Maximin-Minimax Principle to solve game

## Maximin-Minimax Principle

Consider a game with two players $A$ and $B$ in which player $A$ has $m$ strategies (moves) and player $B$ has $n$ strategies (moves). The game can be described in the form of a payoff matrix such that the cell entry `$a_{ij}$`

is the payment to $A$ in $A$'s payoff matrix when $A$ chooses the `$i^{th}$`

strategy and $B$ chooses the `$j^{th}$`

strategy.

The maximin-minimax principle is used for the selection of optimal strategies by two players. This method is also known as a **calculus method**.

## The Minimax Theorem

For every finite two-person zero-sum game,

- there is a number $v$, called the value of the game
- there is a mixed strategy for Player $A$ such that $A$'s
**average gain is at least $v$**no matter what Player $B$ does, and - there is a mixed strategy for Player $B$ such that $B$'s
**average loss is at most $v$**no matter what Player $A$ does.

Player $A$ wishes to maximize his gain while player $B$ wishes to minimize his loss. Since player $A$ would like to maximize his minimum gain, we obtain the **maximin** value for player $A$ and the corresponding strategy is called the **maximin strategy**. Player $A$'s corresponding gain is called the *maximin value* or *lower value* ($\underline{v}$) of the game.

At the same time, player $B$ wishes to minimize his maximum loss, we obtain the **minimax** value for player $B$ and the corresponding strategy is called the **minimax strategy**. Player $B$'s corresponding loss is called the *minimax value* or *upper value* ($\overline{v}$) of the game.

Usually the minimax value is greater than or equal to the maximin value. If the equality holds, i.e., `$\max_{i}\min_{j} a_{ij}=\min_{j}\max_{i} a_{ij}=v$`

then the corresponding pure strategies are called **optimal strategies** and the game is said to have a saddle point.

When these two values (miximin and minimax) are equal, the corresponding strategies are called **optimal strategies**.

The player $A$'s selection is called the *maximin strategy*. Player $B$'s selection is called the *minimax value* or *upper value* ($\overline{v}$) of the game.

Payoff Matrix for Player $A$ is as follows:

Player A \ Player B | $1$ | $2$ | $\cdots$ | $j$ | $\cdots$ | $n$ |
---|---|---|---|---|---|---|

$1$ | $a_{11}$ | $a_{12}$ | $\dots$ | $a_{1j}$ | $\cdots$ | $a_{1n}$ |

$2$ | $a_{21}$ | $a_{22}$ | $\cdots$ | $a_{2j}$ | $\cdots$ | $a_{2n}$ |

$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |

$i$ | $a_{i1}$ | $a_{i2}$ | $\cdots$ | $a_{ij}$ | $\cdots$ | $a_{in}$ |

$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ |

$m$ | $a_{m1}$ | $a_{m2}$ | $\cdots$ | $a_{mj}$ | $\cdots$ | $a_{mn}$ |

Here the payoff matrix is given for player $A$. The player $A$ is called the **maximizing player** and $B$ is called the **minimizing player**.

## Step by step procedure of Maximin-minimax principle

#### Step 1

Select the minimum element of each row of the payoff matrix,

` $$ \begin{equation*} \text{i.e., } \min_{j} a_{ij}, i=1,2,\cdots, m. \end{equation*} $$ `

#### Step 2

Select the maximum element of each column of the payoff matrix,

` $$ \begin{equation*} \text{i.e., } \max_{i} a_{ij}, j=1,2,\cdots, n. \end{equation*} $$ `

#### Step 3

Obtain the maximum value of each row minimum,

` $$ \begin{equation*} \text{i.e., } \max_{i}\min_{j} a_{ij}=\underline{v}. \end{equation*} $$ `

#### Step 4

Obtain the minimum value of each column maximum,

` $$ \begin{equation*} \text{i.e., } \min_{j}\max_{i} a_{ij}=\overline{v}. \end{equation*} $$ `

#### Step 5

If `$\min_{j}\max_{i} a_{ij}=\max_{i}\min_{j} a_{ij}=v$`

, i.e., $\max (\min) = \min (\max)$ then the position of that element is a **saddle point** of the payoff matrix. And the corresponding strategy is called **optimal strategy**.

## Some Important Terms in Game Theory

### Saddle Point

A saddle point of a payoff matrix is that position where the **maximum of row minima** coincides with the **minimum of column maxima**. That is, if some entry $a_{ij}$ of the pay-off matrix has the property that

- $a_{ij}$ is the minimum of the $i^{th}$ row, and
- $a_{ij}$ is the maximum of the $j^{th}$ column,

then `$a_{ij}$`

is the saddle point of the game. In such a situation, player $A$ can win at least `$a_{ij}$`

by choosing `$i^{th}$`

strategy and Player $B$ can loss at most `$a_{ij}$`

by choosing `$j^{th}$`

strategy.

The pay-off at the saddle point `$a_{ij}$`

is called the value of the game and is equal to the *maximin* and *minimax* value of the game. Saddle point is also called an *equilibrium point*.

The saddle point provides the optimum strategies for both the players and value of the game (i.e. gain for player $A$ and loss for player $B$).

### Value of Game

The payoff at the saddle point is called the value of the game and is equal to *maximin* and *minimax* value of the game.

### Fair Game

A game is said to be fair game if $maximin = minimax = 0$.

### Strictly Determinable

A game is said to be strictly determinable game if $maximin=minimax = v$.

## Maximin-Minimax Principle Example 1

Solve the game with the following payoff for player $A$.

