MATHEMATICS II

The lower value is the guaranteed result for Player I, the upper value is the guaranteed result for Player II. Player I, by choosing a row, maximizes the obtained element. Considering this, player needs to analyze each row to find the lower value on each row and then choosing the maximum among those numbers will give him the result. Each row has the lower value -6, -3 and -4 respectively. So max {-3, -4, -6} =-3 is obtained hence the correct answer is B.

The lower value is the guaranteed result for Player I, the upper value is the guaranteed result for Player II. Player II, by choosing a column, minimizes the obtained element. Considering this, player needs to analyze each column to find the upper value on each column and then choosing the minimum among those numbers will give him the result. Each column has the upper value 5 and 6 respectively. So min {5,6}=5 is obtained hence the correct answer is D.

If in the game matrix, there exists an element which is smallest in its row and greatest in its column then this row-column pair is an equilibrium pair. For each row write the lowest element to the right, and for each column write the greatest element below the matrix:

Now, observe that max {-1, 0, 2} = 2 and min {2, 3, 4} = 2. Therefore, (2nd row, 3rd column) is an equilibrium pair. The correct answer is D.

If in the game matrix, there exists an element which is smallest in its row and greatest in its column then this row-column pair is an equilibrium pair. For each row write the lowest element to the right, and for each column write the greatest element below the matrix:

Now, observe that max {-7, -4, -2, -1} = -1 and min {-1, 0, 2} = -1. Therefore, (1st row, 2nd column) is an equilibrium pair. The correct answer is A.

Player II, by choosing a column, minimizes the obtained element. The greatest elements of the columns are 2, 4 and 3 respectively. The smallest among these values is 2 which corresponds to the 1^st column that is the minimax strategy will be 1^st The correct answer is C.

If player II chooses 3rd column he guarantees lost 6, which is maximal among {3, -2, 6}. Hence the correct answer is E.

The lowest element among the 2nd row elements {-4, -10, -7} is -10 and the lowest element among the 4th row elements {-2, 3, 0} is -2. Therefore the guaranteed values/wins for Player I is -10 and -2, summation of them is -12. The correct answer is E.

Player I, by choosing a row, maximizes the obtained element. The lowest elements of the rows are -2, -1 and 0 respectively. The greatest among those values is 0 on the 3^rd row which will be the maximin strategy for Player I. The correct answer is C.

If the Player II chooses the 3rd column, the guaranteed value for him will be the maximal element among {2, -8, -2, 0, -7} which is 2. The correct answer is D.

If Player I chooses 2d row he guarantees win -8, which is minimal among {-8, -2, 5}. Hence the correct answer is A.

For each row write the lowest element to the right, and for each column write the greatest element below the matrix

maximum {2,1} = 2, minimum {4, 2, 7} = 2. There exists an equilibrium pair (1st row, 2nd column). The number 2, which lies in the intersection of 1st row and 2nd column, is the lowest in the 1st row, is the greatest in the 2nd column.