Math 484.1.  December 5, 2013.   Midterm 3.
15 problems, 5 pts each.   Write your name here ____________________________  and return the page even if you wrote nothing else on it.

Write details here. Choose one answer for each question on scantron.

41.  The value of  matrix game  with payoff matrix

 3 3 2 2 6 2 0 1 2 1 2 2

is  (A) 2.5, (B) 8/3,  (C)   2.6,  (D) 1.5, (E) none of the above.

Answer: (E). There is a saddle point in Row 1, Column 3 (or 4).

It can be found by domination.

42.  The value of  matrix game  with payoff matrix

 3 3 2 2 0 2 0 1 2 1 3 2

is  (A) 1.5, (B) 2,  (C)   3,  (D) 3.5, (E) none of the above.

Answer: (B). There is a saddle point in Row 1, Column 4.

43.  The value of  matrix game  with payoff matrix

 0 3 -2 2 -3 0 1 1 2 -1 0 2

is  (A) 0, (B) 1,  (C)   2,  (D) 3, (E) none of the above.

Answer: (A).The last column goes by domination.

The remaining 3 by 3 matrix game is symmetric.

44.  The value of the matrix game  with payoff matrix

 1 3 5 6 2 2

is  (A) 2.5, (B) 8/3,  (C)   2.6,  (D) 3.5,  (E) none of the above.

Answer: (B). The last column goes by domination.

Then slopes can be used to find optimal strategies.

45.  Consider the matrix game with payoff matrix

 3 3 2 2 0 2 0 1 2 1 1 0

An optimal strategy for the column player is

(A) [1, 1, 1, 1]/3,   (B)  [1,2, 1]/4,   (C)  [2, 1, 1,1]/5,    (D) [1,1, 2,1]/5 , (E) none of the above.

Answer: (E). (A) and (B) are not mixed strategies.  There are saddle points with the payoff 2.

Using (C) or (D), the column player pays more than 2 against the first row.

46.  Consider the matrix game with payoff matrix

 0 3 -2 2 -3 0 1 1 2 -1 0 2

An optimal strategy for the column player is

(A) [1, 1, 1, 0]/3, (B)    [1,2, 1, 1]/4 ,    (C)  [2, 1, 3,0] /6   (D) [1,2, 3,0]/6, (E) none of the above.

Answer: (D).The last column goes by domination.

The remaining 3 by 3 matrix game is symmetric.

The strategy (D) gives 0 for the column player.

47. Consider the matrix game with payoff matrix

 0 3 0 2 -3 0 -1 1 -2 -1 0 2

An optimal strategy for the column player is

(A) [1, 1, 1,1]/3, (B)    [1,2, 1]/4 ,    (C)  [1, 0, 1, 0] /2   (D) [1,1, 1, 2]/5,   (E) none of the above.

Answer: (C).  (A) and (B) are not mixed strategies.  There are saddle points  with the payoff  0.

Using (C) (which is a mixture of 2 optimal pure strategies), the column player pays at most 0.

Using (D), the column player pays more then 0 against the first row.

48. The mean of 10 numbers  -1,0, -2,  0, 1, 0, 3, 0, 0, 0 is

(A) -1, (B) 0, (C) 1, (D) 2, (E) none of the above.

Answer: (E). The mean is 0.1.

49. The central value for 10 numbers  -1,0, -2,  0, 1, 0, 3, 0, 0, 0 is

(A) -1, (B) 0, (C) 1, (D) 2, (E) none of the above.

50. The midrange for 10 numbers  -1,0, -2,  0, 1, 0, 3, 0, 0, 0 is

(A) -1, (B) 0, (C) 1, (D) 2, (E) none of the above.

Answer: (E). The midrange is 1/2.

51.  The mode for 10 numbers  -1,0, -2,  0, 1, 0, 3, 0, 0, 0 is

(A) -1, (B) 0, (C) 1, (D) 2, (E) none of the above.

52.   The least-squares solution (x, y)  for the system

x+y = 8,  x + 2y = 1,  x+3y = 7   is

(A) (3,1), (B) (6.5, 0.5),  (C)  (19/3, 1/2), (D) (1, 0),  (E) none of the above.

53. The optimal solution (x, y)  for  (x + y - 8)2 + (x +2y -1 )2 + (x + 3y - 7)2 -> min

is  (A) (3,1), (B) (6.5, 0.5),  (C)  (19/3, 1/2), (D) (1, 0),  (E) none of the above.

54.     |2x| +  |x-1| +  |x - 3| +  |5x- 5| ->  min.

The optimal solution is  (A) 0, (B)  1,  (C)  2, (D) 3,  (E) none of the above.

Answer: (B).  It is the median of 0, 0, 1, 1, 1, 1,1, 1,3.

55.   max( |2x|,   |x-1| ,   |x - 3| ,   |x- 5| ) ->  min.

The optimal solution is (A) 1, (B) 1.5,  (C) 1.8, (D) 5/3,  (E) none of the above.