MGS 9920: Practice Decision Tree Problem 1. Decision Tree and Incorporating New
Information (Posterior Probabilities) GM has entered into a VEBA agreement to settle its
Other Postemployment Benefits (OPEB) to its employees. However, there is a risk in this
agreement that the settlement may be denied or be approved by the court in the future. GM
estimates the probability of a court approval is 0.7. After considering the inputs of the labor
union, General Motors considers an alternative to VEBA, to pursue a Mandatory Agreement.
The payoffs (in millions of $) for each of the decision alternative are shown in the table below.
Probabilities
Decision Alternatives
Stay with New VEBA
Pursue Mandatory
Agreement
0.7
0.3
States of nature
Approve (A)
Deny (D)
29
-12
3
11
For this problem, if we want to do a decision tree,
a. How many decision nodes are required?
b. How many branches come out of each decision node?
c. How many event nodes are required?
d. How many branches come out of each event node?
e. Draw the decision tree. Label each branch completely including probabilities and
payoffs.
f. Solve the decision tree and find the best decision alternative or strategy.
1.1. GM wants to hire a campaigning firm to improve the odds of court approval for this
particular New VEBA agreement. If hired, the firm will provide a report whether the campaign
this year is going to be Successful or Unsuccessful. For the above GM problem and the decision
tree, consider the reliabilities of the campaigning firm given below, and compute the posterior
probabilities, draw the revised decision tree (a blank tree is provided below), and compute EVPI,
EVSI, and the efficiency of sample information.
Incorporating New Information
Successful =
Unsuccessful=
Approve =
Deny =
Successful
Unsuccessful
RELIABILITIES
Approve
0.80
Deny
0.70
PRIOR PROBABILITIES
Approve
Deny
0.70
0.30
JOINT & MARGINAL PROBABILITIES
Approve
Deny
Successful
Unsuccessful
POSTERIOR PROBABILITIES
Approve
Deny
Successful
Unsuccessful
Incorp. New info.
=
=
Approve
29
Deny
-12
Stick with VEBA
Pursue Mandatory
Successful
Approve
3
Deny
11
=
=
Unsuccessful
Approve
Deny
29
-12
Stick with VEBA
Pursue Mandatory
Approve
Deny
EVSI =
Maximum possible
expected return
with sample
information
maximum possible
expected return
without sample
information
EVPI =
Efficiency of sample information = EVSI / EVPI =
=
3
11
Practice Decision Tree Problem 2. Decision Tree and Incorporating New Information
(Posterior Probabilities) (Refer to the previous FEMA problem). After considering the
guidance provided by their internal advisory board, FEMA (Federal Emergency Management
Agency) dropped the alternative of ‘low preparation’ and changed the probabilities of the states
of nature. The modified savings (payoffs) table is provided below.
Probabilities
Decision Alternatives
High Preparation
Medium Preparation
0.4
0.6
States of nature
Weak (W)
Strong (S)
-9
26
10
16
For this problem, if we want to do a decision tree,
a. How many decision nodes are required?
b. How many event nodes are required?
c. How many branches come out of each event node?
d. Draw the decision tree. Label each branch completely including probabilities and
payoffs.
e. Solve the decision tree and find the best preparation strategy.
2.1. For the above FEMA problem and the decision tree, consider the reliabilities given below,
and compute the posterior probabilities, draw the revised decision tree (a blank tree is provided
below), and compute EVPI, EVSI, and the efficiency of sample information.
Bayes' Theorem
0.4
Weak
-9
High Preparation
0
-9
12
-9
0.6
Strong
26
26
26
2
13.6
0.4
Weak
10
Medium Preparation
0
10
13.6
10
0.6
Strong
16
16
16
Encouraging =
Discouraging =
Weak =
Strong =
RELIABILITIES
Weak
Encouraging
0.60
Discouraging
Strong
0.70
PRIOR PROBABILITIES
Weak
Strong
0.40
0.60
JOINT & MARGINAL PROBABILITIES
Weak
Strong
Encouraging
Discouraging
POSTERIOR PROBABILITIES
Weak
Strong
Encouraging
Discouraging
MGS 9920
=
=
Weak
-9
Strong
26
Weak
10
Strong
16
High Preparation
Medium Preparation
Encouraging
=
=
Discouraging
Weak
-9
Strong
26
Weak
10
Strong
16
High Preparation
Medium Preparation
EVSI =
Maximum possible
expected return
with sample
information
maximum possible
expected return
without sample
information
EVPI =
Efficiency of sample information = EVSI / EVPI =
=