# Definition:Payoff Table

## Definition

Let $G$ be a two-person game.

A **payoff table** for $G$ is an array which specifies the payoff to each player for each strategy of both players.

$G$ is completely defined by its **payoff table**.

$\text B$ | ||

$\text A$ | $\begin {array} {r {{|}} c {{|}} } & \text L & \text R \\ \hline \text T & w_1, w_2 & x_1, x_2 \\ \hline \text B & y_1, y_2 & z_1, z_2 \\ \hline \end {array}$ |

The two numbers in the entry formed by row $r$ and column $c$ are the payoffs when the row player's moves is $r$ and the column player's moves is $c$.

The first component given is the payoff to the row player.

If the names of the players are $1$ and $2$, the convention is that the row player is player $1$ and the column player is player $2$.

If the names of the players are $\text A$ and $\text B$, the convention is that the row player is player $\text A$ and the column player is player $\text B$.

### Payoff Table for Zero-Sum Game

Let $G$ be a two-person zero-sum game.

A **payoff table** for $G$ is an array which specifies the payoff to (conventionally) the maximising player for each strategy of both players.

As $G$ is zero-sum, there is no need to specify the payoff to the minimising player, as it will be the negative of the payoff to the maximising player.

$\text B$ | ||

$\text A$ | $\begin{array} {r {{|}} c {{|}} } & \text{L} & \text{R} \\ \hline \text{T} & w & x \\ \hline \text{B} & y & z \\ \hline \end{array}$ |

### Entry

Each of the values in a **payoff table** corresponding to the payoff for a combination of a move by each player is called an **entry**.

## Examples

### Arbitrary Example

Consider a two-person game $G$ with players are $\text A$ and $\text B$ such that:

The **payoff table** for player $\text A$ will be in the form:

$\text B$ | ||

$\text A$ | $\begin {array} {r {{|}} c {{|}} } & 1 & 2 & 3 & 4 \\ \hline 1 & a_{1 1} & a_{1 2} & a_{1 3} & a_{1 4} \\ \hline 2 & a_{2 1} & a_{2 2} & a_{2 3} & a_{2 4} \\ \hline 3 & a_{3 1} & a_{3 2} & a_{3 3} & a_{3 4} \\ \hline \end {array}$ |

Here, $a_{i j}$ is the amount $\text A$ wins if $\text A$ makes move $i$ and $\text B$ makes move $j$.

If $G$ is a zero-sum game, then player $\text B$'s **payoff table** will be the same as for player $\text A$, but with $a_{i j}$ replaced with $-a_{i j}$.

## Also known as

A **payoff table** is also known as a **payoff matrix**.

Some sources hyphenate: **pay-off table** or **pay-off matrix**.

## Also see

- Results about
**payoff tables**can be found**here**.

## Sources

- 1994: Martin J. Osborne and Ariel Rubinstein:
*A Course in Game Theory*... (previous) ... (next): $\text I$ Strategic Games: Chapter $2$ Nash Equilibrium: $2.1$: Strategic Games - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**decision theory** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**game theory** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**decision theory** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**game theory**