Statistics for business economics 7th by paul newbold chapter 18

Chapter GoalsAfter 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 app

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Chapter 18

Statistical Decision Theory

Statistics for Business and Economics

7 th Edition

Ch 18-1

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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

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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

18.1

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List Possible Actions or Events

Two Methods

of Listing

Ch 18-4

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Payoff Table

 Form of a payoff table

 M ij is the payoff that corresponds to action a i and state of nature s j

a K

M 11

M 21

M K1

M 12

M 22

M K2

.

M 1H

M 2H

M KH

Ch 18-5

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Payoff Table Example

A payoff table shows actions (alternatives) ,

states of nature , and payoffs

Large factory

Average factory

Small factory

200 90 40

50 120 30

-120 -30 20

Ch 18-6

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Decision Tree Example

Large factory

Small factory Average factory

Strong Economy Stable Economy Weak Economy

Payoffs

200 50 -120

40 30 20

90 120 -30

Ch 18-7

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Decision Making Overview

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

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The Maximin Criterion

 Consider K actions a 1 , a 2 , , a K and H possible states of nature

 More generally, the smallest possible payoff for action a i is given by

 Maximin criterion : select the action a i for which the

corresponding M i* is largest (that is, the action with the greatest minimum payoff )

) M , , M , Min(M

) M , , M , (M

M i *  11 12  1H

Ch 18-9

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Economy Economy Stable Economy Weak

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

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Economy Economy Stable Economy Weak

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

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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

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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 )

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Opportunity Loss Example

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

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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

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Minimax Regret Example

Economy

Stable Economy

Weak Economy

Large factory

Average factory

Small factory

0 110 160

70 0 90

140 50 0

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

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Minimax Regret Example

Economy

Stable Economy

Weak Economy

Large factory

Average factory

Small factory

0 110 160

70 0 90

140 50 0

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

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Decision Making Overview

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

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Payoff Table

 Form of a payoff table with probabilities

 Each state of nature s j has an associated probability P i

a K

M 11

M 21

M K1

M 12

M 22

M K2

.

M 1H

M 2H

M KH

Ch 18-19

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Expected Monetary Value (EMV)

state of nature with

 The expected monetary value of action a i is

ij j iH

H i2

2 i1

1

i ) P M P M P M P M

1 P

H

1j

j 





Ch 18-20

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Expected Monetary

Value Example

 The expected monetary value is the weighted

average payoff, given specified probabilities for each state of nature

Economy

(.3)

Stable Economy

(.5)

Weak Economy

50 120 30

-120 -30 20

Suppose these probabilities have been assessed for these states of nature

Ch 18-21

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Expected Monetary Value

Solution

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

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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

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Add Probabilities and Payoffs

(continued)

Payoffs

Probabilities

200 50 -120

40 30 20

90 120 -30

(.3) (.5) (.2)

Ch 18-24

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Fold Back the Tree

Large factory

Small factory Average factory

200 50 -120

40 30 20

90 120 -30

(.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

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Make the Decision

Large factory

Small factory

Average factory

200 50 -120

40 30 20

90 120 -30

(.3) (.5) (.2)

EV= 61

EV= 81

EV= 31

Maximum EMV= 81

Ch 18-26

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Bayes’ Theorem

 Let s 1 , s 2 , , s H 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

s i will occur, given that A occurs, by P(s i |A) , and the probability of

A , given s i , by P(A|s i )

 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 s i

and is modified to the posterior probability , P(s i |A), given the sample information that event A has occurred

) )P(s s

| P(A )

)P(s s

| P(A )

)P(s s

| P(A

) )P(s s

|

P(A P(A)

) )P(s s

| P(A A)

|

P(s

HH

22

11

ii

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Bayes’ Theorem Example

Stock Choice

(Action)

Percent Return

(Events) Strong

Economy

(.7)

Weak Economy

18.0 12.2

Stock A has a higher EMV

Ch 18-28

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 Permits revising old probabilities based on new information

New Information

Revised Probability

Prior Probability

(continued)

Ch 18-29

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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

F 1 = strong forecast

F 2 = weak forecast

E 1 = strong economy = 0.70

E 2 = weak economy = 0.30 P(F 1 | E 1 ) = 0.90 P(F 1 | E 2 ) = 0.30

(continued)

Ch 18-30

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 Revised Probabilities (Bayes’ Theorem)

3 )

E

| F ( P , 9 )

E

| F (

3 )

E ( P , 7 )

E (

875

)

3 )(.

3 (.

) 9 )(.

7 (.

) 9 )(.

7

(.

) F ( P

) E

| F ( P ) E (

P )

F

| E

(

P

1

1 1

F ( P

) E

| F ( P ) E (

P )

F

| E (

P

1

2 1

(continued)

Ch 18-31

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EMV with Revised Probabilities

Ch 18-32

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Expected Value of Sample Information, EVSI

 Suppose there are K possible actions and H

states of nature, s 1 , s 2 , , s H

 The decision-maker may obtain sample information

Let there be M possible sample results,

A 1 , A 2 , , A M

 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(s s

| P(A )

)P(s s

| P(A )

)P(s s

| P(A )

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Expected Value of Sample Information, EVSI

 For each possible sample result, A i , find the

difference, V i , 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

A i was observed

M M

2 2

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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(s 1 ), P(s 2 ), , P(s H ) are used

 For each possible state of nature, s i , find the difference,

W i , 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 s i will occur

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 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

H H

2 2

1

P(s

 The expected value of perfect information (EVPI) is

(continued)

Ch 18-36

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Expected Value Under Certainty

(Action)

(Events) Strong

Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)

200 90 40

50 120 30

-120 -30 20

Example: Best decision given “Strong Economy” is

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Expected Value Under Certainty

(Action)

(Events) Strong

Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)

200 90 40

50 120 30

-120 -30 20

Ch 18-38

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Expected Value of Perfect Information

Expected Value of Perfect Information (EVPI)

EVPI = Expected profit under certainty

– Expected monetary value of the best decision

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

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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

18.5

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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

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Three Types of Utility Curves

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