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The University of New South Wales
Term 3, 2022
GSOE9210 Engineering Decisions
Sample Final exam
Instructions:
• Time allowed: 2 hours
• Reading time: 10 minutes
• This paper has 19 pages
• Total number of questions: 53 (multiple choice)
• Total marks available: 60 (not all questions are of equal value)
• Allowed materials: UNSW approved calculator, pencil (2B), pen, ruler,
language dictionary (paper)
This exam is closed-book. No books, study notes, or other study ma-
terials may be used
• Provided materials: generalised multiple choice answer sheet, graph
paper (1 page), working out booklet
• Answers should be marked in pencil (2B) on the accompanying multiple
choice answer sheet
• The exam paper may not be retained by the candidate
1
Start of exam
Questions 1 to 7 refer to the problem below.
Recall the school fund-raiser example from lectures. There are two options
for the fund-raising activity: a fête (F) or a sports day (S). The money raised
by each activity depends on the (unpredictable) weather: on a dry day (d)
a fête will make a profit of $150 and a sports day only $120; however, on a
wet day (w) the sports day will earn $85 and the fête only $75.
Suppose Alice has no information about the likelihood of whether any given
day will be dry or wet. The fund-raiser is a once-off event; i.e., it will only
be held once on a particular day.
1. (1 mark ) On any given day, which of the two activities (S or F) will ensure
the greatest lower bound on profit?
a) S only
b) F only
c) both S and F
d) neither S nor F
e) a mixture of S and F
2. (1 mark ) Suppose Alice is more concerned about limiting the maximum
regret—she doesn’t like to miss out on opportunities. Which activity would
Alice prefer?
a) S only
b) F only
c) both S and F
d) neither S nor F
e) a mixture of S and F
For the following questions assume the following:
Suppose now that Alice works for the local branch of the Government’s ed-
ucation department. She is in charge of twelve local schools, and is planning
to hold a single-day fund-raiser in each school on the same day. She can hold
different activities in different schools if she wishes.
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3. (1 mark ) In how many schools should Alice hold a sports day if she wants
to ensure the greatest minimum profit?
a) in none of them
b) in four of them
c) in six of them
d) in eight of them
e) in all twelve of them
4. (1 mark ) In how many schools should a sports day be hosted if limiting the
maximum regret is the main consideration?
a) in none of them
b) in three of them
c) in four of them
d) in six of them
e) in all twelve of them
For the following question, suppose that fund-raising events are held in one
day of each week of every month.
5. (1 mark ) Let p = P (d) be the probability that any given day is dry. Which
is the Bayes action for probability p = 12 ?
a) S only
b) F only
c) both S and F
d) neither S nor F
e) a mixture of S and F
Records kept over the last ten years indicate that, on average, the number
of dry days per month in Alice’s geographic area are as follows:1
Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Dry days 15 13 10 8 6 5 5 7 11 13 14 16
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Note that Alice lives in a very wet area; perhaps a mountain valley.
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6. (2 marks) Alice holds her fund-raisers every month except the one month
in which she takes her annual holidays. If Alice is concerned with limiting
the maximum regret, which of the options below would be the best time for
Alice to take her holidays?
a) Jan or Feb
b) Feb or Sep
c) June or July
d) Apr or Aug
e) Jan or Dec
7. (1 mark ) If Alice were concerned with securing the greatest minimum profit,
in which months should she schedule her holidays?
a) Jan or Feb
b) Feb or Sep
c) June or July
d) Apr or Aug
e) Jan or Dec
Questions 8 to 22 refer to decision table below.
Consider the following decision table for a problem in which the outcomes
are measured in dollars ($).
s1 s2
a1 10 50
a2 40 20
There are two agents, A and B, who are making independent decisions on
which of the possible actions (a1 and a2 ) to take—note that this is not a
game: both agents are choosing separate decisions at different times.
