1. Chapter 21 Statistical Decision Theory
Statistics for
Business and Economics 6th Edition
Chapter 21
Statistical Decision Theory
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

4. List Possible Actions or Events
Two Methods of Listing
Payoff Table
Decision Tree
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

5. Payoff Table Form of a payoff table
Mij is the payoff that corresponds to action ai and state of nature sj
Actions
States of nature s1 s2
. . . sH a1 a2 . aK
M11
M21
MK1
M12
M22
MK2
M1H
M2H
MKH
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

6. Payoff Table Example Profit in $1,000’s (States of nature)
A payoff table shows actions (alternatives), states of nature, and payoffs
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

7. Decision Tree Example Large factory Average factory Small factory
Strong Economy
200
Large factory
Stable Economy 50
Weak Economy
-120
Strong Economy 90
Average factory
Stable Economy
120
Weak Economy
-30
Strong Economy 40
Small factory
Stable Economy 30
Weak Economy 20
Payoffs
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

8. Decision Making Overview
Decision Criteria *
Nonprobabilistic Decision Criteria: Decision rules that can be applied if the probabilities of uncertain events are not known
No probabilities
known
Probabilities
are known
maximin criterion
minimax regret criterion
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

10. Maximin Example The maximin criterion
For each option, find the minimum payoff
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

11. Greatest minimum is to choose Small factory
Maximin Solution
(continued)
The maximin criterion
For each option, find the minimum payoff
Choose the option with the greatest minimum payoff
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 2.
Greatest minimum is to choose Small factory 1.
Minimum Profit
-120
-30 20
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

14. Opportunity Loss Example
Opportunity loss (regret) is the difference between an actual payoff for a decision and the optimal payoff for that state of nature
Payoff Table
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.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

15. Opportunity Loss in $1,000’s
(continued)
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
Payoff Table
Investment Choice
(Alternatives)
Opportunity Loss in $1,000’s
(States of Nature)
Strong Economy
Stable Economy
Weak Economy
Large factory
Average factory
Small factory
110
160 70 90
140 50
Opportunity Loss Table
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

16. Minimax Regret Example
The minimax regret criterion:
For each alternative, find the maximum opportunity loss (or “regret”)
Opportunity Loss Table
Investment Choice
(Alternatives)
Opportunity Loss in $1,000’s
(States of Nature)
Strong Economy
Stable Economy
Weak Economy
Large factory
Average factory
Small factory
110
160 70 90
140 50 1.
Maximum Op. Loss
140
110
160
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

17. Minimax Regret Example
(continued)
The minimax regret criterion:
For each alternative, find the maximum opportunity loss (or “regret”)
Choose the option with the smallest maximum loss
Opportunity Loss Table
Investment Choice
(Alternatives)
Opportunity Loss in $1,000’s
(States of Nature)
Strong Economy
Stable Economy
Weak Economy
Large factory
Average factory
Small factory
110
160 70 90
140 50 1.
Maximum Op. Loss 2.
Smallest maximum loss is to choose Average factory
140
110
160
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

18. Decision Making Overview
Decision Criteria
No probabilities
known
Probabilistic Decision Criteria: Consider the probabilities of uncertain events and select an alternative to maximize the expected payoff of minimize the expected loss *
Probabilities
are known
maximize expected monetary value
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

19. Payoff Table Form of a payoff table with probabilities
Each state of nature sj has an associated probability Pi
Actions
States of nature s1
(P1) s2
(P2)
. . . sH
(PH) a1 a2 . aK
M11
M21
MK1
M12
M22
MK2
M1H
M2H
MKH
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

21. Expected Monetary Value Example
The expected monetary value is the weighted average payoff, given specified probabilities for each state of nature
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

22. Expected Monetary Value Solution
(continued)
Goal: Maximize expected monetary value
Payoff Table:
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
Maximize expected value by choosing Average factory
Expected Values
(EMV) 61 81 31
Example: EMV (Average factory) = 90(.3) + 120(.5) + (-30)(.2)
= 81
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

