quantitative decision techniques 13/04/2009 decision trees and utility theory
TRANSCRIPT
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Quantitative Decision Techniques
13/04/2009Decision Trees and Utility TheoryDecision Trees and Utility Theory
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Chapter Outline
4.1 Introduction
4.2 Decision Trees
4.3 How Probability Values Are Estimated by
Bayesian Analysis
4.4 Utility Theory
4.5 Sensitivity Analysis
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Introduction
Decision trees enable one to look at decisions:
• with many alternatives and states of nature
• which must be made in sequence
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Decision Trees
A graphical representation where:
a decision node from which one of several alternatives may be chosen
a state-of-nature node out of which one state of nature will occur
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Thompson’s Decision Tree Fig. 4.1
1
2
A Decision Node
A State of Nature Node
Favorable Market
Unfavorable Market
Favorable Market
Unfavorable Market
Construct
Large P
lant
Construct Small Plant
Do Nothing
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Five Steps toDecision Tree Analysis
1. Define the problem2. Structure or draw the decision tree3. Assign probabilities to the states of nature4. Estimate payoffs for each possible
combination of alternatives and states of nature
5. Solve the problem by computing expected monetary values (EMVs) for each state of nature node.
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Decision Table for Thompson Lumber
AlternativeState of Nature
Favorable Market ($)
Unfavorable Market ($)
Construct a large plant
200,000 -180,000
Construct a small plant
100,000 -20,000
Do nothing 0 0
Probabilities
0.50 0.50
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Thompson’s Decision Tree Fig. 4.2
A Decision Node
A State of Nature Node
Favorable Market (0.5)
Unfavorable Market (0.5)
FavorableMarket (0.5)
Unfavorable Market (0.5)
Constru
ct Larg
e
Plant
Construct Small Plant
Do Nothing
$200,000
-$180,000
$100,000
-$20,000
0
EMV =$40,000
EMV=$10,000
1
2
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2nd Decision Table for Thompson Lumber
AlternativeState of Nature
Favorable Results
Unfavorable Results
Favorable Market
0.78 0.27
Unfavorable Market
0.22 0.73
Probabilities
0.45 0.55
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Thompson’s Decision Tree -Fig. 3
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Thompson’s Decision Tree -Fig. 4
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Expected Value of Sample Information
Expected value of best
decision with sample
information, assuming no
cost to gather it
Expected value of best
decision without
sample informationEVSI =
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Expected Value of Sample Information
EVSI= EV of best decision with sample information, assuming
no cost to gather it– EV of best decision without sample information= EV with sample info. + cost – EV without sample info.DM could pay up to EVSI for a survey.If the cost of the survey is less than EVSI, it is indeed
worthwhile.
In the example:EVSI = $49,200 + $10,000 – $40,000 = $19,200
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Estimating Probability Values by Bayesian Analysis
• Management experience or intuition
• History
• Existing data
• Need to be able to revise probabilities based upon new data
Posteriorprobabilities
Priorprobabilities New data
Bayes Theorem
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Example:• Market research specialists have told DM that,
statistically, of all new products with a favorable market, market surveys were positive and predicted success correctly 70% of the time.
• 30% of the time the surveys falsely predicted negative result
• On the other hand, when there was actually an unfavorable market for a new product, 80% of the surveys correctly predicted the negative results.
• The surveys incorrectly predicted positive results the remaining 20% of the time.
Bayesian Analysis
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Market Survey Reliability
Actual States of Nature
Result of Survey Favorable
Market (FM)
Unfavorable
Market (UM)
Positive (predicts
favorable market
for product)
P(survey positive|FM) = 0.70
P(survey positive|UM) = 0.20
Negative (predicts
unfavorable
market for
product)
P(survey negative|FM) = 0.30
P(survey negative|UM) = 0.80
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Calculating Posterior Probabilities
P(BA) P(A)P(AB) =
P(BA) P(A) + P(BA’) P(A’)where A and B are any two events, A’ is the complement of A
P(FMsurvey positive) = [P(survey positiveFM)P(FM)] / [P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)]
P(UMsurvey positive) = [P(survey positiveUM)P(UM)] / [P(survey positiveFM)P(FM) + P(survey positiveUM)P(UM)]
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Probability Revisions Given a Positive Survey
Conditional
ProbabilityPosterior
Probability
State
of
Nature
P(Survey positive|State of Nature
Prior
Probability
Joint
Probability
FM 0.70 * 0.50 0.350.450.35 = 0.78
UM 0.20 * 0.500.45
0.10 0.10 = 0.22
0.45 1.00
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Probability Revisions Given a Negative Survey
Conditional
ProbabilityPosterior
Probability
State
of
Nature
P(Survey
negative|State
of Nature)
Prior Probability
Joint Probability
FM 0.30 * 0.50 0.150.55
0.15 = 0.27
UM 0.80 * 0.50 0.400.55
0.40 = 0.73
0.55 1.00
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Utility Theory
• Utility assessment assigns the worst outcome a utility of 0, and the best outcome, a utility of 1.
