Question

Solve the (4x3) game with pay off matrix.

8 5 8

[A] = 8 6 5

7 4 5

6 5 6

 At each stage, clearly explain the steps involved.

Answer 1

If all the elements of Column-i are greater than or equal to the corresponding elements of any other Column-j, then the Column-i is dominated by the Column-j and it is removed from the matrix

So, we can remove 1st column (col1 $\ge$ col2)

 If all the elements of a Row-i are less than or equal to the corresponding elements of any other Row-j, then the Row-i is dominated by the Row-j and it is removed from the matrix.

So, we can remove 3rd row (row3 $\le$ row2), and 4th row (row4 $\le$ row1).

Then we get matrix:

5 8

8 6

This game has no saddle point.

Then:

A play’s (p₁,p₂):

$p_1=\frac{d-c}{(a+d)-(b+c)}$ , $p_2=1-p_1$

B play’s (q₁,q₂):

$q_1=\frac{d-b}{(a+d)-(b+c)}$ , $q_2=1-q_1$

Value of the game:

$V=\frac{ad-bc}{(a+d)-(b+c)}$

From payoff matrix

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

we have:

$a=5,b=8,c=8,d=6$

$p_1=\frac{6-8}{(5+6)-(8+8)}=\frac{2}{5}$ , p₂=3/5

$q_1=\frac{6-8}{(5+6)-(8+8)}=\frac{2}{5}$ , q₂=3/5

$V=\frac{5\cdot6-8\cdot8}{(5+6)-(8+8)}=\frac{34}{5}$