24. Add Probabilities and Payoffs
(continued)
Strong Economy
(.3)
200
Large factory
Stable Economy
(.5) 50
Weak Economy
(.2)
-120
(.3)
Strong Economy 90
Average factory
(.5)
Stable Economy
120
(.2)
Weak Economy
-30
Decision
(.3)
Strong Economy 40
Small factory
(.5)
Stable Economy 30
(.2)
Weak Economy 20
States of nature
Probabilities
Payoffs
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

25. Fold Back the Tree Large factory Average factory Small factory
Strong Economy
(.3)
EMV=200(.3)+50(.5)+(-120)(.2)=61
200
Large factory
Stable Economy
(.5) 50
Weak Economy
(.2)
-120
(.3)
Strong Economy
EMV=90(.3)+120(.5)+(-30)(.2)=81 90
Average factory
(.5)
Stable Economy
120
(.2)
Weak Economy
-30
(.3)
Strong Economy
EMV=40(.3)+30(.5)+20(.2)=31 40
Small factory
(.5)
Stable Economy 30
(.2)
Weak Economy 20
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

26. Make the Decision Large factory Average factory Small factory
Strong Economy
(.3)
EV=61
200
Large factory
Stable Economy
(.5) 50
Weak Economy
(.2)
-120
(.3)
Strong Economy
EV=81 90
Maximum
EMV=81
Average factory
(.5)
Stable Economy
120
(.2)
Weak Economy
-30
(.3)
Strong Economy
EV=31 40
Small factory
(.5)
Stable Economy 30
(.2)
Weak Economy 20
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

28. 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

29. Expected Value of Perfect Information, EVPI
(continued)
The expected value of perfect information (EVPI) is
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

30. Expected Value Under Certainty
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
Expected value under certainty
= expected value of the best decision, given perfect information
Value of best decision for each event:
Example: Best decision given “Strong Economy” is “Large factory”
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

31. Expected Value Under Certainty
(continued)
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
Now weight these outcomes with their probabilities to find the expected value:
200(.3)+120(.5)+20(.2)
= 124
Expected value under certainty
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

32. Expected Value of Perfect Information
Expected Value of Perfect Information (EVPI)
EVPI = Expected profit under certainty
– Expected monetary value of the best decision
Recall: Expected profit under certainty = 124
EMV is maximized by choosing “Average factory”,
where EMV = 81
so: EVPI = 124 – 81
= 43
(EVPI is the maximum you would be willing to spend to obtain perfect information)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

33. Bayes’ Theorem Example
Consider the choice of Stock A vs. Stock B
Stock Choice
(Action)
Percent Return
(Events)
Strong Economy
(.7)
Weak Economy
(.3)
Stock A 30
-10
Stock B 14 8
Expected Return:
Stock A has a higher EMV
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

34. Bayes’ Theorem Example
(continued)
Prior
Permits revising old probabilities based on new information
Probability
New
Information
Revised
Probability
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

35. Bayes’ Theorem Example
(continued)
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.
F1 = strong forecast
F2 = weak forecast
E1 = strong economy = 0.70
E2 = weak economy = 0.30
P(F1 | E1) = P(F1 | E2) = 0.30
Prior probabilities from stock choice example
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

36. Bayes’ Theorem Example
(continued)
Revised Probabilities (Bayes’ Theorem)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

37. EMV with Revised ProbabilitiesPi
Event
Stock A
xijPi
Stock B
.875
strong 30
26.25 14
12.25
.125
weak
-10
-1.25 8
1.00
Σ = Σ = 11.25
Revised probabilities
EMV Stock B = 11.25
EMV Stock A = 25.0
Maximum EMV
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

38. 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:
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

39. Expected Value of Sample Information, EVSI
(continued)
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

40. Utility
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

41. Utility
(continued)
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

42. Three Types of Utility Curves$ $ $
Risk Aversion
Risk Seeker
Risk-Neutral
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.

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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.