Readings
Readings
Chapter 13
Decision Analysis
BA 452 Lesson C.3 Statistical Decision Making
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Overview
Overview
BA 452 Lesson C.3 Statistical Decision Making
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Overview
Decision Formulation lists alternatives, uncertain states of nature (hot, cold, …),
and resulting consequences. Decision Formulation is especially important
when a decision is unprecedented.
Decision Making without Probabilities assigned to the states of nature is
possible for optimists (who assume the best happens) and for pessimists (who
assume the worst happens).
Expected Value is the average consequence of a sequence of decisions,
according to the central limit theorem. Hence, people facing repeated decisions
should maximize expected value.
Backward Induction finds your optimal sequence decisions by making your last
decision first. — So, before your first cigarette, think about your last.
Decision Tree Formulation pictures a decision with nodes and branches that
lists alternatives, uncertain states of nature, and consequences. Decision
Trees are useful for a sequence of decisions.
BA 452 Lesson C.3 Statistical Decision Making
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Decision Formulation
Decision Formulation
BA 452 Lesson C.3 Statistical Decision Making
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Decision Formulation
Overview
Decision Formulation lists alternatives, uncertain states of
nature (hot, cold, …), and resulting consequences.
Decision Formulation is especially important when a
decision is unprecedented (outside your experience).
BA 452 Lesson C.3 Statistical Decision Making
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Decision Formulation


Decision theory and decision analysis help people (including
business people) make better decisions.
• They identify the best decision to take.
• They assume an ideal decision maker:
• Fully informed about possible decisions and their
consequences.
• Able to compute with perfect accuracy.
• Fully rational.
Decisions can be difficult in two different ways:
• The need to use game theory to predict how other people will
respond to your decisions.
• The consequence of decisions, good and bad, are stochastic.
• That is, consequences depend on decisions of nature.
BA 452 Lesson C.3 Statistical Decision Making
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Decision Formulation



A decision problem is characterized by decision
alternatives, states of nature (decisions of nature), and
resulting payoffs.
The decision alternatives are the different possible
actions or strategies the decision maker can employ.
The states of nature refer to possible future events (rain
or sun) not under the control of the decision maker.
• States of nature should be defined so that they are
mutually exclusive (one or the other) and collectively
exhaustive (one will happen).
• There will be either rain or sun, but not both.
BA 452 Lesson C.3 Statistical Decision Making
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Decision Formulation




The consequence resulting from a specific combination
of a decision alternative and a state of nature is a payoff.
Payoffs can be expressed in terms of profit, cost, time, or
distance.
For a single (one-shot) decision, a payoff table shows
payoffs for all combinations of decision alternatives and
states of nature.
For a sequence of decisions, a game tree shows payoffs
for all combinations of decision alternatives and states of
nature.
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities
Decision Making without
Probabilities
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities
Overview
Decision Making without Probabilities assigned to the
states of nature is possible for optimists (who assume the
best happens) and for pessimists (who assume the worst
happens).
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities

Two commonly used criteria for decision making do not
require probability information regarding the likelihood of
the states of nature:
• the optimistic approach.
• the conservative (or pessimistic) approach.
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities



The optimistic approach would be used by an optimistic
decision maker.
The decision with the largest possible payoff is chosen.
• For example, play the lottery whenever possible.
• Other examples?
If the payoff table were in terms of costs, the decision
with the lowest cost would be chosen.
• For example, buy the cheapest car (and hope it gets
you to work).
• Other examples?
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities


The conservative approach would be used by a
conservative (or pessimistic) decision maker.
For each decision the minimum payoff is listed and then
the decision corresponding to the maximum of these
minimum payoffs is selected. (Hence, the minimum
possible payoff is maximized.)
• For example, drive a Hummer at low speeds on surface streets
because you might die if you drive a Honda at high speeds on
the freeway.

