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