Risk, Feasibility and Benefit/Cost
Analysis
Burns, Chapter 6
Recitation
• What are the components of any decision
problem?
• Which of these have to be mutually exclusive?
• What is the optimist criterion?
• What is the pessimist criterion?
• What is the inbetweenist criterion?
• What decision environment do all of these
assume?
RULE OF INSUFFICIENT REASON
1) For each row, add-up all of the payoffs in that row
and record the result in a column to the right labeled
ROW SUM.
2) Examine the column labeled ROW SUM to the right
and pick the alternative with the largest payoff in that
column.
REGRETTIST CRITERION
1) Form the regret table.
1) For each row in the regret table, find the largest
regret number in the row and record that in a column to
the right labeled ROW MAXIMUM.
2) Examine the column labeled ROW MAXIMUM to the
right and pick the alternative with the smallest regret in
that column.
Regret Criterion
Bid Project 1
$40,000
-$2,000
Bid Project 2
$100,000
-$10,000
Bid Project 3
$50,000
-$8,000
Bid Project 4
$60,000
-$15,000
Don’t bid on anything
0
0
PAYOFF TABLE
COLUMN MAX
WIN
$100,000
LOSE
0
REGRET TABLE
WIN
LOSE
ROW MAXIMUM
Bid Project 1
$60,000
$2,000
$60,000
Bid Project 2
$0
$10,000
$10,000
Bid Project 3
$50,000
$8,000
$50,000
Bid Project 4
$40,000
$15,000
$40,000
Do Nothing
$100,000
0
$100,000
Let’s assume a RISK environment
• Now we know what exactly??
• DMUR – Decision Making Under Risk
DMUR--Expected Value
PAYOFF TABLE
WIN
LOSE
Probability
DECISION
ALTERNATIVES
Bid Project 1
.7
.3
$40,000
-$2,000
$27,400
Bid Project 2
$100,000
-$10,000
$67,000
Bid Project 3
$50,000
-$8,000
$32,600
Bid Project 4
$60,000
-$15,000
$37,500
Do Nothing
$0
$0
0
$0
$70,000
Expected Payoff $100,000
of
Perfect
Information,
EPPI
Expected Value
Question: How much is perfect information worth? What is its value?
DMUR -- Expected Regret
REGRET TABLE
WIN
LOSE
Expected Regret
Probabilities
DECISION
ALTERNATIVES
Bid Project 1
.7
.3
$60,000
$2,000
$42,600
Bid Project 2
$0
$10,000
$3,000
Bid Project 3
$50,000
$8,000
$37,400
Bid Project 4
$40,000
$15,000
$32,500
Do Nothing
$100,000
0
$70,000
Expected Value & Regret
PAYOFF TABLE
WIN
LOSE
Expected Value
Probability
DECISION
ALTERNATIVES
Bid Project 1
.7
.3
$40,000
-$2,000
$27,400
Bid Project 2
$100,000
-$10,000
$67,000 
Bid Project 3
$50,000
-$8,000
$32,600
Bid Project 4
$60,000
-$15,000
$37,500
Do Nothing
0
0
0
Column Maxima
$100,000
$0
$70,000
REGRET TABLE
WIN
LOSE
Probabilities
DECISION
ALTERNATIVES
Bid Project 1
.7
.3
$60,000
$2,000
$42,600
Bid Project 2
$0
$10,000
$3,000 
Bid Project 3
$50,000
$8,000
$37,400
Bid Project 4
$40,000
$15,000
$32,500
Do Nothing
$100,000
0
$70,000
Expected Regret
Notes
• For any alternative, the expected value and
expected regret numbers sum to the
expected payoff of perfect information
• The expected value and expected regret
criteria always select the same alternative,
because when the former is maximized, the
latter is minimized
Expected Payoff of Perfect
Information, EPPI
• Calculated by finding the largest payoff in
each column and then taking the products
with the column probabilities and summing
these products
• The EPPI is the best we could do if we had
perfect information
Go/No Go Decision
Payoff Table
SUCCESS
FAILURE
EXPECTED VALUE
Probabilities:
.2
.8
Do Project
$3,000,000
-$1,000,000
-$200,000
Don’t do project
0
0
0
Column Maxima
$3,000,000
0
$600,000 = EPPI
Expected Payoff of Perfect
Information, EPPI
• By definition, EPPI = ∑pi * max(Pij)
• The product of the state probability with
the maximum payoff in that column,
summing those products
• = $600,000
• The EPPI is the payoff to us of the additional
information
• The additional information is assumed to
be ‘perfect’ here
Expected Value of Perfect
Information, EVPI
• by definition, EVPI = EPPI - EV*
• EVPI = $600,000 - 0 = $600,000
• The EVPI is the value to us of the additional
information
• The value is the “best we could do with the
additional information” minus the “best we
could do without the additional
information”
Joint, Marginal and Conditional
Probabilities
• Consider two events A and B, occurring
chronologically, temporally as event A and
then event B
• The following relationships are true
–
–
–
–
P(A∙B) = P(B/A) ∙P(A) = P(A/B)∙P(B)
P(A), and P(B) are marginal probabilities
P(A∙B) = P(B∙A) are joint probabilities
P(A/B) , P(B/A) are conditional probabilities
The Steps
1. Solution without additional information
2. Compute initial probabilities—P(S), P(F),
P(PS/AS), P(PF/AS), P(PS/AF), P(PF/AF)