If the payoff were in terms of costs, the maximum costs
would be determined for each decision and then the
decision corresponding to the minimum of these
maximum costs is selected. (Hence, the maximum
possible cost is minimized.)
• For example, buy a television with proven low maintenance cost.
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities
Question: Consider the following problem with three
decision alternatives and three states of nature with the
following payoff table representing profits:
States of Nature
s1
s2
s3
d1
4
4
-2
d2
0
3
-1
d3
1
5
-3
Decisions
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities
States of Nature
s1
s2
s3
d1
4
4
-2
d2
0
3
-1
d3 1 5
An optimistic decision maker would use
the maximax approach. Choose the decision that has
the largest single value in the payoff table.
-3
D
e
c
i
s
i
o
n
s
Maximaxd
ecision
Decision
d1
d2
d3
Maximum
Payoff
4
3
5
Maximax
payoff
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities
States of Nature
s1
s2
s3
d1
4
4
-2
d2
0
3
-1
d3 1 5
A conservative decision maker would use
the maximin approach. List the minimum payoff for
each decision. Choose the decision with the maximum
of these minimum payoffs.
-3
D
e
c
i
s
i
o
n
s
Maximin
decision
Decision
d1
d2
d3
Minimum
Payoff
-2
-1
-3
Maximin
payoff
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities
Here is a practical application:
 Pittsburgh Development Corporation (PDC) bought land
for a new condominium complex. There decision is the
size of the complex:
• d1 = a small complex with 30 condominiums
• d2 = a medium complex with 60 condominiums
• d3 = a large complex with 90 condominiums
 The future demand for condominiums is uncertain. It
depends on the result of an election. There are two
possibilities for the election, and so for demand:
• s1 = strong demand for the condominiums
• s2 = weak demand for the condominiums
BA 452 Lesson C.3 Statistical Decision Making
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Decision Making without Probabilities
The minimum information needed to complete the
formulation of the problem is the payoff table for
profit. Find maximax and maximin solutions.
States of Nature
s1
s2
d1
Decisions d2
d3
8
14
20
7
5
-9
BA 452 Lesson C.3 Statistical Decision Making
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Expected Value
Expected Value
BA 452 Lesson C.3 Statistical Decision Making
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Expected Value
Overview
Expected Value is the average consequence of a
sequence of decisions, according to the central limit
theorem. Hence, people facing repeated decisions
should maximize expected value.
BA 452 Lesson C.3 Statistical Decision Making
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Expected Value


Most decision makers are neither optimists nor
pessimists (at least not in the extreme form just
discussed).
Decision making then requires probability
information regarding the likelihood of the states of
nature.
BA 452 Lesson C.3 Statistical Decision Making
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Expected Value

