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Two-person Zero-sum Game with Pure Strategies

To identify the saddle point and value of game the following procedure to be adopted on

the payoff matrix :

 

(i) Identify the minimum from each row and place a symbol * in that cell/entry. Take the maximum of these minima.

 

(ii) Identify the maximum from each column and place a symbol x in that cell/entry. Take the minimum of these maxima.

(iii) If both the symbols * and x occurs in a/an cell/entry, then that cell/entry is called saddle point and the value in that cell/entry is called value of the game (v).

 

Also v = Maximum (row minima) = Minimum (column maxima). There may be more than one saddle point but the value of the game is unique

 

Example 1 . Solve the following game :

 

Player B

A1 1 5 4 2
PLAYER A A2 2 3 5 3
A3 3 4 5 3

Solution. The calculations are displayed in the following table :

Player B

                        B 1

A1    1*

Player A    A2  2*

A3  3*X

Max.                 3

B2    B3

5x     4

3      5x

4      5x

5       5

B4       Min.

2          1

3x       2

3*x      3

3

Max. (Row Min.) = 3,

Min. (Column Max.) = 3

 

In the above game, there are two saddle points at (A3, B1) and (A3, B4).

The value of the game is 3. Here the optimal strategy for player A is A3 and the optimal strategy for player B is B 1 and B4.

 

Example 2. Determine the solution of the following game :

Solution. In the given game, player A has 4 strategies and player B has 5 strategies. The calculations are displayed in the following table :

Max. (Row Min.) = 4, Min. (Col Max.) = 4

In this game there is one saddle point at (A2, B2)

The value of the game is 4.

The optimal strategy for player A is A2

and the optimal strategy for player· B is B2.