This chapter discusses statistical decision theory and making decisions under uncertainty. It covers constructing payoff tables and decision trees to list possible actions and outcomes. It introduces several decision criteria for selecting the best action, including maximin, minimax regret, and expected monetary value. Expected monetary value calculates the weighted average payoff using the probabilities of different outcomes. Bayes' theorem allows revising probabilities based on new sample information. The key goals are describing decision making processes and criteria for evaluating options under uncertainty.

1. Chapter 18
Statistical Decision Theory
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Statistics for
Business and Economics
7th Edition
Ch. 18-1

2. Chapter Goals
After completing this chapter, you should be
able to:
 Describe basic features of decision making
 Construct a payoff table and an opportunity-loss table
 Define and apply the expected monetary value criterion for
decision making
 Compute the value of sample information
 Describe utility and attitudes toward risk
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-2

3. Steps in Decision Making
 List Alternative Courses of Action
 Choices or actions
 List States of Nature
 Possible events or outcomes
 Determine ‘Payoffs’
 Associate a Payoff with Each Event/Outcome
combination
 Adopt Decision Criteria
 Evaluate Criteria for Selecting the Best Course
of Action
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-3
18.1

4. List Possible Actions or Events
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Payoff Table Decision Tree
Two Methods
of Listing
Ch. 18-4

5. Payoff Table
 Form of a payoff table
 Mij is the payoff that corresponds to action ai and
state of nature sj
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Actions
States of nature
s1 s2 . . . sH
a1
a2
.
.
.
aK
M11
M21
.
.
.
MK1
M12
M22
.
.
.
MK2
. . .
. . .
.
.
.
. . .
M1H
M2H
.
.
.
MKH
Ch. 18-5

6. Payoff Table Example
A payoff table shows actions (alternatives),
states of nature, and payoffs
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Action)
Profit in $1,000’s
(States of nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
Ch. 18-6

7. Decision Tree Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Large factory
Small factory
Average factory
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Payoffs
200
50
-120
40
30
20
90
120
-30
Ch. 18-7

8. Decision Making Overview
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
No probabilities
known
Probabilities
are known
Decision Criteria
Nonprobabilistic Decision Criteria:
Decision rules that can be applied
if the probabilities of uncertain
events are not known
*
 maximin criterion
 minimax regret criterion
Ch. 18-8
18.2

9. The Maximin Criterion
 Consider K actions a1, a2, . . ., aK and H possible states of nature
s1, s2, . . ., sH
 Let Mij denote the payoff corresponding to the ith action and jth state
of nature
 For each action, find the smallest possible payoff and denote the
minimum M1
* where
 More generally, the smallest possible payoff for action ai is given by
 Maximin criterion: select the action ai for which the
corresponding Mi
* is largest (that is, the action with the
greatest minimum payoff)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
)M,,M,Min(MM 1H1211
*
1 
)M,,M,(MM 1H1211
*
i 
Ch. 18-9

10. Maximin Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Alternatives)
Profit in $1,000’s
(States of Nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
1.
Minimum
Profit
-120
-30
20
The maximin criterion
1. For each option, find the minimum payoff
Ch. 18-10

11. Maximin Solution
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Alternatives)
Profit in $1,000’s
(States of Nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
1.
Minimum
Profit
-120
-30
20
The maximin criterion
1. For each option, find the minimum payoff
2. Choose the option with the greatest minimum payoff
2.
Greatest
minimum
is to
choose
Small
factory
(continued)
Ch. 18-11

12. Regret or Opportunity Loss
 Suppose that a payoff table is arranged as a
rectangular array, with rows corresponding to
actions and columns to states of nature
 If each payoff in the table is subtracted from the
largest payoff in its column . . .
 . . . the resulting array is called a regret table, or
opportunity loss table
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-12

13. Minimax Regret Criterion
 Consider the regret table
 For each row (action), find the maximum
regret
 Minimax regret criterion: Choose the action
corresponding to the minimum of the
maximum regrets (i.e., the action that
produces the smallest possible opportunity
loss)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-13

