \paren {\map g {a, \omega_a} \succsim_i^* \map g {b, \omega_b} }$

where $\succsim_i$ is the preference relation for player $i$.

Sources

• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): $\text I$ Strategic Games: Chapter $2$ Nash Equilibrium: $2.1$: Strategic Games