Homework 7: Decision-Making under Risk,
Due Thursday 10/22 at 12pm (noon)
CSCI 510, Fall 2015
30 points
Solution v3
Rubric v2
The MedFRS project has selected Company L to develop, produce, and field test the
First Responder Workstations (FRWs), and train the first-responder personnel in
their use.
Company L has identified two candidate approaches for developing and producing
the FRWs. The Bold approach B will give them a profit of $600K if it succeeds, but a
loss of $100K if it fails. The Conservative Approach C will give them a profit of
$200K for sure.
All quantities shown, other than probabilities, are in $K.
1. (12 points). Initial analysis
a. Construct the payoff matrix for the decision situation.
Approach
B (Bold)
C (Conservative)
State of Nature
Success
Failure
600
-100
200
200
(2 points: 0.5 point for each cell with payoff value)
b. Determine the best decisions and payoffs under:
(From section 19.2 pp 280-282.)
i. the Maximin rule;
Payoff = Max[MinB, MinC]
= Max [Min (600, -100), Min (200, 200)]
= Max [-100, 200]
= 200, from Approach C.
(2 points: 1 for equation form, 0.5 for correct payoff, 0.5 for correct decision)
ii. the Maximax rule; and
Payoff = Max[MaxB, MaxC]
= Max [Max (600, -100), Max (200, 200)]
= Max [600, 200]
= 600, from Approach B.
(2 points: 1 for equation form, 0.5 for correct payoff, 0.5 for correct decision)
iii. the Laplace rule.
EVLaplace (Approach) = 0.5 * Success_Payoff (Appproach)
+ 0.5 * Failure_Payoff (Approach).
EVLaplace (B) = 0.5 * 600 + 0.5 * (-100)
= 300 – 50 = 250.
EVLaplace (C) = 0.5 * 200 + 0.5 * 200 = 200.
Decision = B with payoff 250, because 250 > 200.
(3 points: 1 for equation form, 1 for correct payoff, 1 for correct decision)
c. Determine the breakpoint probability of Bold approach success above
which the Bold approach will be preferable.
(From section 19.3 pp 282-283.)
Let PSuc = the probability of success; then
Payoff (B)
= PSuc * 600 + (1 - PSuc) * (-100)
= PSuc * 700 - 100, and
Payoff (C)
= PSuc * 200 + (1 - PSuc) * 200
= 200.
The breakeven probability is the value of PSuc where
Payoff (B) = Payoff (C).
Solving:
PSuc * 700 - 100 = 200;
PSuc = 300 / 700 = 0.428571 approximately.
(3 points: 2 for equation form/method (multiple methods possible), 1 for correct
answer)
2. (12 points). Value of perfect information
(From section 20.7 pp 294-295.)
a. Determine the Expected Value of Perfect Information (EV)perfect info:
the payoff if Company L had perfect information about the success or
failure of the Bold approach. Assume that the original success or
failure probabilities were equally likely at 0.5 each.
(From equation 20-7.)
(EV)perfect info = P(Success) * max (values of Success outcomes)
+ P(Failure) * max (values of Failure outcomes)
= 0.5 * max (600, 200) + 0.5 * max (-100, 200)
= 0.5 * 600 + 0.5 * 200
= 400.
(5 points: 3 for equation form, 2 for correct answer)
b. Determine the Expected Value of No Information (EV)no info: the
payoff if Company L had no information about the success or failure of
the Bold approach. Assume that the original success or failure
probabilities were equally likely at 0.5 each.
(From equation 20-6.)
(EV)no info
= max (EV(B), EV(C))
= max (0.5 * 600 + 0.5 * (-100), 200)
= max (250, 200)
= 250.
(5 points: 3 for equation form, 2 for correct answer)
c. Determine the Expected Value of Acquiring Perfect Information
(EVPI): the maximum investment in acquiring information on the
success of the Bold approach that would be worth doing.
(From equation 20-8.)
EVPI = (EV)perfect info - (EV)no info
= 400 - 250
= 150.
(2 points: 1 for correct answer according to student numbers, 1 for numerically
correct answer)
3. (6 points). RRL for prototypes
Company L also identifies two levels of prototyping that would buy
information to reduce the risk of proceeding with no information. Prototype
X would cost $20K and reduce the risk of B failing from $100K to $50K.
Prototype Y would cost $40K and reduce the risk from $100K to $20K.
(From Risk Analysis lecture (EC-11) slide 40.)
a. Compute the Risk Reduction Leverage (RRL) for Prototypes X and Y.
RRL = Risk Reduction Leverage; RE = Risk Exposure.
RRL = (REBefore – REAfter) / Risk Reduction Cost.
REBefore = 100.
RRL(X) = (100 – 50) / 20 = 2.5.
RRL(Y) = (100 – 20) / 40 = 2.0.
(2 points: 1 for equation form, 1 for correct answer)
b. Identify which is the better investment.
Since RRL(X) > RRL(Y), Prototype X is the better investment.
(2 points: 1 for correct answer according to student numbers, 1 for numerically
correct answer)