Value of the game-game theory, Mathematics

Value Of The Game

The game value refers to the average pay off per play of the game over an extended period of time

                                                          1766_Value Of The Game.png

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:

                                                        140_Value Of The Game 1.png

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 2nd column to win 1 point similarly if X plays along with his 2nd row then Y will play his 3rd column to win 7 points and if x plays along with his 3rd 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 1st column

X will play his 1st row to lose 1 point, If Y plays with his 2nd column

X will play his 1st row to win 4 points, If Y plays with his 3rd column

X will play his 1st row to win 2 points, If Y plays with his 4th column

Hence player Y will make the best of the situation by playing his 2nd 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:

                                                                              111_Value Of The Game 2.png

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

 

Posted Date: 2/19/2013 5:22:49 AM | Location : United States







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