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Chapter Seventeen
Decision Theory
McGraw-Hill/Irwin
Copyright © 2003 by The McGraw-Hill Companies, Inc. All rights reserved.
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18
Decision Theory
17.1
17.2
17.3
17.4
Bayes’ Theorem
Introduction to Decision Theory
Decision Making Using Posterior Probabilities
Introduction to Utility Theory
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17.1 Bayes’ Theorem: An Example,
AIDS Testing
Question: Suppose that a person selected randomly for testing, tests
positive for AIDS. The test is known to be highly accurate (99.9% for
people who have AIDS, 99% for people who don’t.) What is the
probability that the person actually has AIDS?
Answer:
Surprisingly, much lower than most of us would guess!
The Facts :
AIDS Incidence Rate : 6 cases per 1000 Americans
P(AIDS)  0.006
P( AIDS )  0.994
Testing Accuracy :
P(POS|AIDS )  0.999
P(POS|AIDS )  0.01
Solution : P(AIDS|POS )
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An Example, AIDS Testing (Continued)
P(AIDS|POS ) 
P(AIDS  POS)

P(POS)
P(AIDS  POS)
P(AIDS  POS)  P( AIDS  POS)

P(AIDS)P(P OS|AIDS)
P(AIDS)P(P OS|AIDS)  P( AIDS )P(POS| AIDS )

( 0.006 )( 0.999 )
0.005994

( 0.006 )( 0.999 )  ( 0.994 )( 0.01 ) 0.005994  0.00994

0.005994
0.015934
( Bayes ' Theorem)
 0.3762
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Bayes’ Theorem
S1, S2, …, Sk represents k mutually exclusive possible states of nature,
one of which must be true.
P(S1), P(S2), …, P(Sk) represents the prior probabilities of the k
possible states of nature.
If E is a particular outcome of an experiment designed to determine
which is the true state of nature, then the posterior (or revised)
probability of a state Si, given the experimental outcome E, is:
P(Si  E)
P(Si|E) =
P(E)
P(Si )P(E|S i )

P(E)
P(Si )P(E|S i )

