Answer to Question #173495 in Operations Research for ANJU JAYACHANDRAN

Question

2. Solve the 4( × )3 game with pay off matrix. (10) 

6 5 6

7 4 5

8 6 5

8 5 8

A

 At each stage, clearly explain the steps involved.

Answer 1

B₁  B₂  B₃

A $\begin{vmatrix} 6 & 5 & 6\\ 7 & 4 & 5 \\ 8 &6 & 5 \\ 8 & 5 & 8 \\ \end{vmatrix}$

Now we need to solve the game using the above payoff matrix.

A₁ = First row of the payoff matrix

A₂ = Second row of the payoff matrix

A₃ = Third row of the payoff matrix

A₄ = Fourth row of the payoff matrix

Now,

A_m = Strategy opted by A m = 1, 2, 3, 4

B_n = Strategy opted by B n = 1, 2, 3

We need to check for each strategy opted by A or B what money does B needs to pay A and vice versa.

Steps involved:

1) We will find the maximum value for each column and minimum value for each row.

2) The row minimum value indicate the payments made from A to B.

3) The column maximum value indicates the payments from B to A.

Minimum value for A₁ = 5

Minimum value for A₂ = 4

Minimum value for A₃ = 5

Minimum value for A₄ = 5

Maximum value for B₁ = 8

Maximum value for B₂ = 6

Maximum value for B₃ = 8

FOR PLAYER A:

1) The minimum value for 1^st row indicates that if player A has chosen strategy A₁ irrespective of what player B has chosen he has to pay 5 units to B.

2) The minimum value for 2^nd row indicates that if player A has chosen strategy A₂ irrespective of what player B has chosen he has to pay 4 units to B.

3) The minimum value for 3^rd row indicates that if player A has chosen strategy A₃ irrespective of what player B has chosen he has to pay 5 units to B.

4) The minimum value for 4^th row indicates that if player A has chosen strategy A₄ irrespective of what player B has chosen he has to pay 5 units to B.

FOR PLAYER B:

1) The maximum value for 1^st column indicates that if player B has chosen strategy B₁ irrespective of what player A has chosen he has to pay 8 units to A.

2) The maximum value for 2^nd column indicates that if player B has chosen strategy B₂ irrespective of what player A has chosen he has to pay 6 units to A.

3) The maximum value for 3^rd column indicates that if player B has chosen strategy B₃ irrespective of what player A has chosen he has to pay 8 units to A.

Hence, player A should choose strategy A₂ and player B should choose strategy B₂ so that their loss minimized.