3. Do Bayesian Revision
4. Find solution assuming consultant
predicts SUCCESS
5. Find solution assuming consultant
predicts FAILURE
6. Using the selected alternative in Steps 4
and 5 calculate the EPSI and the EVSI
EPSI –Expected Payoff of Sample
Information
• EPSI = ∑ Probj * highest payoff within each
scenario
• Does NOT assume the information is
‘perfect’
EVSI = Expected Value of Sample
Information
• EVSI = EPSI – EV*
• The best we could do with the additional
information minus the best we could do
without the additional information
• Does NOT assume ‘perfect’ information
Decision making without
additional information
Payoff Table
SUCCESS
FAILURE
EXPECTED VALUE
Probabilities:
.2
.8
Do Project
$3,000,000
-$1,000,000
-$200,000
Don’t do project
0
0
0 
Column Maxima
$3,000,000
0
$600,000 = EPPI
Don’t do the project and realize a payoff of zero
Project number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
29
Actual state
Success
Failure
Failure
Failure
Failure
Failure
Success
Failure
Failure
Failure
Success
Failure
Failure
Failure
Failure
Success
Failure
Failure
Failure
Failure
Consultant’s prediction
SUCCESS
FAILURE
FAILURE
SUCCESS
FAILURE
FAILURE
SUCCESS
FAILURE
FAILURE
SUCCESS
SUCCESS
FAILURE
FAILURE
FAILURE
FAILURE
FAILURE
FAILURE
FAILURE
FAILURE
FAILURE
.6
.15
.15
P(PS/AS)=.75
.4
P(AS)=.2
.25
.75
P(PF/AS)=.25
.1
.05
.05
.1
.06667
P(PA/AF)=.125
P(AF)=.8
.93333
.7
Forward Looking Tree
Tree
.7
P(PF/AF)=.875
Backward Looking
Payoff Table assuming
consultant predicts
success
SUCCESS
FAILURE
EXPECTED VALUE
Revised Probabilities:
.6
.4
Do Project
$3,000,000
-$1,000,000
$1,400,000 
Don’t do project
0
0
0
Payoff Table assuming
consultant predicts
failure
SUCCESS
FAILURE
EXPECTED VALUE
Revised Probabilities:
.066667
.9333333
Do Project
$3,000,000
-$1,000,000
-$733,333
Don’t do project
0
0
0
Expected Payoff of Sample
Information, EPSI
• Sample information is never perfect
information
• By definition, EPSI = .25*$1,400,000 + .75*$0
= $350,000
• Clearly, the payoff of sample information is
a lot less than the payoff of perfect
information
Expected Value of Sample
Information, EVSI
• by definition, EVSI = EPSI - EV*
• EVSI = $350,000 - 0 = $350,000
• The value is the “best we could do with the
additional information” minus the “best we
could do without the additional
information”
Expected value and Expected
regret
• will sum to the same number
– what is that number
• Will always choose the same alternative??
• Optimal expected regret is always equal to
_____
Homework--Problem 6-12
Decision Trees
• The Payoff Table approach is useful for a
single decision situation.
• Many real-world decision problems consist
of a sequence of dependent decisions.
• Decision Trees are useful in analyzing
multi-stage decision processes.
• Characteristics of the Decision Tree
– A Decision Tree is a chronological representation
of the decision process
– There are two types of nodes
• Decision nodes (represented by squares)
• State of nature nodes (represented by circles).
– The root of the tree corresponds to the present
time.
Characteristics of Decision
Trees
– The tree is constructed outward into the future
with branches emanating from the nodes.
• A branch emanating from a decision node
corresponds to a decision alternative. It includes a
cost or benefit value.
• A branch emanating from a state of nature node
corresponds to a particular state of nature, and
includes the probability of this state of nature.
Probability Trees
• Describe the process for using these to do
Bayesian revision
Probability Trees
• Construct backward-looking tree
– Find joint probabilities at the end nodes by
taking the product of all probabilities leading
out to the end node
• Construct forward-looking tree
– Move joint probabilities to their appropriate
end nodes
– Calculate marginal probabilities of indicator
states
– Calculate posterior conditional probabilities
Solving Decision Trees
• Are they solved backwards, forwards,
sideways?
• What must every end-node have attached
to it?
• What must every arc emanating from a
chance node have attached to it?
• How are chance nodes handled?
• How are decision nodes handled?
The EVSI using Decision Trees
• What is the Definition for the EVSI?
• How do you calculate it using decision
trees?
BILL GALLEN DEVELOPMENT COMPANY
– B. G. D. plans to do a commercial development on a property.
– Relevant data
•
•
•
•
Asking price for the property is 300,000 dollars.
Construction cost is 500,000 dollars.
Selling price is approximated at 950,000 dollars.