Consider agent A first. Agent A’s utility function for money is logarithmic
(with base 2); i.e., u(x) = log2 (x − a), where a ∈ R is a parameter to be
determined.
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8. (1 mark ) If u(10) = 0, which alternative below best describes the utility
function u(x)?
a) log(x)
b) log(x − 1)
c) log(x + 9)
d) log(x − 9)
e) none of the above
9. (1 mark ) Let p = P (s1 ). If p = 12 , which of the following statements is
correct?
a) a1 has greater expected dollar value than a2
b) a2 has greater expected dollar value than a1
c) both actions have the same expected dollar value
d) a1 is dominated
e) none of the above
10. (2 marks) For p = 12 , which of the following statements is true?
a) A prefers a1 to a2
b) A prefers a2 to a1
c) A is indifferent between the two actions
d) A prefers neither action
e) none of the above
11. (1 mark ) For which value(s) of p would A be indifferent between the two
actions?
a) p = 0
b) 0 < p ⩽ 41
c) 14 < p ⩽ 12
d) 12 < p < 34
e) 43 ⩽ p
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12. (1 mark ) For p = 12 , the certainty equivalent of a1 is closest to . . .
a) $0
b) $10
c) $15
d) $25
e) $45
13. (1 mark ) For p = 12 , the certainty equivalent of a2 is closest to . . .
a) $0
b) $10
c) $15
d) $25
e) $45
14. (1 mark ) For p = 12 , what is the approximate value of the risk premium of
a1 ?
a) $0
b) −$10
c) −$6
d) $15
e) $20
15. (1 mark ) For p = 21 , what is the approximate value of the risk premium of
a2 ?
a) $0
b) −$10
c) −$3
d) $3
e) $10
For agent B all we know is that she is indifferent between a certain $20 and
10% chance of $50 and 90%of $10. She is also indifferent between $40 and
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the lottery 10 : $50| 10 : $10 .
Assume in the following questions that p = P (s1 ) = 21 .
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16. (1 mark ) Which of the following statements is true?
a) B prefers a1 to a2
b) B prefers a2 to a1
c) B is indifferent between the two actions
d) B prefers neither action
e) none of the above
17. (2 marks) Assume that utilities for dollar values other than those given
can be linearly interpolated. For a utility scale in the range [0, 10], which
expression below best represents u(x) for $20 ⩽ x ⩽ $40?
a) x − 10
1
b) 10 x−1
2
c) 5 x − 10
d) 4 − 4x
e) 41 x − 4
18. (1 mark ) The certainty equivalent of a1 is closest to . . .
a) $20
b) $25
c) $30
d) $35
e) $40
19. (1 mark ) The certainty equivalent of a2 is closest to . . .
a) $0
b) $10
c) $15
d) $25
e) $45
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20. (1 mark ) What is the approximate value of the risk premium of a1 ?
a) $0
b) −$10
c) −$6
d) $15
e) $20
21. (1 mark ) What is the approximate value of the risk premium of a2 ?
a) $0
b) −$10
c) −$3
d) $3
e) $10
22. (1 mark ) For which value of p = P (s1 ) would B be indifferent between the
two actions?
1
a) 10
b) 15
c) 25
d) 35
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e) 10
Questions 23 to 26 refer to the problem below.
Two friends agree to “meet at the park”, but subsequently each realises
that there are two identical parks (A and B) nearby. Each friend has to
decide, independently, to which park to go to meet their friend. The game
is modelled by the matrix below.
A B
111 0, 0
A 1,
000 1, 1
B 0,
8
23. (1 mark ) How many plays survive simplification by elimination of dominated
strategies?
a) none
b) one
c) two
d) three
e) four
24. (1 mark ) How many equilibrium points does this game have?
a) none
b) one
c) two
d) three
e) four
25. (1 mark ) How many Pareto optimal plays are there in this game?
a) none
b) one
c) two
d) three
e) four
26. (1 mark ) Suppose Alice believes that the probability of Bob going to park A
is p = PB (A). Which value of p would leave Alice indifferent between going
to either park?
a) p = 0
b) p = 14
c) p = 13
d) p = 12
e) for any p ∈ [0, 1]
Questions 27 to 30 refer to problem below.
