Ken Black QA ch19

8/3/2019 Ken Black QA ch19

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Business Statistics, 5th ed.

by Ken Black

Chapter 19

DecisionAnalysis

Discrete Distributions

PowerPoint presentations prepared by Lloyd Jaisingh,Morehead State University


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Learning Objectives

Learn about decision making under certainty, under

uncertainty, and under risk.

Learn several strategies for decision-making underuncertainty, including expected payoff, expected opportunity

loss, maximin, maximax, and minimax regret.

Learn how to construct and analyze decision trees.

Understand aspects of utility theory.

Learn how to revise probabilities with sample information.


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Three Variables

in Decision Analysis Problems

Decision Alternatives are the various choices oroptions available to the decision maker in any

given problem situation

States of Nature are the occurrences of naturethat can effect the outcome of the decision. Theyare beyond the decision makers control

Payoffs are the benefits or rewards that resultfrom selecting a particular decision alternative.

They are often expressed in dollars, but may bestated in other units, such as market share


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Decision Table

1 2 3

1 1 1 1 2 1 3 1

2 2 1 2 2 2 3 2

3 3 1 3 2 3 3 3

1 2 3

s s s s

d P P P P

d P P P P

d P P P P

d P P P P

n

n

n

n

m m m m m n

, , , ,

, , , ,

, , , ,

, , , ,

States of Nature

Decision

Alternatives

where: sj= state of nature

dj = decision alternative

Pi,j = payoff for decision iunder statej


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Example: Decision Table

for an Investor

Stagnant

Slow

Growth

Rapid

Growth

Stocks (500)$ 700$ 2,200$Bonds (100)$ 600$ 900$

CDs 300$ 500$ 750$

Mixture (200)$ 650$ 1,300$Annual payoffs for an investment of $10,000


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Decision-Making Under Certainty

The states of nature are known.

Stagnant

Slow

Growth

Rapid

GrowthStocks (500)$ 700$ 2,200$Bonds (100)$ 600$ 900$

CDs 300$ 500$ 750$Mixture (200)$ 650$ 1,300$

Annual payoffs for an investment of $10,000

The economy

will grow

rapidly.

Invest in stocks.

The economy

will grow

rapidly.

Invest in stocks.


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Criteria for Decision Making

Under Uncertainty

Maximax payoff: Choose the best of thebest

Maximin payoff: Choose the best of theworst

Hurwicz payoff: Use a weighted average ofthe extremes

Minimax regret: Minimize the maximumopportunity loss


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Maximax Criterion

1. Identify the maximum payoff for each alternative.

2. Choose the alternative with the largest maximum.

Stagnant

Slow

Growth

Rapid

Growth Maximum

StocksBonds

CDs

Mixture

(500)$ 700$ 2,200$ 2,200$(100)$ 600$ 900$ 900$

300$ 500$ 750$ 750$

(200)$ 650$ 1,300$ 1,300$


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Maximin Criterion

1. Identify the minimum payoff for each alternative.

2. Choose the alternative with the largest minimum.

Stagnant

Slow

Growth

Rapid

Growth Minimum

Stocks (500)$ 700$ 2,200$ (500)$Bonds (100)$ 600$ 900$ (100)$

CDs 300$ 500$ 750$ 300$

Mixture (200)$ 650$ 1,300$ (200)$


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Hurwicz Criterion

1. Identify the maximum payoff for each alternative.

2. Identify the minimum payoff for each alternative.

3. Calculate a weighted average of the maximum and the minimum using and (1 - ) for weights.

4. Choose the alternative with the largest weighted average.

Stagnant

Slow

Growth

Rapid

Growth Maximum Minimum

Weighted

Average

Stocks (500)$ 700$ 2,200$ 2,200$ (500)$ 1,390$

Bonds (100)$ 600$ 900$ 900$ (100)$ 600$CDs 300$ 500$ 750$ 750$ 300$ 615$

Mixture (200)$ 650$ 1,300$ 1,300$ (200)$ 850$

=.7 1 =.3


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Decision Alternatives

for Various Values ofStocks Bonds CDs Mixture

Max Min Max Min Max Min Max Min

1- 2,200 -500 900 -100 750 300 1,300 -2000.0 1.0 -500 -100 300 -200

0.1 0.9 -230 0 345 -500.2 0.8 40 100 390 100

0.3 0.7 310 200 435 250

0.4 0.6 580 300 480 400

0.5 0.5 850 400 525 550

0.6 0.4 1120 500 570 7000.7 0.3 1390 600 615 850

0.8 0.2 1660 700 660 1000

0.9 0.1 1930 800 705 1150

1.0 0.0 2200 900 750 1300


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Graph of Hurwicz Criterion

Selections for Various Values of

-500

0

500

1000

1500

2000

2500

0.0 0.2 0.4 0.6 0.8 1.0

Stocks

Mixture

Bonds

CDs


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Investment Example:

