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.
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?
Solution.
We use the maximin (minimax) principle to analyze the game.
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 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 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.
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.
A mixed strategy game can be solved by following methods: