Definition:Payoff Table

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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:

player $\text A$ make make any of $3$ moves
player $\text B$ make make any of $4$ moves.

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

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