Selected Regrets

Stagnant

Slow

Growth

Rapid

Growth

Stocks (500)$ 700$ 2,200$

Bonds (100)$ 600$ 900$CDs 300$ 500$ 750$

Mixture (200)$ 650$ 1,300$

I invested in CDs.

Then the economy

grew rapidly. I am

out $1,450.

I invested in stocks.

Then the economy

stagnated. I regret notinvesting in CDs. I am

$800 down from where

I could have been.

I invested in stocks, and

the economy grew slowly.

I have no regrets.


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Investment Example:

Opportunity Loss Table

Stagnant

Slow

Growth

Rapid

Growth

Stocks 800 0 0Bonds 400 100 1,300

CDs 0 200 1,450

Mixture 500 50 900


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Investment Example:

Calculating Opportunity Loss

Stagnant

Slow

Growth

Rapid

Growth

Stocks (500)$ 700$ 2,200$

Bonds (100)$ 600$ 900$

CDs 300$ 500$ 750$

Mixture (200)$ 650$ 1,300$

Payoff Table

Stagnant

Slow

Growth

Rapid

GrowthStocks 800 0 0

Bonds 400 100 1,300

CDs 0 200 1,450

Mixture 500 50 900

Opportunity Loss Table

OLi,j = Max(column j) - Pi,j


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Minimax Regret

1. Identify the maximum regret for each alternative.2. Choose the alternative with the least maximum regret.

Stagnant

Slow

Growth

Rapid

Growth Maximum

Stocks 800 0 0 800

Bonds 400 100 1,300 1,300

CDs 0 200 1,450 1,450Mixture 500 50 900 900


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Decision Making under Risk

Probabilities of the states of nature have beendeterminedDecision-Making under uncertainty: probabilities of

the states of nature are unknownDecision-Making under risk: probabilities of the

states of nature are known (have been estimated)

Decision Trees Expected Monetary Value of Alternatives


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Decision Table with States of Nature

Probabilities for Investment Example

Stagnant

.25

Slow

Growth

.45

Rapid

Growth

.30

Stocks (500)$ 700$ 2,200$

Bonds (100)$ 600$ 900$

CDs 300$ 500$ 750$

Mixture (200)$ 650$ 1,300$

Probabilities


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Decision Tree

for the Investment Example

Stocks

Bonds

CDs

Mixture

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

-$500

$700

$2,200

-$100

$600

$900

$300

$500

$750

-$200

$650

$1,300

Decision

Node

Chance

Node


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Expected Monetary Value Criterion

[ ]EMV

where

i i jj

n

jd X P= =

,

:

1

= decision alternative i

= the probability of state j

= the payoff for decision i in state j

i

j

i, j

dP

X


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EMV Calculations


[ ] ( ) ( ) ( ) ( ) ( ) ( )

[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )

[ ] ( ) ( ) ( ) ( ) ( ) ( )

EMV stocks

EMV bondsEMV CDs

EMV mixture

= + + =

= + + =

= + + =

= + + =

. . .

. . .

. . .

. . . .

25 500 45 700 3 2200 850

25 100 45 600 30 900 51525 300 45 500 30 750 525

25 200 45 650 30 1300 63250


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Decision Tree with Expected Monetary

Values for the Investment Example

Stocks

Bonds

CDs

Mixture

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Slow growth (.45)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Stagnant (.25)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

Rapid Growth (.30)

-$500

$700

$2,200

-$100

$600

$900

$300

$500

$750

-$200

$650

$1,300

$850

$515

$525

$623.50


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EMV Criterion


1. Calculate the expected monetary value of each alternative.

2. Choose the alternative with the largest EMV.

StagnantSlow

GrowthRapid

Growth

Expected

MonetaryValue

0.25 0.45 0.30

Stocks (500)$ 700$ 2,200$ 850.00$

Bonds (100)$ 600$ 900$ 515.00$

CDs 300$ 500$ 750$ 525.00$

Mixture (200)$ 650$ 1,300$ 632.50$


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Expected Monetary Payoff with Perfect

