# Payoff Matrix

In game theory, a payoff matrix is a table in which strategies of one player are listed in rows and those of the other player in columns and the cells show payoffs to each player such that the payoff of the row player is listed first.

Payoff of a game is incremental gain/benefit or loss/cost that accrue to a player by executing its strategy given the strategy of the other player. The payoff depends on the context of the game. For example, firms deciding about their advertising budgets worry about their revenue, firms undertaking new investment in plant and machinery are interested in finding their rate of return and so on.

A payoff matrix is an important tool in game theory because it summarizes the necessary information and helps us determine whether a dominant strategy and/or a Nash equilibrium exist. It has application in oligopoly models, etc.

## Presentation

If the row player has n strategies and the column player has m strategies the number of cells in the matrix must be n × m and a total number of 2 × n × m payoff values must be there.

A payoff matrix lists the name of the row player to the left of the matrix and the name of the column player above the matrix. The strategies of the row player form the rows of the matrix and the strategies of the column player form its column. The payoff to the row player is always listed first in each cell but the actual presentation may vary as follows:

- The payoffs may be separated using a comma such that the payoffs to the row player appears to the left of the comma and the payoffs to the column player are listed to the right of the comma.
- Alternatively, the row payoff is listed in the bottom left of each cell and column payoff is shown in the upper right corner of the cell. Sometimes a diagonal is drawn inside the cell to separate the two payoffs. The example below illustrate different presentation methods.

## Example

Let’s consider two telecommunication operators: Row and Column. Currently, they share the market equally. There is a $50 million worth of untapped market. If Row expands its network, it will be able to capture the whole $50 million revenue if Column doesn’t expand its network, and vice versa. Similarly, if both expand their network, Row will get $20 million additional revenue and Column $30 million, but if no one expands its network, both gets zero additional revenue.

Let’s create a payoff matrix for this game.

- There are two players: Row and Column and each has two strategies i.e. to expand or not to expand. Hence, there must be four cells in the matrix.
- We list Row as the player whose strategies are listed in rows in red and Column as the player whose strategies are tabulated in columns in blue.
- The upper left cell corresponds a strategy in which both firms expand. In such an eventuality, Row gets $20 million (which appears first) and Column gets $30 million (which appears last).
- The lower left cell corresponds to a strategy when Row does not expand but Column expands. The payoff to Row and Column in this case is 0 and $50 million respectively. This payoff reverses in the upper right corner which represents payoff when Row expands but Column doesn’t.
- The lower right cell represents a situation in which neither firm moves to capture the market, and both get a payoff of zero.

The following table shows the different ways in which the payoff matrix may be presented.

Payoffs represent revenue gain in millions of USD |
Column | ||
---|---|---|---|

Expand | Not Expand | ||

Row | Expand | 20,30 | 50,0 |

Not Expand | 0,50 | 0,0 |

Payoffs represent revenue gain in millions of USD |
Column | ||
---|---|---|---|

Expand | Not Expand | ||

Row | Expand | 30 20 |
0 50 |

Not Expand | 50 0 |
0 0 |

Payoffs represent revenue gain in millions of USD |
Column | ||
---|---|---|---|

Expand | Not Expand | ||

Row | Expand | 30 20 |
0 50 |

Not Expand | 50 0 |
0 0 |

Written by Obaidullah Jan, ACA, CFA and last revised on