Variance application costs 30,000 dollars in fees and expenses
– There is only 40% chance that the variance will be approved.
– If B. G. D. purchases the property and the variance is denied, the property can
be sold for a net return of 260,000 dollars.
– A three month option on the property costs 20,000 dollars, which will allow
B.G.D. to apply for the variance.
• A consultant can be hired for 5000 dollars.
– P (Consultant predicts approval | approval granted) = 0.70
– P (Consultant predicts denial | approval denied) = 0.80
SOLUTION
• Construction of the Decision Tree
– Initially the company faces a decision about hiring the
consultant.
– After this decision is made more decisions follow regarding
• Application for the variance.
• Purchasing the option.
• Purchasing the property.
0
3
2
Buy land
-300,000
4
Apply for variance
-30,000
11
Apply for variance
-30,000
1
6
Build
-500,000
7
120,000
Sell
950,000
8
5
9
13
Buy land
-300,000
14
Sell
260,000
Build
-500,000
-70,000
10
15
Sell
950,000
16
100,000
12
17
-50,000
0
3
g
in
oth
on
D
2
nt
lta
u
ns
0
Do
t
no
o
ec
hir
Buy land
-300,000
Pu
rch
-20 ase
op
,00
tio
0
n
4
1
Hi
re
-50
00
11
co
ns
Apply for variance
-30,000
Apply for variance
-30,000
5
12
ed
rov
p
p
A
0.4
Den
ied
0.6
ed
rov
p
p
A
0.4
Den
ied
6
Build
-500,000
7
9
13
Buy land
-300,000
14
8
950,000
Sell
260,000
Build
-500,000
0.6
ult
120,000
Sell
an
t
18
Let us consider
This is where
the decision
we are at
to this
hire stage
a consultant
-70,000
10
15
Sell
950,000
16
100,000
17
-50,000
2
-5000
20
1
19
Buy land
21
-300,000
28
18
35
-300,000
-30,000
Apply for variance
-30,000
-5000
36
Buy land
Apply for variance
37
44
Apply for variance
-30,000
Apply for variance
-30,000
23
Build
-500,000
24
Sell
950,000
115,000
25
0.70
?
22
0.30
?
26
Sell
260,000
-75,000
27
The consultant serves as a source for additional information
about denial or approval of the variance.
Therefore, at this point we need to calculate the
posterior probabilities for the approval and denial
of the variance application
Posterior Probability of approval | consultant predicts approval) = 0.70
Posterior Probability of denial | consultant predicts approval) = 0.30
• The rest of the Decision Tree is built in a
similar manner.
• A complete picture can be obtained from
WINQSB.
• Determining the Optimal Strategy
– Work backward from the end of each branch.
– At a state of nature node, calculate the
expected value of the node.
– At a decision node, the branch that has the
highest ending node value is the optimal
decision.
– The highest ending node value is the value for
the decision node.
115,000
23
58,000
0.70
?
22
0.30
?
75,000
26
115,000
Build
-500,000
-75,000
115,000
115,000
24
Sell
950,000
-75,000
-75,000
Sell
260,000
With 58,000 as the chance node value,
we continue backward to evaluate
the previous nodes.
115,000
115,000
115,000
25
-75,000
-75,000
-75,000
27
WINQSB DecisionTree input screen
Decision Tree Evaluation
and strategy determination
Hire the
consultant
(go to
node 18)
If the
consultant
predicts an
approval
(indicated by
node 19)
If the
variance
is approved
(indicated by
node 23)
then buy
the land and
apply for
the variance.
Wait for the
results.
We proceed by the same manner
and complete the strategy.
…then
build and
sell.
Decision Making and Utility
• Introduction
– The expected value criterion may not be
appropriate if the decision is a one-time
opportunity with substantial risks.
– Decision makers do not always choose
decisions based on the expected value
criterion.
• A lottery ticket has a negative net expected return.
• Insurance policies cost more than the present
value of the expected loss the insurance company
pays to cover insured losses.
Example
• Suppose you play a coin-tossing game in which
there is a .55 prob of heads, a .45 prob of tails.
• The stakes: you win $100,000 if the coin comes
up heads; you lose $100,000 if the coin comes up
tails
• The game has an expected return of $10,000
• WOULD YOU PLAY???
• Three types of Decision Makers
– Risk Averse -Prefers a certain outcome to a chance
outcome having the same expected value.
– Risk Taking - Prefers a chance outcome to a certain
outcome having the same expected value.
– Risk Neutral - Is indifferent between a chance
outcome and a certain outcome having the same
expected value.
Risk Preferences
• Risk averse
• Risk neutral--most large firms are thought to be risk
neutral
– same result as Expected value or payoff--no need for utility
conversion
• Risk seeking??
– Small firms do not get to be large firms without being willing
to take on some very large, but calculated risks.
– Would you say Microsoft’s market behavior is risk seeking?
• Yet this company turned $4.4 billion in profits in 1999 on just $14.4
billion revenues (ranked 284 of 500 firms)
• GM produced $3 billion in profits on $161 billion in revenues (ranked 1
out of 500 firms)