# Definition:Game

## Definition

In the context of game theory, a game is any social situation which involves more than one person.

More specifically, a game is a problem of strategy between two or more parties in which all parties involved seek (usually) to maximise their outcome.

A game $G$ can be specified by:

its players
the moves available to each player
the preference relation for each player:

that is:

$G = \sequence {N, \sequence {A_i}, \sequence {\succsim_i} }$

$G$ can also be specified by:

its players
the moves available to each player
the payoff function for each player:

that is:

$G = \sequence {N, \sequence {A_i}, \sequence {u_i} }$

### Rules

A game is specified completely by its rules.

They prescribe for each player a model of rational choice:

$(1): \quad$ A set $A$ of moves from which the player may choose
$(2): \quad$ A set $C$ of consequences of each of those moves
$(3): \quad$ A consequence function $g: A \to C$ which maps a consequence to each action
$(4): \quad$ A preference relation $\succsim$, which is a total ordering on $C$.

### Player

Each of the parties involved in a game are called players.

### Strategy

A strategy is a complete plan of action that defines what a player will do under all circumstances in a game.

### Payoff

The payoff of a game is the reward or punishment made to a player at the end of the game as a result of combination of the various strategies employed.

## Also see

• Results about games can be found here.