\begin{definition}[Definition:Payoff Function]
Let $G$ be a game.
Let $N$ be the set of players of $G$.
Let $A$ be the set of moves available to player $i \in N$.
A '''payoff function''' on $A$ is a mapping $u_i$ from $A$ to the real numbers $\R$:
:$u_i: A \to \R$
defined by the condition:
:$\forall a, b \in A: a \succsim_i b \iff u_i \left({a}\right) \ge u_i \left({b}\right)$
where $\succsim_i$ denotes the preference relation for player $i$.
\end{definition}