P(S1 )P(E|S1 )+P(S 2 )P(E|S 2 )+ ...+P(Sk )P(E|S k )
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17.2 Introduction to Decision Theory
Example: A developer must decide how large a luxury
condominium complex to build – small, medium, or large – when
profitability depends on the level of future demand – low or high –
for luxury condominiums.
Elements of Decision Theory
States of nature: Set of potential future conditions that
affects decision results. (e.g. low demand versus high
demand)
Alternatives: Set of alternative actions for the decision
maker to chose from. (e.g. small, medium, large)
Payoffs: Set of payoffs for each alternative under each
potential state of nature, often summarized in a payoff table.)
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Payoff Tables and Decision Trees
Example: Condominium Complex Situation
Decision Tree
Payoff Table
States of Nature
Alternatives
Low
High
8
8
Small
5
15
Medium
-11
22
Large
(payoffs in millions of dollars)
Decision Tree Legend
Branch
Decision point (node)
State of nature (node)
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Decision Making Under Uncertainty
Condominium Example
Maximin: Identify the minimum
(or worst) possible payoff for each
alternative and select the alternative
that maximizes the worst possible
payoff. Pessimistic.
Condominium Example
Maximax: Identify the maximum
(or best) possible payoff for each
alternative and select the alternative
that maximizes the best possible
payoff. Optimistic.
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Decision Making Under Risk
Expected value criterion: Using prior probabilities for the states of
nature, compute the expected payoff for each alternative and select
the alternative with the largest expected payoff.
Condominium Example
Expected payoff under certainty
Expected payoff under risk
Expected value of perfect information (EVPI)
EVPI = expected payoff under certainty – expected value under risk
= 17.8 – 12.1 = 5.7
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17.3 Decision Making Using Posterior
Probabilities
Example 17.3: The Oil Drilling Case
Decision Tree and Payoff Table for Prior Analysis
E(Payoff|Drill) = -190,000
E(Payoff|Do Not Drill) = 0*
Decision: Do Not Drill
Suppose now that the oil company can obtain sample information in
the form of a seismic study that can yield three possible readings –
low, medium and high.
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Computing Posterior Probabilities,
Given Sample Information
Example: The Oil Drilling Case
Prior and conditional probabilities
Prior Probabilities of Oil
P(Oil)
O
none
0.7
i
some
0.2
l
much
0.1
x
An application of
Bayes Theorem
Seismic Experiment Accuracy: P(Reading|Oil)
Reading
P(high|Oil) P(medium|Oil)
O
none
0.04
0.05
i
some
0.02
0.94
l
much
0.96
0.03
P(low|Oil)
0.91
0.04
0.01
=
Joint Probabilities: P(Oil&Reading) = P(Oil)P(Reading|Oil)
Reading
P(Oil&high) P(Oil&medium)
P(Oil&low)
O
none
0.028
0.035
0.637
i
some
0.004
0.188
0.008
l
much
0.096
0.003
0.001
Total
0.128
0.226
0.646
P(Reading)
P(high)
P(medium)
P(low)
Posterior Probabilities: P(Oil|Reading) = P(Oil&Reading)/P(Reading)
Reading
P(Oil|high) P(Oil|medium)
P(Oil|low)
O
none
0.21875
0.15487
0.98607
i
some
0.03125
0.83186
0.01238
l
much
0.75000
0.01327
0.00155
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Posterior Decision Tree
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Expected Payoffs, Given Sample
Information
Posterior Probabilities: P(Oil|Reading) = P(Oil&Reading)/P(Reading)
Reading
P(Oil|high)
P(Oil|medium)
P(Oil|low)
O
none
0.21875
0.15487
0.98607
i
some
0.03125
0.83186
0.01238
l
much
0.75000
0.01327
0.00155
Drilling Decision Payoff Table: Payoff(Decision|Oil)
Decision
drill
no drill
O
none
-700,000
0
i
some
500,000
0
l
much
2,000,000
0
Conditional Payoffs for Drilling: Payoff(drill|Reading) = Payoff(drill|Oil)P(Oil|Reading)
[Conditional Payoffs for No Drilling are Zero for all Readings: Payoff(no drill|Reading) = 0]
Reading
high
medium
low
O
none
-153,125
-108,407
-690,248
i
some
15,625
415,929
6,192
l
much
1,500,000
26,549
3,096
Total
1,362,500
334,071
-680,960
Pay(drill|Read)
Pay(drill|high) Pay(drill|medium)
Pay(drill|low)
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Summary of Posterior Analysis and the
Expected Value of Sample Information
Summary of Posterior Analysis (Optimal decision and payoff for each reading)
Reading
low
medium
high
-680,960
334,071
1,362,500
Pay(drill|Read)
0
0
0
Pay(no drill|Read)
no drill
drill
drill
Decision
0
334,071
1,362,500
Exp(Pay|Read)
Expected Payoff of Sampling (EPS)
Exp(Pay|Read)
P(Reading)
Product
high
1,362,500
0.128
174,400
Reading
medium
334,071
0.226
75,500
Expected Payoff of No Sampling (EPNS)
Expected payoff for optimal decision from prior analysis, 0 (Do Not Drill)
Expected Value of Sample Information (EVSI)
EVSI = EPS - EPNS
low
0
0.646
0
249,900
EPS
0
EPNS
249,900
EVSI
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17.4 Example: Utility Theory
Investment 1 Profits
Profit
Prob Prof x Prob
$50,000
0.7
35000
$10,000
0.1
1000
-$20,000
0.2
-4000
Expected Profit
$32,000
Investment 2 Profits
Profit
Prob Prof x Prob
$40,000
0.6
24000
$30,000
0.2
6000
-$10,000
0.2
-2000
Expected Profit
$28,000
No Investment Profit
Profit
Prob Prof x Prob
$0
1
0
Expected Profit
$0
Utilities
Profit
$50,000
$40,000
$30,000
$10,000
$0
-$10,000
-$20,000
Utility
1.00
0.95
0.90
0.75
0.60
0.45
0.00
Investment 1 Utilities
Utility
Prob Util x Prob
1.00
0.7
0.700
0.75
0.1
0.075
0.00
0.2
0.000
Expected Utility
0.775
Investment 2 Utilities
Utility
Prob Util x Prob
0.95
0.6
0.570
0.90
0.2
0.180
0.45
0.2
0.090
Expected Utility
0.840
No Investment Utility
Utility
Prob Util x Prob
0.60
1
0.600
Expected Utility
0.600
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Utilities
Utilities are measures of the relative value of varying dollar payoffs
for an individual decision maker and thus capture the decision
maker’s attitude toward risk. Under certain mild assumptions about
rational behavior, decision makers should replace dollar payoffs with
their respective utilities and maximize expected utility.
Example Utility Curves
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Decision Theory
17.1 Bayes’ Theorem
Summary:
17.2 Introduction to Decision Theory
17.3 Decision Making Using Posterior Probabilities
17.4 Introduction to Utility Theory
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