14. Opportunity Loss Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Alternatives)
Profit in $1,000’s
(States of Nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
The choice “Average factory” has payoff 90 for “Strong Economy”. Given
“Strong Economy”, the choice of “Large factory” would have given a
payoff of 200, or 110 higher. Opportunity loss = 110 for this cell.
Opportunity loss (regret) is the difference between an
actual payoff for a decision and the optimal payoff for
that state of nature
Payoff
Table
Ch. 18-14

15. Opportunity Loss
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Alternatives)
Profit in $1,000’s
(States of Nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
(continued)
Investment
Choice
(Alternatives)
Opportunity Loss in $1,000’s
(States of Nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
0
110
160
70
0
90
140
50
0
Payoff
Table
Opportunity
Loss Table
Ch. 18-15

16. Minimax Regret Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Alternatives)
Opportunity Loss in $1,000’s
(States of Nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
0
110
160
70
0
90
140
50
0
Opportunity Loss Table
The minimax regret criterion:
1. For each alternative, find the maximum opportunity
loss (or “regret”)
1.
Maximum
Op. Loss
140
110
160
Ch. 18-16

17. Minimax Regret Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Alternatives)
Opportunity Loss in $1,000’s
(States of Nature)
Strong
Economy
Stable
Economy
Weak
Economy
Large factory
Average factory
Small factory
0
110
160
70
0
90
140
50
0
Opportunity Loss Table
The minimax regret criterion:
1. For each alternative, find the maximum opportunity
loss (or “regret”)
2. Choose the option with the smallest maximum loss
1.
Maximum
Op. Loss
140
110
160
2.
Smallest
maximum
loss is to
choose
Average
factory
(continued)
Ch. 18-17

18. Decision Making Overview
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
No probabilities
known
Probabilities
are known
Decision Criteria
*
Probabilistic Decision Criteria:
Consider the probabilities of
uncertain events and select an
alternative to maximize the
expected payoff of minimize the
expected loss
 maximize expected monetary value
Ch. 18-18
18.3

19. Payoff Table
 Form of a payoff table with probabilities
 Each state of nature sj has an associated
probability Pi
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Actions
States of nature
s1
(P1)
s2
(P2)
. . . sH
(PH)
a1
a2
.
.
.
aK
M11
M21
.
.
.
MK1
M12
M22
.
.
.
MK2
. . .
. . .
.
.
.
. . .
M1H
M2H
.
.
.
MKH
Ch. 18-19

20. Expected Monetary Value (EMV)
Criterion
 Consider possible actions a1, a2, . . ., aK and H states
of nature
 Let Mij denote the payoff corresponding to the ith action
and jth state and Pj the probability of occurrence of the
jth state of nature with
 The expected monetary value of action ai is
 The Expected Monetary Value Criterion: adopt the
action with the largest expected monetary value
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall


H
1j
ijjiHHi22i11i MPMPMPMP)EMV(a 
1P
H
1j
j 
Ch. 18-20

21. Expected Monetary
Value Example
 The expected monetary value is the weighted
average payoff, given specified probabilities for
each state of nature
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Alternatives)
Profit in $1,000’s
(States of Nature)
Strong
Economy
(.3)
Stable
Economy
(.5)
Weak
Economy
(.2)
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
Suppose these
probabilities
have been
assessed for
these states of
nature
Ch. 18-21

22. Expected Monetary Value
Solution
Example: EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2)
= 81
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Action)
Profit in $1,000’s
(States of nature)
Strong
Economy
(.3)
Stable
Economy
(.5)
Weak
Economy
(.2)
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
Expected
Values
(EMV)
61
81
31
Maximize
expected
value by
choosing
Average
factory
(continued)
Payoff Table:
Goal: Maximize expected monetary value
Ch. 18-22

23. Decision Tree Analysis
 A Decision tree shows a decision problem,
beginning with the initial decision and ending
will all possible outcomes and payoffs
Use a square to denote decision nodes
Use a circle to denote uncertain events
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-23

24. Add Probabilities and Payoffs
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Large factory
Small factory
Decision
Average factory
States of nature
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
(continued)
PayoffsProbabilities
200
50
-120
40
30
20
90
120
-30
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
Ch. 18-24

25. Fold Back the Tree
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Large factory
Small factory
Average factory
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
200
50
-120
40
30
20
90
120
-30
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
EMV=200(.3)+50(.5)+(-120)(.2)=61
EMV=90(.3)+120(.5)+(-30)(.2)=81
EMV=40(.3)+30(.5)+20(.2)=31
Ch. 18-25

26. Make the Decision
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Large factory
Small factory
Average factory
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
Strong Economy
Stable Economy
Weak Economy
200
50
-120
40
30
20
90
120
-30
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
(.3)
(.5)
(.2)
EV=61
EV=81
EV=31
Maximum
EMV=81
Ch. 18-26

27. Bayes’ Theorem
 Let s1, s2, . . ., sH be H mutually exclusive and collectively
exhaustive events, corresponding to the H states of nature of a
decision problem
 Let A be some other event. Denote the conditional probability that
si will occur, given that A occurs, by P(si|A) , and the probability
of A , given si , by P(A|si)
 Bayes’ Theorem states that the conditional probability of si, given
A, can be expressed as
 In the terminology of this section, P(si) is the prior probability of si
and is modified to the posterior probability, P(si|A), given the
sample information that event A has occurred
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
))P(ss|P(A))P(ss|P(A))P(ss|P(A
))P(ss|P(A
P(A)
))P(ss|P(A
A)|P(s
HH2211
iiii
i



Ch. 18-27
18.4

28. Bayes’ Theorem Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Stock Choice
(Action)
Percent Return
(Events)
Strong
Economy
(.7)
Weak
Economy
(.3)
Stock A 30 -10
Stock B 14 8
Consider the choice of Stock A vs. Stock B
Expected
Return:
18.0
12.2
Stock A has a
higher EMV
Ch. 18-28

29. Bayes’ Theorem Example
 Permits revising old
probabilities based on new
information
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
New
Information
Revised
Probability
Prior
Probability
(continued)
Ch. 18-29

30. Bayes’ Theorem Example
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Additional Information: Economic forecast is strong economy
 When the economy was strong, the forecaster was correct
90% of the time.
 When the economy was weak, the forecaster was correct 70%
of the time.
Prior probabilities
from stock choice
example
F1 = strong forecast
F2 = weak forecast
E1 = strong economy = 0.70
E2 = weak economy = 0.30
P(F1 | E1) = 0.90 P(F1 | E2) = 0.30
(continued)
Ch. 18-30

31. Bayes’ Theorem Example
 Revised Probabilities (Bayes’ Theorem)
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
3.)E|F(P,9.)E|F(P 2111 
3.)E(P,7.)E(P 21 
875.
)3)(.3(.)9)(.7(.
)9)(.7(.
)F(P
)E|F(P)E(P
)F|E(P
1
111
11 


125.
)F(P
)E|F(P)E(P
)F|E(P
1
212
12 
(continued)
Ch. 18-31

32. EMV with
Revised Probabilities
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
EMV Stock A = 25.0
EMV Stock B = 11.25
Revised
probabilities
Pi Event Stock A xijPi Stock B xijPi
.875 strong 30 26.25 14 12.25
.125 weak -10 -1.25 8 1.00
Σ = 25.0 Σ = 11.25
Maximum
EMV
Ch. 18-32

33. Expected Value of
Sample Information, EVSI
 Suppose there are K possible actions and H
states of nature, s1, s2, . . ., sH
 The decision-maker may obtain sample information.
Let there be M possible sample results,
A1, A2, . . . , AM
 The expected value of sample information is
obtained as follows:
 Determine which action will be chosen if only the prior
probabilities were used
 Determine the probabilities of obtaining each sample
result:
))P(ss|P(A))P(ss|P(A))P(ss|P(A)P(A HHi22i11ii  
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-33

34. Expected Value of
Sample Information, EVSI
 For each possible sample result, Ai, find the
difference, Vi, between the expected monetary value
for the optimal action and that for the action chosen if
only the prior probabilities are used.
 This is the value of the sample information, given that
Ai was observed
MM2211 )VP(A)VP(A)VP(AEVSI  
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
Ch. 18-34

