CH11 - 1 MP CHAPTER 11 SOLUTIONS MP SECTION 11.1 Row Min...

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MP CHAPTER 11 SOLUTIONS MP SECTION 11.1 Row Min ----------------- 1. 2 2 2 ----------------- 1 3 1 ----------------- Col. Max 2 3 Since max (row min) = min (column max) = 2, row choosing row 1 and column choosing column 1 is a saddle point. Value of the game is 2 units to the row player. 2. Row Min ----------------- 4 5 5 8 4 ----------------- 6 7 6 9 6 ----------------- 5 7 5 4 4 ----------------- 6 6 5 5 5 ----------------- Column Max 6 7 6 9 Since max (row min) = min (col max) = 6, row choosing row 2 and column choosing column 1 or column 3 are saddle points. Value of game is 6 units to the row player. 3. Since the Witch wants to maximize the length of Max's route, we let the Witch be the row player. We obtain the following reward matrix: Max Goes To Witch Leaves Unblocked Atlanta Nashville Row Min ------------------------------------ Atl-St.L and Nash.-St. L. 1600 1800 1600 ------------------------------------ Atl.-NO and Nash.-St. L 1700 1800 1700 ------------------------------------ Atl.-St. L and Nash.-NO 1600 1400 1400 ------------------------------------ Atl.-NO and Nash.-NO 1700 1400 1400 ------------------------------------ Column Maximum 1700 1800 Since max(row min) = min(column max) = 1700 we find that Max going to Atlanta and the Witch leaving the Atlanta-New Orleans and Nashville-St. Louis roads unblocked (or blocking Atlanta-St. Louis and Nashville-New Orleans) is a saddle point. Total length of Max's trip will be 1700 miles. 4. Since a saddle point is a row minimum, it is the smallest number in its row, and since a saddle point is a column maximum, a saddle point must be the largest number in its column. Now suppose that a ij is the smallest number in row i and 1
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the largest number in column j. We will show that the row player choosing row i and the column player choosing column j yields a saddle point. Let the element of the reward matrix in row i and column j be a ij . We begin by showing that (1) max (row minimum) min(column maximum). Suppose a ij = smallest entry in row i. Then for any k, a ij a ik . Thus a ij max a ik . i Thus for any k (row i minimum) = a ij (column k maximum). Thus for any k (1) max (row minimum) (column k maximum) and (1) max (row minimum) min (column maximum) Now suppose that a ij is the smallest element in row i and the largest element in column j. We will show that a ij yields a saddle point. Clearly, (2) max (row min) a ij and (3) min (column maximum) a ij . Combining (2) and (3) with (1) we obtain a ij max(row min) min (column max) a ij . The last set of inequalities shows that a ij = max (row minimum) = min (column maximum), so a ij does yield a saddle point if it is the smallest number in row i and the largest number in column j.
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CH11 - 1 MP CHAPTER 11 SOLUTIONS MP SECTION 11.1 Row Min...

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