In the previous lesson, we have discussed the method of solution of a game without a saddle point.
The Principle Of Dominance
In the previous lesson, we have
discussed the method of solution of a game without a saddle point. While
solving a game without a saddle point, one comes across the phenomenon of the
dominance of a row over another row or a column over another column in the
pay-off matrix of the game. Such a situation is discussed in the sequel.
In a given pay-off matrix A, we
say that the ith row dominates the kth row if
In such a situation player A will
never use the strategy corresponding to kth row, because he will gain less for
choosing such a strategy.
Similarly, we say the pth column
in the matrix dominates the qth column if 
In this case, the player B will
loose more by choosing the strategy for the qth column than by choosing the
strategy for the pth column. So he will never use the strategy corresponding to
the qth column. When dominance of a row ( or a column) in the pay-off matrix
occurs, we can delete a row (or a column) from that matrix and arrive at a
reduced matrix. This principle of dominance can be used in the determination of
the solution for a given game. Let us consider an illustrative
example involving the phenomenon of dominance in a game.
Problem 1 Solve the
game with the following pay-off matrix 
Solution First consider the minimum of
each row.
The following condition holds: Max {row minima} ≠ min {column
maxima} Therefore we see that there is no saddle point for the game under
consideration.
We see that each element in
column III is greater than the corresponding element in column II. The choice
is for player B. Since column II dominates column III, player B will discard
his strategy 3. Now we
have the reduced game
For this matrix again, there is
no saddle point. Column II dominates column IV. The choice is for player B. So
player B will give up his strategy 4 The game
reduces to the following:
This matrix
has no saddle point. The third row dominates the first
row. The choice is for player A. He will give up his strategy 1 and retain
strategy 3. The game reduces to the following:
Again, there is no saddle point.
We have a 2x2 matrix. Take this matrix as
Then we have a = 3, b = 4, c = 6 and d = 3. Use the formulae for p, 1-p,
r, 1-r and V.



Tags : Operations Management - Game Theory, Goal Programming & Queuing Theory
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