# Pure Strategy & Mixed Strategy : Game Theory

The simplest type of game is one where the best strategies for both players are pure strategies. This is the case if and only if, the pay-off matrix contains a saddle point.

To illustrate, consider the following pay-off matrix concerning zero sum two person game.

## Example: Pure Strategy in Game Theory

Player B
Player A   I II III IV V
I -2 0 0 5 3
II 4 2 1 3 2
III -4 -3 0 -2 6
IV 5 3 -4 2 -6

What is the optimal plan for both the players?

"The best plan is to profit by the folly of others." -Pliny the Elder

Solution.

We use the maximin (minimax) principle to analyze the game.

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Player B
Player A   I II III IV V Minimum
I -2 0 0 5 3 -2
II 4 2 1 3 2 1
III -4 -3 0 -2 6 -4
IV 5 3 -4 2 -6 -6
Maximum   5 3 1 5 6

Select minimum from the maximum of columns.
Minimax = 1
Player A will choose II strategy, which yields the maximum payoff of 1.

Select maximum from the minimum of rows.
Maximin = 1
Similarly, player B will choose III strategy.

Since the value of maximin coincides with the value of the minimax, therefore, saddle point (equilibrium point) = 1.

The amount of payoff at an equilibrium point is also known as value of the game.

The optimal strategies for both players are: Player A must select II strategy and player B must select III strategy. The value of game is 1, which indicates that player A will gain 1 unit and player B will sacrifice 1 unit.

## Mixed Strategy: Game Theory

Mixed strategy means a situation where a saddle point does not exist, the maximin (minimax) principle for solving a game problem breaks down. The concept is illustrated with the help of following example.

### Example: Mixed Strategy in Game Theory

Two companies A and B are competing for the same product. Their different strategies are given in the following pay-off matrix:

Company B
Company A   I II III
I -2 14 -2
II -5 -6 -4
III -6 20 -8

Determine the optimal strategies for both the companies.

Solution.

First, we apply the maximin (minimax) principle to analyze the game.

Company B
Company A   I II III Minimum
I -2 14 -2 -2
II -5 -6 -4 -6
III -6 20 -8 -8
Maximum   -2 20 -2

Minimax = -2
Maximin = -2

There are two elements whose value is –2. Hence, the solution to such a game is not unique.

In the above problem, there is no saddle point. In such cases, the maximin and minimax principle of solving a game problem can't be applied. Under this situation, both the companies may resort to what is known as mixed strategy.

In a mixed strategy, each player moves in a random fashion.

A mixed strategy game can be solved by following methods:

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