DOMINANCE PROPERTY

Q1. Solve

                        Player B
                          B1    B2    B3
Player A   A1   [ 1      7        2
                 A2     6      2        7
                 A3     5      1        6]

Soln:
STEP 1:
Find out the maximin and minimax values

 [ 1      7        2     1
    6      2        7    2     maximin=2
    5      1        6]   1

   6        7        7
minimax=6

maximin != minimax
Therefore no saddle point.

STEP II:
Now compare each row and check whether they are minimum values. If so then delete that row.

Compare A1 and A2 , here few values are minimum and few are maximum.
Now compare  A2 and A3, A3 has minimum values therefore delete A3.

By dominance Property,       B1       B2        B3
                                    A1    1           7         2
                                    A2     6          2         7
STEP III:
Now compare each column and check whether they are maximum values. If so then delete that column.

Compare B1 and B2 , here few values are minimum and few are maximum.
Now compare  B1 and B3, B3 has maximum values therefore delete B3.

          B1         B2
A1     1             7        1
A2     6             2        2         maximin=2
           6             7
minimax=6

maximin!=minimax
Therefore no saddle point

STEP IV:
                a22-a12
p1=--------------------------
       (a22+a11)-(a12+a21)

p1=1/2     p2=1/2

             a22-a21
q1=--------------------------
       (a22+a11)-(a12+a21)

q1=2/5    q2=3/5

                            a11a22-a21a12
value of game=-------------------------- =4
                        (a22+a11)-(a12+a21)

SA= [ A1       A2        A3
          1/2        1/2        0]

SB=[B1       B2         B3
        2/5      3/5          0]