Definition:Profile

Definition

Let $G$ be a game.

Let $N$ be the set of players of $G$.

Let $V$ be a variable defining some aspect of each of the players.

The family of values of $V$ is referred to as a profile, denoted:

$\family {x_i}_{i \mathop \in N}$

If the fact that $i \in N$ is understood, then $\family {x_i}$ can be used.

Let $x = \left\langle{x_i}\right\rangle_{i \mathop \in N}$ be a profile of $V$.

Then $x_{-i}$ is used to denote $x$ for all players except for $i$:

$x_{-i} := \left\langle{x_j}\right\rangle_{j \mathop \in N \setminus \left\{ {i}\right\} }$