Solve the following game graphically

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Presentation transcript:

Solve the following game graphically 1 2 3 4 5 1 2 -5 5 -1 8 -4 6 A

Solution. 1 2 3 4 5 -5 5 -1 8 -4 6 1 2

If B select strategy Expected pay off of A Since, Maximax ≠ Maximin Thus, players will use the mixed strategy Since, we do not have saddle point Let p1 the probability of Mr A selecting strategy I & Hence (1- p1) be the probability of Mr A selecting stratety2 If B select strategy Expected pay off of A 1 2 3 4 5 -5 (p1)+8 (1-p1)= -13p1 +8 5 (p1) -4 (1-p1)= 9p1-4 0 (p1) +-1(1-p1)= p1-1 -1 (p1) + 6 (1-p1)= -7p1+6 8 (p1) +-5 (1-p1)= 13p1-5

we plot these values on the graph given below: 11 10 9 8 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 11 10 9 8 7 6 5 4 3 2 1 -1 -2 -3 -4 -5 B5 B1 13p1-5 9p1-4 -7p1+6 B4 B3 p1-1 Q R B11 -13p1+8 Maximin B1 P LOWER ENVELOP

Since R is the maximin point and here B, B3 interesect Since R is the maximin point and here B, B3 interesect. These strategies will be selected & the resultant matrix is the produced below. 1 2 Odds 1 9 5 Odds 1 13 14 -5 8 -1

V = a1 (b1-b2)+ b1 (a1-a2) ______________________= (b1-b2)+ (a1-a2) (-5×9)+(8×5) --------------- =-5/14 9+5

Probability of selecting strategies No. Probablility of selecting strategies No 1 2 9/14 5/14 3 4 5 1/14 13/14