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]

GAME THEORY : (WITHOUT SADDLE POINT)

Q1.Solve the following Pay off Matrix

                   B1         B2
     A1   [   5              1
     A2       3              4]

Soln:

STEP I:
Find the row minima and colmn maxima, and find out the maximin amd minimax values.

[5      1       1
 3      4]      3        maximin=3

 5       4
minimax=4

maximin != minimax
Therefore there is no saddle point.

              a22-a12
p1=   --------------------
        (a11+a22)-(a12+a21)

p1=(4-1) / (5+4)-(1+3)

p1=3/5      p2=1-p1=1-3/5=2/5  
p2 = 2/5


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

q1=1/5    q2=4/5

                            a22a11-a12a21
Value of game= ---------------------------   =17/5
                             (a11+a22)-(a12+a21)

Optimum Strategies are given by
       
           [ A1        A2
SA=     3/5        2/5]

             [B1       B2
SB =       1/5      4/5]

GAMES (2 x 2) WITHOUT SADDLE POINT

Consider a 2 x 2 , 2 person zero sum game without any saddle point having the pay off matrix for player as

          B1       B2
A1  [ a11        a12
A2    a21        a22]

The optimum strategy for A is given as
              [  A1       A2
 SA  =       p1        p2]

The optimm strategy for B is given as
            [ B1        B2
SB =      q1         q2]

             a22-a12
p1= ---------------------
       (a11+a22)-(a12+a21)

p1+p2=1
p2=1-p1


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

q1+q2=1
q2=1-q1

                              a11a22 - a12a21
Value of game  = -----------------------------------
                              (a11+a22)-(a12+a21)