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)