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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
for the Investment Example
[ ] ( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( )
[ ] ( ) ( ) ( ) ( ) ( ) ( )
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
$623.50
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EMV Criterion
for the Investment Example
1. Calculate the expected monetary value of each alternative.
2. Choose the alternative with the largest EMV.
StagnantSlow
GrowthRapid
Growth
Expected
MonetaryValue
0.25 0.45 0.30
Stocks (500)$ 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
for the Investment Example
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
for the Investment Example
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
for the Investment Example
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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