Information for the Investment Example

Stagnant.25

Slow

Growth.45

Rapid

Growth.30

Stocks (500)$ 700$ 2,200$

Bonds (100)$ 600$ 900$CDs 300$ 500$ 750$

Mixture (200)$ 650$ 1,300$

Expected Monetary Payoff with Perfect Information

= ($300)(.25) + ($700)(.45) + ($2200)(.30)

= $1050


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Expected Value of Perfect Information


Expected Value of Perfect Information

= Expected Monetary Payoff with Perfect Information - Max(EMV[di])

= $1050 - $850

= $200


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Utility

The degree of pleasure or displeasure adecision-maker has in being involved in theoutcome selection process given the risks

and opportunities availableRisk-AvoiderRisk-NeutralRisk-Taker


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Risk Neutral: Indifferent to Owning a

or b

a

b

$100,000

-$0

.5

.5

$50,000

00


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Risk Avoider: Indifferent to Owning a

or b

a

b

$100,000

-$0

.5

.5

$20,000

00


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Risk Taker: Indifferent to Owning a or

b

a

b

$100,000

-$0

.5

.5

$70,000

00


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Utility Curves for Game Players

Chance of

Winning

the Contest

Monetary Payoff

Risk-Avoider

Risk

Neutral

Risk-Taker


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Revising Probabilities

in Light of Sample Information

Bayes Rule

Expected Value of Sample Information


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Decision Table

for Investment Problem

No

Growth

(.65)

Rapid

Growth

(.35)

Bonds 500$ 100$

Stocks (200)$ 1,100$


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Expected Monetary Value Criterion


NoGrowth

RapidGrowth

Expected

MonetaryValue

0.65 0.35

Bonds 500$ 100$ 360.00$Stocks (200)$ 1,100$ 255.00$


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Decision Tree


Stocks

Bonds

No Growth (.65)

No Growth (.65)

Rapid Growth (.35)

Rapid Growth (.35)

$500

$100

-$200

$1,100

EMV=$360

EMV=$255

$360


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Historical Performance

of Economic Forecaster

Actual State of Economy

No Growth(s1)

Rapid Growth(s2)

Forecaster PredictsNo Growth (F1) .80 .30Forecaster PredictsRapid Growth (F2) .20 .70

P(Fi|sj)


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Bayes Rule

P X YP Y X P X

P Y X P X P Y X P X P Y X P X

i

i i

n n

( | )( | ) ( )

( | ) ( ) ( | ) ( ) ( | ) ( )

=

+ + 1 1 2 2


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Revision Based on a Forecast

of No Growth (F1)

State of

Economy

Prior

Probabilities

Conditional

Probabilities

Joint

Probabilities

Revised

Probabilities

No

Growth

(s 1)

P(s 1) = .65 P(F1| s 1) = .80 P(F1 s 1) = .520 .520/.625 = .832

Rapid

Growth

(s 2)

P(s 2) =.35 P(F1| s 2) = .30 P(F1 s 2) = .105 .105/.625 = .168

P(F1) = .625

P(sj|F1)


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Revision Based on a Forecast

of Rapid Growth (F2)

State of

Economy

Prior

Probabilities

Conditional

Probabilities

Joint

Probabilities

Revised

Probabilities

No

Growth

(s 1)

P(s 1) = .65 P(F2| s 1) = .20 P(F2 s 1) = .130 .130/.375 = .347

Rapid

Growth

(s 2)

P(s 2) =.35 P(F2| s 2) = .70 P(F2 s 2) = .245 .245/.375 = .653

P(F1) = .375

P(sj|F2)


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Decision Tree for the Investment Example

After Revision of Probabilities

Stocks

Bonds

No Growth (.832)

No Growth (.832)

Rapid Growth (.168)

Rapid Growth (.168)

$500

$100

-$200

$1,100

$432.80

$18.40

$432.80

Stocks

Bonds

No Growth (.347)

No Growth (.347)

Rapid Growth (.653)

Rapid Growth (.653)

$500

$100

-$200

$1,100

$238.80

$648.90

$648.90

Forecast

No Growth

(.625)

Forecast

Rapid Growth

(.375)

$513.84Buy

Forecast


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Expected Value of Sample Information for

the Investment Example

Expected value of sample information

= expected monetary value with information

- expected monetary value without information

= $513.84 - $360= $153.84


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Ken Black QA ch19

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