Value Of The Game
The game value refers to the average pay off per play of the game over an extended period of time
In this game player X will play his first/initial row on each play of the game. Player y will have to play first or initial column on each play of the game in order to minimize his looses hence this game is in favour of X and he wins 3 points on all play of the game.It is a game of pure strategy and the value of the game is 3 points in favour of X
Illustration
Find out the optimum strategies for the two players X and Y and determine the value of the game from the given pay off matrix:
Strategy suppose the worst and acts accordingly
If X plays first
If X plays first along with his row one then Y will play along with his 2^{nd} column to win 1 point similarly if X plays along with his 2^{nd} row then Y will play his 3^{rd} column to win 7 points and if x plays along with his 3^{rd} row then Y will play his fourth column to win 9 points
In this game X cannot win then he should adopt first row strategy in order to minimize losses
This decision rule is identified as 'maximum strategy' that is X chooses the highest of these minimum pay offs
By using the same reasoning from the point of view of y
X will play his 3rd row to win 4 points, If Y plays with his 1^{st} column
X will play his 1st row to lose 1 point, If Y plays with his 2^{nd} column
X will play his 1st row to win 4 points, If Y plays with his 3^{rd} column
X will play his 1st row to win 2 points, If Y plays with his 4^{th} column
Hence player Y will make the best of the situation by playing his 2^{nd} column that is a 'Minimax strategy'
This type of game is also a game of pure strategy and the value of the game is -1 as win of 1 point per game to y by using matrix notation, the solution is displayed below:
In this case the game value is -1
Minimum of the column maximums is -1
Maximum of the row is also is -1
That is X's strategy is maximim strategy
Y's strategy is Minimax strategy