Definition:Payoff Function

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 $i \in N$.

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 $i$.

Also known as
This is also known as a utility function, but the latter has also been defined as a mapping from the set of consequences $C$ to $\R$.

It can also be referred to informally as a utility scale.

Also see

 * Definition:Model of Rational Choice
 * Definition:Rules of Game


 * Definition:Preference Relation
 * Definition:Utility Function