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

Sources

• 1991: Roger B. Myerson: Game Theory ... (previous) ... (next): $1.1$ Game Theory, Rationality, and Intelligence
• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): $2.1$: Strategic Games