Alice and Bob have agreed to meet for lunch. Alice prefers restaurant A
and Bob prefers restaurant B. Unfortunately, they didn’t specify at which
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restaurant they were to meet. This ‘game’ is modelled by the following game
matrix.
a b
A 2,
211 0, 0
B 0,
000 1, 2
27. (1 mark ) How many plays survive simplification by elimination of dominated
strategies?
a) none
b) one
c) two
d) three
e) four
28. (1 mark ) How many equilibrium points does this game have?
a) none
b) one
c) two
d) three
e) four
29. (1 mark ) How many Pareto optimal plays are there in this game?
a) none
b) one
c) two
d) three
e) four
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30. (1 mark ) Suppose Alice believes that the probability of Bob going to restau-
rant A is p = PB (a). Which value of p would leave Alice indifferent between
going to either restaurant?
a) p = 0
b) p = 14
c) p = 13
d) p = 12
e) for any p ∈ [0, 1]
Questions 31 to 33 refer to the problem below.
Alice and Bob, who are tennis partners, agreed to play this weekend. There
are two tennis courts near them, A and B, but they didn’t specify at which
court they would play. Court A is closer to both. This ‘game’ is modelled
by the following game matrix.
a b
A 2,
222 0, 0
B 0,
000 1, 1
31. (1 mark ) How many equilibrium points does this game have?
a) none
b) one
c) two
d) three
e) four
32. (1 mark ) How many Pareto optimal plays are there in this game?
a) none
b) one
c) two
d) three
e) four
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33. (1 mark ) Suppose Alice believes that the probability of Bob going to court
A is p = PB (a). Which value of p would leave Alice indifferent between going
to either court?
a) p = 0
b) p = 41
c) p = 13
d) p = 12
e) for any p ∈ [0, 1]
Questions 34 to 36 refer to problem below.
Alice sells magazines. She advertises her business by sending out promotional
leaflets to her customers. She has printed three types of leaflet (A, B, or C),
but she can only afford to send one leaflet per customer. Her market—the
customers to which she sells her magazines—is segmented into two categories,
s1 and s2 .
Her average sales, per 100 leaflets sent, are shown in the table below.
s1 s2
A 0 19
B 15 5
C 10 12
34. (1 mark ) For the decision problem described by the table above, Alice’s
guaranteed minimum average sales per hundred leaflets, if she didn’t know
to which segment her customers belong when she sent out her leaflets, is:
a) 65
12
b) 75
12
c) 85
12
d) 95
12
e) none of the above
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35. (1 mark ) Let p = P (s1 ) be the probability that a customer belongs to seg-
7
ment s1 . If p = 10 , which leaflet would be most profitable?
a) A
b) B
c) C
d) a non-pure mixture of A and C
e) none of the above
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36. (2 marks) Assume p = 10 , as in the previous question. Suppose Alice could
hire an oracle who could predict to which segment each customer belongs
with complete accuracy. If each unit sold makes a profit of $10, what is the
highest rate, in dollars per 100 leaflets/customers, which Alice should pay
for the oracle’s service?
a) $29
b) $42
c) $23
d) $37
e) none of the above
Questions 37 to 43 refer to zero-sum game matrix below.
b1 b2 b3 b4
a1 4 2 5 2
a2 2 1 −1 −2
a3 3 2 4 2
a4 −6 0 6 1
-6.0
37. (1 mark ) Which plays by the row player are best responses to column player’s
b3 ?
a) a1 only
b) a2 only
c) a3 only
d) a4 only
e) there are multiple best responses
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38. (1 mark ) Which plays by the row player are best responses to column player’s
b2 ?