35. Expected Value of
Perfect Information, EVPI
Perfect information corresponds to knowledge of which
state of nature will arise
 To determine the expected value of perfect
information:
 Determine which action will be chosen if only the prior
probabilities P(s1), P(s2), . . ., P(sH) are used
 For each possible state of nature, si, find the difference,
Wi, between the payoff for the best choice of action, if it
were known that state would arise, and the payoff for
the action chosen if only prior probabilities are used
 This is the value of perfect information, when it is known
that si will occur
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-35

36.  Another way to view the expected value of perfect
information
Expected Value of Perfect Information
EVPI = Expected monetary value under certainty
– expected monetary value of the best alternative
Expected Value of
Perfect Information, EVPI
HH2211 )WP(s)WP(s)WP(sEVPI  
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
 The expected value of perfect information (EVPI) is
(continued)
Ch. 18-36

37. Expected Value Under Certainty
 Expected
value under
certainty
= expected
value of the
best
decision,
given perfect
information
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Action)
Profit in $1,000’s
(Events)
Strong
Economy
(.3)
Stable
Economy
(.5)
Weak
Economy
(.2)
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
Example: Best decision
given “Strong Economy” is
“Large factory”
200 120 20
Value of best decision
for each event:
Ch. 18-37

38. Expected Value Under Certainty
 Now weight
these outcomes
with their
probabilities to
find the
expected value:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Investment
Choice
(Action)
Profit in $1,000’s
(Events)
Strong
Economy
(.3)
Stable
Economy
(.5)
Weak
Economy
(.2)
Large factory
Average factory
Small factory
200
90
40
50
120
30
-120
-30
20
200 120 20
(continued)
200(.3)+120(.5)+20(.2)
= 124
Expected
value under
certainty
Ch. 18-38

39. Expected Value of
Perfect Information
Expected Value of Perfect Information (EVPI)
EVPI = Expected profit under certainty
– Expected monetary value of the best decision
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
so: EVPI = 124 – 81
= 43
Recall: Expected profit under certainty = 124
EMV is maximized by choosing “Average factory”,
where EMV = 81
(EVPI is the maximum you would be willing to spend to obtain
perfect information)
Ch. 18-39

40. Utility Analysis
 Utility is the pleasure or satisfaction
obtained from an action
 The utility of an outcome may not be the same for
each individual
 Utility units are arbitrary
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-40
18.5

41. Utility Analysis
 Example: each incremental $1 of profit does not
have the same value to every individual:
 A risk averse person, once reaching a goal,
assigns less utility to each incremental $1
 A risk seeker assigns more utility to each
incremental $1
 A risk neutral person assigns the same utility to
each extra $1
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
Ch. 18-41

42. Three Types of Utility Curves
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
$ $ $
Risk Aversion Risk Seeker Risk-Neutral
Ch. 18-42

43. Maximizing Expected Utility
 Making decisions in terms of utility, not $
 Translate $ outcomes into utility outcomes
 Calculate expected utilities for each action
 Choose the action to maximize expected utility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-43

44. The Expected Utility Criterion
 Consider K possible actions, a1, a2, . . ., aK and H states
of nature.
 Let Uij denote the utility corresponding to the ith action and
jth state and Pj the probability of occurrence of the jth state
of nature
 Then the expected utility, EU(ai), of the action ai is
 The expected utility criterion: choose the action to maximize
expected utility
 If the decision-maker is indifferent to risk, the expected utility
criterion and expected monetary value criterion are equivalent


H
1j
ijjiHHi22i11i UPUPUPUP)EU(a 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-44

45. Chapter Summary
 Described the payoff table and decision trees
 Defined opportunity loss (regret)
 Provided criteria for decision making
 If no probabilities are known: maximin, minimax regret
 When probabilities are known: expected monetary value
 Introduced expected profit under certainty and the
value of perfect information
 Discussed decision making with sample
information and Bayes’ theorem
 Addressed the concept of utility
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-45