• A standard gamble is used to determine utility values: When you are indifferent, the utility values are equal.
• Choose the alternative with the maximum expected utility EU(ai) = u(ai) = u(vij) P(j)
j
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Utility Theory
$5,000,000
$0
$2,000,000
Accept Offer
Reject Offer
Red(0.5)
Blue(0.5)
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Utility Assessment
• Utility assessment assigns the worst outcome a
utility of 0, and the best outcome, a utility of 1.
• A standard gamble is used to determine utility
values.
• When you are indifferent, the utility values are
equal.
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Standard Gamble for Utility Assessment
Best outcomeUtility = 1
Worst outcomeUtility = 0
Other outcomeUtility = ??
(p)
(1-p)Alternative 1
Alternative 2
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Figure 4.7$10,000U($10,000) = 1.0
0U(0)=0
$5,000U($5,000)=p=0.80
p= 0.80
(1-p)= 0.20Invest in
Real Estate
Invest in Bank
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Utility Assessment (1st approach)
v*u(v*) = 1
x1u(x1) = 0.5
x2u(x2) = 0.75
(0.5)
(0.5)Lottery ticket
Certain money
Best outcome (v*)u(v*) = 1
Worst outcome (v–)u(v–) = 0
Certain outcome (x1)u(x1) = 0.5
(0.5)
(0.5)Lottery ticket
Certain money
x1u(v*) = 0.5
Worst outcome (v–)u(v–) = 0
x3u(x3) = 0.25
(0.5)
(0.5)Lottery ticket
Certain money
In the example:u(-180) = 0 and u(200) = 1X1= 100 u(100) = 0.5X2 = 175 u(175) = 0.75X3 = 5 u(5) = 0.25
I II
III
0
0.2
0.4
0.6
0.8
1
-200 -150 -100 -50 0 50 100 150 200
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Utility Assessment (2nd approach)
Best outcome (v*)u(v*) = 1
Worst outcome (v–)u(v–) = 0
Certain outcome (vij)u(vij) = p
(p)
(1–p)Lottery ticket
Certain money
In the example:u(-180) = 0 and u(200) = 1
For vij=–20, p=%70 u(–20) = 0.7
For vij=0, p=%75 u(0) = 0.75
For vij=100, p=%90 u(100) = 0.9
0
0.2
0.4
0.6
0.8
1
-200 -150 -100 -50 0 50 100 150 200
Utilities STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketExpected
UtilityConstruct large plant 1 0 0.6Construct small plant 0.9 0.7 0.82Do nothing 0.75 0.75 0.75PROBABILITIES 0.6 0.4
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Sample Utility Curve
00.10.20.30.40.50.60.70.80.91
$- $2,000 $4,000 $6,000 $8,000 $10,000
Monetary Value
Util
ity
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Preferences for Risk
Monetary Outcome
RiskA
void
er
RiskSe
ekerRisk
Indi
ffere
nce
Uti
lity
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Example
Point up (0.45)
Point down (0.55)
$10,000
-$10,000
0
Alternative 1
Play the game
Alternative 2Do not play the game
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Utility Curve for Example
0
0.1
0.20.3
0.4
0.5
0.6
0.70.8
0.9
1
-$20,000 -$10,000 $0 $10,000 $20,000 $30,000
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Using Expected Utilities in Decision Making
Tack landspoint up (0.45)
Tack lands point down (0.55)
0.30
0.05
0.15
Alternative 1
Play the game
Alternative 2Don’t play
Utility