Question

How to find optimal row and optimal column in a nonstrictly determined game explain with at least 2 examples (one must be at least 3x3 matrix) and how to find the value (in a determined game, I remember, the value is the saddle value). will rate positive if u make the cut.

Answer 1

Let us consider a game matrix first

for first row miminum value will be =7

for second row the minimum value will be =5

Now for column

you have to find the maximum values that a column have suppose that is the most amount that they have to pay

so for first column the maximum value will be =9

so for second column the maximum value will be =11

now we can see there is no shaddle point in the game and we have to find the game value

so we have to find oddments first (oddment is the difference between the elements of other row or coulmn )

Oddment for row 1=11-5=6

Oddment for row 2=9-7=2

Oddment for column 1=11-7=4

Oddment for column 2=9-5=4

probablity value for the player for row1

probablity value for the player for row2

probablity value for the player column1

probablity value for the player for column1

So the optimal row for the row player is denoted by the probablity of the rows

So the optimal column for the column player is denoted by the probablity of the column

So the

For finding the game value take the values of 1st column or second column (whatever you take it will give u the same answer )and multiply it with the respective oddments of the row then dived the totsl value of oddment of row .

you can do it for column also you will get the same result

so the game value

ii) Now lets discuss for an 3 by 3 matrix

we have to find the minimum value for the rows

for first row miminum value will be =-3

for second row miminum value will be =-4

for third row miminum value will be =--4

the maximin value=-3

Now for the column

we have to find the maximum value

so for first column the maximum value will be =5

so for first column the maximum value will be =5

so for first column the maximum value will be =4

Now the minimax value will be =4

Since there is no saddle point so for finding the game value and optimal row and column we have to reduce the row and column by dominance property

so for reducing the row add the element of particular row so that will be in this case

for row1= 5-3+3=5

for row2=-4+5+4=5

for row 3=4-4-3=-3 since the mimimum value is from the row third so we have to compare the element of row third to the respective elements of the other row if all the values are lesser than the other row then that row will be dominated (deleted ) by the other row .

so as we compair the values of third row 1 st row we will found that the all valuees are less than the respective element s of the first row so the third row is dominated by the first row so after thayt we get a matrix like

now as per the above rule we cant dominate any row .so lets dominate the column now

so for that add all the column first

for coulmn 1=5-4=1

for coulmn 2=5-3=2

for coulmn 3=3+4=7

so for dominating a column we have to compare the all particular values of that particular column with the respective values of other column if the values are higher then that column is dominated by that coulmn so in this case the highest value we got in the third column so we will compare the values of third column to the values of the other two column

in this case we find that not all the valuse of a particular column is higher then the othen the all values of any other column

so in such situation for dominating a column we have to calculate the average of the values of the other column and compare with that coulmn so in this case

average of 5 and -3 is =5-3/2=1

average of -4 and 5 is =5-4/2=0.5

as we see the values of third column 3>1 and 4 >0.5 so column 3 is dominated by these column so after dominitating the third column we get the matrix like

Now we got the matrix in the form of 2*2 so the all the properties and values can be determined as like the previous one discussed for in starting for 2* 2 matrix

for all the matrix having no shaddle point should be reduced first by dominance property and than can be solved .