Expected value approach
• Once probabilistic information regarding the
states of nature is assessed, one may use the
expected value (EV) approach.
• Here the expected return for each decision is
calculated by summing the products of the
payoff under each state of nature and the
probability of the respective state of nature
occurring.
• The decision yielding the best expected return is
chosen.
BA 452 Lesson C.3 Statistical Decision Making
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Expected Value
Question: Burger King is considering opening a new
restaurant on Main Street. It has three different models,
each with a different seating capacity. Burger Prince
estimates that the average number of customers per hour
will be 80, 100, or 20. Here is the payoff table for the three
models:
Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120
Model A
Model B
Model C
$10,000
$ 8,000
$ 6,000
$15,000
$18,000
$16,000
$14,000
$12,000
$21,000
BA 452 Lesson C.3 Statistical Decision Making
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Expected Value
Choose the model with largest expected value if the
probabilities of states s1, s2, and s3 are .4, .2, and .4
BA 452 Lesson C.3 Statistical Decision Making
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Expected Value
Answer: Compute expected values when the
probabilities of states s1, s2, and s3 are .4, .2, and .4:
Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120
Model A
Model B
Model C
$10,000
$ 8,000
$ 6,000
$15,000
$18,000
$16,000
$14,000
$12,000
$21,000
EV(A) = .4(10,000) + .2(15,000) + .4(14,000) = $12,600
EV(B) = .4(8,000) + .2(18,000) + .4(12,000) = $11,600
EV(C) = .4(6,000) + .2(16,000) + .4(21,000) = $14,000
Choose the model with largest EV, Model C.
BA 452 Lesson C.3 Statistical Decision Making
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Backward Induction
Backward Induction
BA 452 Lesson C.3 Statistical Decision Making
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Backward Induction
Overview
Backward Induction finds your optimal sequence decisions
by making your last decision first. — So, before your first
cigarette, think about your last.
BA 452 Lesson C.3 Statistical Decision Making
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Backward Induction
Use backward induction (and draw a decision tree, if
needed) to help make the following sequential decisions,
where later decisions depend on earlier decisions:
1. Should Dell Computer invest $5 million in research to create a
faster external hard drive for computers?
• The probability of a successful project is 0.5.
• If the project were successful, it requires a new $20 million
production facility to make the new products.
• If the new products were made, demand and revenues are
uncertain:
• With probability 0.5, demand is high and revenue is $59
million.
• With probability 0.3, demand is med. and revenue is $45
million.
• With probability 0.2, demand is low and revenue is $35
million.
2. If the project were successful, should Dell Computer sell its rights in
the project for $25 million?
BA 452 Lesson C.3 Statistical Decision Making
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Backward Induction
Step 1: Replace uncertain payoffs with expected value:
1. Should Dell Computer invest $5 million in research to create a
faster external hard drive for computers?
• The probability of a successful project is 0.5.
• If the project were successful, it requires a new $20 million
production facility to make the new products.
• If the new products were made, expected revenue
= 0.5 x $59 million + 0.3 x $45 million + 0.2 x $35 million
= $50 million, so expected profits are $25 million.
•
•
•
With probability 0.5, demand is high and revenue is $59 million.
With probability 0.3, demand is med. and revenue is $45 million.
With probability 0.2, demand is low and revenue is $35 million.
2. If the project were successful, should Dell Computer sell its rights in
the project for $25 million?
BA 452 Lesson C.3 Statistical Decision Making
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Backward Induction
Step 2: Make the second decision:
1. Should Dell Computer invest $5 million in research to create a
faster external hard drive for computers?
• The probability of a successful project is 0.5.
• If the project were successful, it requires a new $20 million
production facility to make the new products.
• If the new products were made, expected revenue
= 0.5 x $59 million + 0.3 x $45 million + 0.2 x $35 million
= $50 million, so expected profits are $25 million.
•
•
•
With probability 0.5, demand is high and revenue is $59 million.
With probability 0.3, demand is med. and revenue is $45 million.
With probability 0.2, demand is low and revenue is $35 million.
2. If the project were successful, should Dell Computer sell its rights in
the project for $25 million? Do not sell rights in the project (and
earn profit $(25-5) = $20 million, since keeping the rights is worth
$25 million profit.
BA 452 Lesson C.3 Statistical Decision Making
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Backward Induction
Step 3: Replace uncertain payoffs with expected value:
1. Should Dell Computer invest $5 million in research to create a
faster external hard drive for computers?
Expected profit = 0.5 x $25 million (successful) + 0.5 x (-$5 million) =
$10 million, which is positive and so worth the investment.
• The probability of a successful project is 0.5.
• If the project were successful, it requires a new $20 million
production facility to make the new products.
• If the new products were made, expected revenue
= 0.5 x $59 million + 0.3 x $45 million + 0.2 x $35 million
= $50 million, so expected profits are $25 million.
•
•
•
With probability 0.5, demand is high and revenue is $59 million.
With probability 0.3, demand is med. and revenue is $45 million.
With probability 0.2, demand is low and revenue is $35 million.
2. If the project were successful, should Dell Computer sell its rights in
the project for $25 million? Do not sell rights in the project (and
earn profit $(25-5) = $20 million, since keeping the rights is worth
$25 million profit.
BA 452 Lesson C.3 Statistical Decision Making
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Backward Induction
Decision Tree
The backward induction steps leading to the conclusion to invest the $5 million
in research can be seen by drawing a decision tree, where nodes are either
your decisions (moves) or nature’s decisions (moves). Draw and solve a
decision tree for the original decision problem:
1. Should Dell Computer invest $5 million in research to create a faster
external hard drive for computers?
