Question
Solve game with pay off matrix is 4 x 3.
A=⎣⎡887656458556⎦⎤
Solution
Dominance rule to reduce the size of the payoff matrix;
Using dominance property;
playerB
B1 B2 B3
playerAA1A2A3A4 ⎣⎡887656458556⎦⎤
row−4≤row−1,so remove row−4,(A4≤A1:6≤8,5≤5,6≤8)
playerB
B1 B2 B3
playerAA1A2A3 ⎣⎡887564855⎦⎤
row−3≤row−2,so remove row−3,(A3≤A2:7≤8,4≤6,5≤5)
playerB
B1 B2 B3
playerAA1A2 [885685]
column−1≥column−3,so remove column−1.(B1≥B3:8≥8,8≥5)
playerB
B2 B3
playerAA1A2 [5685]
now we solve last matrix
for the solution, we write difference of elements of row to front of other row
similarly we write difference of elements of column to below of other column
such that
playerB
B2 B3
playerAA1A2 [5685]∣6−5∣∣5−8∣
|8-5| |5-6|
playerB
B2 B3
playerAA1A2 [5685]13
|3| |1|
playerB
B2 B3
playerAA1A2 [5685]131+311+33
|3| |1|
3+13 3+11
playerB
B2 B3
playerAA1A2 [5685]134143
|3| |1|
43 41
Hence
strategy of A={41,43,0,0}
strategy of B={0,43,41}
value of game =(5×41)+(6×43)
=45+418=423
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