chapter 18 statistical decision theory copyright © 2010 pearson education, inc. publishing as...

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


18.1

List Possible Actions or Events


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


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


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


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


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)


)M,,M,Min(MM 1H1211*1

)M,,M,(MM 1H1211*i

Ch. 18-9

Maximin Example


Investment Choice

(Alternatives)


(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


Investment Choice

(Alternatives)


(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


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)


Opportunity Loss Example


Investment Choice

(Alternatives)


(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


Investment Choice

(Alternatives)


(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


Investment Choice

(Alternatives)


(States of Nature)

Strong 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

140110160

Ch. 18-16

Minimax Regret Example


Investment Choice

(Alternatives)


(States of Nature)

Strong 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

140110160

2.

Smallest maximum loss is to choose

Average factory

(continued)

Ch. 18-17

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

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

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


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


Investment Choice

(Alternatives)


(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


Investment Choice

(Action)


(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


Add Probabilities and Payoffs


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


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


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


))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


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


Permits revising old probabilities based on new information


NewInformation

RevisedProbability

PriorProbability

(continued)

Ch. 18-29



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

EMV with Revised Probabilities


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


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


(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


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


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


Investment Choice

(Action)


(Events)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)


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:


Investment Choice

(Action)


(Events)

Strong Economy

(.3)

Stable Economy

(.5)

Weak Economy

(.2)


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


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


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


(continued)

Ch. 18-41

Three Types of Utility Curves


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


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


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


chapter 18 statistical decision theory copyright © 2010 pearson education, inc. publishing as...

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