•
The probability of a successful project is 0.5.
•
If the project were successful, it requires a new $20 million production
facility to make the new products.
•
If the new products were made, demand and revenues are uncertain:
•
With probability 0.5, demand is high and revenue is $59 million.
•
With probability 0.3, demand is med. and revenue is $45 million.
•
With probability 0.2, demand is low and revenue is $35 million.
2. If the project were successful, should Dell Computer sell its rights in the
project for $25 million?
BA 452 Lesson C.3 Statistical Decision Making
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Decision Tree
Decision Tree
BA 452 Lesson C.3 Statistical Decision Making
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Decision Tree
Overview
Decision Tree Formulation pictures a decision with nodes
and branches connecting nodes that lists alternatives,
uncertain states of nature (hot, cold, …), and resulting
consequences. Decision Trees are especially useful for a
sequence of decisions.
BA 452 Lesson C.3 Statistical Decision Making
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Decision Tree
Decision Tree Example: Consider
the following decision tree, where
square nodes are decision nodes
and round nodes are chance
nodes.
1. Each chance node has
branches marked with their
probabilities. For example,
branch C occurs with
probability .25.
2. At the end of each branch is its
payoff. For example, branch S
ends in payoff 300, and branch
D ends in payoff 20.
3. Use backward induction and
expected-payoff maximization
to determine the optimal initial
choice of A or B. Explain.
BA 452 Lesson C.3 Statistical Decision Making
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Review Questions
Review Questions
 You should try to answer some of the following
questions before the next class.
 You will not turn in your answers, but students may
request to discuss their answers to begin the next class.
 Your upcoming Final Exam will contain some similar
questions, so you should eventually consider every
review question before taking your exams.
BA 452 Lesson C.3 Statistical Decision Making
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Review 1: Backward Induction
Review 1: Backward Induction
BA 452 Lesson C.3 Statistical Decision Making
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Review 1: Backward Induction
Question: Make the following decisions:
1. Dante development Corporation is considering bidding
on a contract for a new office building complex.
• The cost of preparing a bid is $200,000.
• If you bid, you will win the contract with probability
.8.
• If you win the contract, you pay $2,000,000 to be a
partner in the project.
2. If you win the contract, you consider selling your share
in the project for $2,100,000.
• If you do not sell, there is uncertain revenue.
• 70% chance of revenue $3,000,000.
• 20% chance of revenue $2,500,000.
• 10% chance of revenue $0.
BA 452 Lesson C.3 Statistical Decision Making
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Review 1: Backward Induction
Answer:
Step 1: Replace uncertain payoffs with expected value:
1. Dante development Corporation is considering bidding on a
contract for a few office building complex.
• The cost of preparing a bid is $200,000.
• If you bid, you will win the contract with probability .8.
• If you win the contract, you pay $2,000,000 to be a partner in
the project.
2. If you win the contract, you consider selling your share in the
project for $2,100,000.
• If you do not sell, there is uncertain revenue.
• 70% chance of revenue $3,000,000.
• 20% chance of revenue $2,500,000.
• 10% chance of revenue $0.
• If you do not sell, expect revenue
= .7 x
$3M + .2 x $2.5M + .1 x $0 = $2,600,000.
BA 452 Lesson C.3 Statistical Decision Making
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Review 1: Backward Induction
Step 2: Make the second decision:
1. Dante development Corporation is considering bidding
on a contract for a few office building complex.
• The cost of preparing a bid is $200,000.
• If you bid, you will win the contract with probability
.8.
• If you win the contract, you pay $2,000,000 to be a
partner in the project.
2. If you win the contract, do not sell your share. Your
expected revenue is $2,600,000.
BA 452 Lesson C.3 Statistical Decision Making
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Review 1: Backward Induction
Step 3: Replace uncertain payoffs with expected value:
1. Dante development Corporation is considering bidding
on a contract for a few office building complex.
• The cost of preparing a bid is $200,000.
• If you bid, you will win the contract with probability
.8.
• If you win the contract, you pay $2,000,000 to be a
partner in the project, expect revenue $2,600,000,
and so expect profit $400,000 = $2.6M - $0.2M $2M.
• If you loose the contract, expect profit = -$200,000
• If you bid, expect profit
= .8 x $400,000 + .2 x (-$200,000) = $280,000.
BA 452 Lesson C.3 Statistical Decision Making
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Review 1: Backward Induction
Make your decisions:
1. Dante development Corporation is considering bidding
on a contract for a few office building complex.
• If you bid, expect profit
= .8 x $400,000 + .2 x (-$200,000) = $280,000.
• Since expected profit is positive if you bid, and zero
if you do not bid, then bid.
2. If you win the contract, do not sell your share.
BA 452 Lesson C.3 Statistical Decision Making
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Review 1: Backward Induction
Decision Tree
The backward induction steps leading to the conclusion to bid on the
contract and not sell your share if you win the bid can be seen by
drawing a decision tree, where nodes are either your decisions (moves)
or nature’s decisions (moves). Draw and solve a decision tree for the
original decision problem:
1. Dante development Corporation is considering bidding on a
contract for a new office building complex.
• The cost of preparing a bid is $200,000.
• If you bid, you will win the contract with probability .8.
• If you win the contract, you pay $2,000,000 to be a partner in
the project.
2. If you win the contract, you consider selling your share in the
project for $2,100,000.
• If you do not sell, there is uncertain revenue.
• 70% chance of revenue $3,000,000.
• 20% chance of revenue $2,500,000.
• 10% chance of revenue $0.
BA 452 Lesson C.3 Statistical Decision Making
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BA 452
Quantitative Analysis
End of Lesson C.3
BA 452 Lesson C.3 Statistical Decision Making
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