# 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:

- $(2): \quad$ 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.

## Sources

- 1956: Steven Vajda:
*The Theory of Games and Linear Programming*... (previous) ... (next): Chapter $\text{I}$: An Outline of the Theory of Games: $2$ - 1983: Morton D. Davis:
*Game Theory*(revised ed.) ... (previous) ... (next): $\S 1$: An Overview - 1991: Roger B. Myerson:
*Game Theory*... (previous) ... (next): $1.1$ Game Theory, Rationality, and Intelligence - 1993: Richard J. Trudeau:
*Introduction to Graph Theory*... (previous) ... (next): $1$. Pure Mathematics: Games - 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $1.2$: Games and Solutions - 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $2.1$: Strategic Games