a) a1 only
b) a2 only
c) a3 only
d) a4 only
e) there are multiple best responses
39. (1 mark ) Which plays by the row player are best responses to column player’s
b1 ?
a) a1 only
b) a2 only
c) a3 only
d) a4 only
e) there are multiple best responses
40. (1 mark ) Which plays by the column player are best responses to row player’s
a2 ?
a) b1 only
b) b2 only
c) b3 only
d) b4 only
e) there are multiple best responses
41. (1 mark ) How many saddle points does this game have?
a) none
b) one
c) two
d) three
e) four
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42. (1 mark ) After simplification, how many strategies are left for the row player?
a) none
b) one
c) two
d) three
e) four
43. (1 mark ) After simplification, how many strategies are left for the column
player?
a) none
b) one
c) two
d) three
e) four
Questions 44 to 47 refer to the game matrix below.
b1 b2 b3
a1 2,
266 0, 4 4, 4
a2 3,
333 0, 0 1, 5
a3 1,
111 3, 5 2, 3
44. (1 mark ) Which plays by the row player are best responses to the column
player’s b1 ?
a) a1 only
b) a2 only
c) a3 only
d) there are two best responses
e) there are more than two best responses
15
45. (1 mark ) Which plays by the column player are best responses to the row
player’s a3 ?
a) b1 only
b) b2 only
c) b3 only
d) there are two best responses
e) there are more than two best responses
46. (2 marks) Which plays by the row player are best responses to the column
player’s mixed action 13 b1 13 b2 13 b3 ?
a) a1 only
b) a2 only
c) a3 only
d) there are two best responses
e) there are more than two best responses
47. (1 mark ) Which plays by the column player are best responses to the row
player’s mixed action 12 a1 41 a2 14 a3 ?
a) b1 only
b) b2 only
c) b3 only
d) there are two best responses
e) there are more than two best responses
Questions 48 to 53 refer to the problem below.
16
9
P 1
4 m
S
10 D
Consider the football situation shown above, where Alice (blue #10) has
three options:
P pass to her team-mate (blue #9);
D dribble closer to goal before shooting; or
S shoot from where she is.
The chances of scoring if Alice passes (P) to her team-mate are 3 in 10. Her
chances of scoring by first dribbling closer (D) to goal and then shooting are
5 in 10. Her chances of scoring by shooting from where she is (S) are 2 in 10.
Bob, the goal-keeper (yellow #1), can choose to move (m) toward the ball
as shown to reduce Alice’s scoring chances to 1 in 10 if she dribbles, at the
expense of increasing her scoring chances by passing and shooting respectively
to 5 and 3 in 10.
48. (1 mark ) Which is Alice’s Maximin pure action?
a) P
b) D
c) S
d) both P and D
e) none of the above
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49. (1 mark ) Which is Bob’s Maximin pure action?
a) m
b) m
c) both m and m
d) neither m nor m
e) none of the above
50. (2 marks) How many pure strategy equilibria does this game have?
a) 0
b) 1
c) 2
d) 3
e) none of the above
51. (2 marks) Assuming that this situation were repeated many times (i.e.,
mixed strategies are allowed), the lowest value to which Bob could restrict
Alice’s best response is:
a) 7 in 10
b) 6 in 10
c) 5 in 10
d) 4 in 10
e) none of the above
52. (1 mark ) Let p = P (m) be the probability that the goal-keeper will move.
Which value of p would restrict Alice’s best response to the least chance of
scoring?
a) p = 13
b) p = 35
c) p = 23
d) p = 25
e) none of the above
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53. (1 mark ) If mixtures are allowed for both players, which of the following is
an equilibrium?
a) ( 13 P 23 D, 31 m 23 m)
b) (P, m)
c) (D, 13 m 23 m)
d) ( 12 P 12 D, 32 m 13 m)
e) none of the above
End of exam
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