Player $A$ / Player $B$ | $B_1$ | $B_2$ | $B_3$ | $B_4$ |
---|---|---|---|---|

$A_1$ | 8 | 10 | 9 | 14 |

$A_2$ | 10 | 11 | 8 | 12 |

$A_3$ | 13 | 12 | 14 | 13 |

#### Solution

We have two players $A$ and $B$. Player $A$ has three strategies namely $A_1$, $A_2$ and $A_3$ while player $B$ has four strategies namely $B_1$, $B_2$, $B_3$ and $B_4$.

B1 | B2 | B3 | B4 | |
---|---|---|---|---|

A1 | 8 | 10 | 9 | 14 |

A2 | 10 | 11 | 8 | 12 |

A3 | 13 | 12 | 14 | 13 |

#### Step 1

For each row of the payoff matrix, select the minimum element and call it *RowMin*.

` $$ \begin{equation*} \text{i.e., } \min_{j} a_{ij}, i=1,2,\cdots, m. \end{equation*} $$ `

B1 | B2 | B3 | B4 | RowMin | |
---|---|---|---|---|---|

A1 | 8 | 10 | 9 | 14 | 8 |

A2 | 10 | 11 | 8 | 12 | 8 |

A3 | 13 | 12 | 14 | 13 | 12 |

#### Step 2

For each column of the payoff matrix, select the maximum element and call it *ColMax*.

` $$ \begin{equation*} \text{i.e., } \max_{i} a_{ij}, j=1,2,\cdots, n. \end{equation*} $$ `

B1 | B2 | B3 | B4 | RowMin | |
---|---|---|---|---|---|

A1 | 8 | 10 | 9 | 14 | 8 |

A2 | 10 | 11 | 8 | 12 | 8 |

A3 | 13 | 12 | 14 | 13 | 12 |

ColMax | 13 | 12 | 14 | 14 |

#### Step 3

From each RowMin, obtain the maximum value, i.e., $Max(RowMin)$.

` $$ \begin{equation*} \text{i.e., } \max_{i}\min_{j} a_{ij}=\underline{v}. \end{equation*} $$ `

Thus $Max(min) = Max(8, 8, 12)=12$

#### Step 4

For each ColMax, obtain the minimum value, i.e. $Min(ColMax)$.

` $$ \begin{equation*} \text{i.e., } \min_{j}\max_{i} a_{ij}=\overline{v}. \end{equation*} $$ `

Thus $Min(max)=Min(13, 12, 14, 14)=12$.

#### Step 5

$Max(min) =12$ and $Min(max)=12$. Since the $Max(min)= Min(max)=12$ for the game, the game has a saddle point.

Thus optimal strategy for Player $A$ is `$A_{3}$`

and the optimal strategy for Player $B$ is `$B_{2}$`

.

The value of the game for player $A$ is $12$ and for player $B$ is $-12$.

## Maximin-Minimax Principle Example 2

Solve the game with the following payoff for player $A$.

Player $A$ \ Player $B$ | $B_1$ | $B_2$ | $B_3$ |
---|---|---|---|

$A_1$ | 15 | 2 | 3 |

$A_2$ | 6 | 5 | 7 |

$A_3$ | -7 | 4 | 0 |

#### Solution

We have two players $A$ and $B$. Player $A$ has three strategies and player $B$ has four strategies.

B1 | B2 | B3 | |
---|---|---|---|

A1 | 15 | 2 | 3 |

A2 | 6 | 5 | 7 |

A3 | -7 | 4 | 0 |

#### Step 1

Select the minimum element of each row of the payoff matrix,

` $$ \begin{equation*} \text{i.e., } \min_{j} a_{ij}, i=1,2,\cdots, m. \end{equation*} $$ `

B1 | B2 | B3 | RowMin | |
---|---|---|---|---|

A1 | 15 | 2 | 3 | 2 |

A2 | 6 | 5 | 7 | 5 |

A3 | -7 | 4 | 0 | -7 |

#### Step 2

For each column of the payoff matrix, select the maximum element and call it *ColMax*.

` $$ \begin{equation*} \text{i.e., } \max_{i} a_{ij}, j=1,2,\cdots, n. \end{equation*} $$ `

B1 | B2 | B3 | RowMin | |
---|---|---|---|---|

A1 | 15 | 2 | 3 | 2 |

A2 | 6 | 5 | 7 | 5 |

A3 | -7 | 4 | 0 | -7 |

ColMax | 15 | 5 | 7 |

#### Step 3

From each RowMin, obtain the maximum value, i.e., $Max(RowMin)$.

` $$ \begin{equation*} \text{i.e., } \max_{i}\min_{j} a_{ij}=\underline{v}. \end{equation*} $$ `

Thus $Max(min) = Max(2, 5, -7)=5$

#### Step 4

For each ColMax, obtain the minimum value, i.e. $Min(ColMax)$.

` $$ \begin{equation*} \text{i.e., } \min_{j}\max_{i} a_{ij}=\overline{v}. \end{equation*} $$ `

Thus $Min(max)=Min(15, 5, 7)=5$.

#### Step 5

$Max(min) =5$ and $Min(max)=5$. Since the $Max(min)= Min(max)=5$ for the game, the game has a saddle point.

Thus optimal strategy for Player $A$ is `$A_{2}$`

and the optimal strategy for Player $B$ is `$B_{2}$`

.

The value of the game for player $A$ is $5$ and for player $B$ is $-5$.

## Endnote

In this tutorial, you learned about the Maximin-minimax Principle to solve a two-person zero-sum game.

To learn more about different methods to solve a game please refer to the following tutorials:

Let me know in the comments if you have any questions on **Maximin-minimax principle to solve game** and your thought on this article.