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Chapter 18
Statistical Decision Theory
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Statistics for Business and Economics
7th Edition
Ch. 18-1
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
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
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
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
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
20Ch. 18-6
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
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
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 a i 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
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 criterion1. For each option, find the minimum payoff
Ch. 18-10
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 criterion1. 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
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
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
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
20The 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
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
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
140110160
Ch. 18-16
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”)
2. Choose the option with the smallest maximum loss
1.
Maximum Op. Loss
140110160
2.
Smallest maximum loss is to choose
Average factory
(continued)
Ch. 18-17
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
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
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
1jijjiHHi22i11i MPMPMPMP)EMV(a
1PH
1jj
Ch. 18-20
Expected MonetaryValue 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
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 factoryAverage factorySmall factory
2009040
50120 30
-120-30 20
Expected Values
(EMV)618131
Maximize expected value by choosing Average factory
(continued)
Payoff Table:
Goal: Maximize expected monetary value
Ch. 18-22
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
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
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
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
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(s i|A) , and the probability of A , given si , by P(A|si)
Bayes’ Theorem states that the conditional probability of s i, given A, can be expressed as
In the terminology of this section, P(s i) 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(AA)|P(s
HH2211
iiiii
Ch. 18-27
18.4
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
Bayes’ Theorem Example
Permits revising old probabilities based on new information
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
NewInformation
RevisedProbability
PriorProbability
(continued)
Ch. 18-29
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
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
11111
125.)F(P
)E|F(P)E(P)F|E(P
1
21212
(continued)
Ch. 18-31
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
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
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
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
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
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 factoryAverage factorySmall factory
2009040
50120 30
-120-30 20
Example: Best decision given “Strong Economy” is “Large factory”
200 120 20Value of best decision for each event:
Ch. 18-37
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 factoryAverage factorySmall factory
2009040
50120 30
-120-30 20
200 120 20
(continued)
200(.3)+120(.5)+20(.2) = 124
Expected value under certainty
Ch. 18-38
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
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
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
Three Types of Utility Curves
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ut i
lity
$ $ $
Uti
lity
Ut i
lity
Risk Aversion Risk Seeker Risk-Neutral
Ch. 18-42
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
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
1jijjiHHi22i11i UPUPUPUP)EU(a
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 18-44
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