Math 484.2.  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

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

is  (A) 2.5, (B) 8/3,  (C)   1,  (D) 1.5, (E) none of the above.

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

It can be found by domination.

42.  The value of  matrix game  with payoff matrix

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

43.  The value of  matrix game  with payoff matrix

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

is  (A) -1, (B) 1.5,  (C)   2,  (D) 3, (E) none of the above.

Answer: (E).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

 5 3 5 3 5 2

is  (A) 2.9, (B) 8/3,  (C)   2.6,  (D) 3.8,  (E) none of the above.

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

Then slopes can be used to find optimal strategies.

45.  Consider the matrix game with payoff matrix

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

Answer: (C). (A) , (B), and (D) are not mixed strategies.  There is a saddle points  in the last column.

46.  Consider the matrix game with payoff matrix

 0 4 -2 2 -4 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: (E).The last column goes by domination.

The remaining 3 by 3 matrix game is symmetric. So the value of game is 0. (B) is not a mixed strategy.

Using any of given strategies, the column player pays more than 0 in the worst case.

47. Consider the matrix game with payoff matrix

 1 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: (E).  (A) and (B) are not mixed strategies.  There is a  saddle point  with the payoff  0.

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

48. The mean of 11 numbers  0, -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 1/11.

49. The central value of 11 numbers  0, -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 of 11 numbers  0, -1,0, -2,  0, 1, 0, 3, 0, 0, 0 is

(A) -1, (B) 0, (C) 0.5, (D) 2, (E) none of the above.

51. The mode of 11 numbers  0, -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 = 1,  x + 2y = 1,  x+3y = 1   is

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

Answer: (D). (D) is an exact solution for the system so it is the best fit in every sense.

53. The optimal solution (x, y)  for  (x + y -1)2 + (x - 2y -1) 2  + (x + 2y - 1)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| +  |6x- 6| ->  min.

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