Post on 23-Mar-2020
Dr. Y. İlker TOPCUwww.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info
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Dr. Özgür KABAKweb.itu.edu.tr/kabak/
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Introduction One dimensional (single criterion) decision
making Single stage vs. multi stage decision making
Decision analysis is an analytical and systematic way to tackle problems
A good decision is based on logic: rational decision maker (DM)
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Components of Decision Analysis A state of nature is an actual event that may occur in
the future. A payoff matrix (decision table) is a means of
organizing a decision situation, presenting the payoffs from different decisions/alternatives given the various states of nature.
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Basic Steps in Decision Analysis1) Clearly define the problem at hand2) List the possible alternatives3) Identify the possible state of natures4) List the payoff (profit/cost) of alternatives
with respect to state of natures5) Select one of the mathematical decision
analysis methods (models)6) Apply the method and make your decision
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Decision-Making Environments Type 1: Decision-making under certainty
DM knows with certainty the payoffs of every alternative .
Type 2: Decision-making under uncertainty
DM does not know the probabilities of the various states of nature. Actually s/he knows nothing!
Type 3: Decision-making under risk
DM does know the probabilities of the various states of nature.
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Decision Making Under Certainty Instead of state of natures, a true state is known
to the decision maker before s/he has to make decision
The optimal choice is to pick an alternative with the highest payoff
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Decision Making Under Uncertainty Maximax
Maximin
Criterion of Realism
Equally likelihood
Minimax
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Decision Table / Payoff Matrix
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketConstruct large plant $200,000 ($180,000)Construct small plant $100,000 ($20,000)Do nothing $0 $0
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 8
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
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Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
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Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
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Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$162,000
Construct small plant$100,000($20,000)$88,000
Do nothing$0$0$0
Optimism-pessimism index (a)0.9MAX. W.A.=$162,000
DECISION :Construct large plant
&A
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Minimax regret
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
REGRET VALUESMaximum in row
$0$180,000$180,000
$100,000$20,000$100,000
$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
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Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
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EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Monetary Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
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EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
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MaximaxChoose the alternative with the maximum optimistic
level
ok = {oi} = { {vij}}
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketMaximum in
row (ok)Construct large plant $200,000 ($180,000) $200,000Construct small plant $100,000 ($20,000) $100,000Do nothing $0 $0 $0
m
i 1max
=
m
i 1max
=
m
j 1max
=
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MaximinChoose the alternative with the maximum security
level
sk = {si} = { {vij}}m
i 1max
=
m
i 1max
=
m
j 1min
=
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketMinimum in
row (sk)Construct large plant $200,000 ($180,000) ($180,000)Construct small plant $100,000 ($20,000) ($20,000)Do nothing $0 $0 $0
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Criterion of RealismHurwicz suggested to use the optimism-pessimism index (α)Choose the alternative with the maximum weighted average
of optimistic and security levels
{α oi + (1 – α) si} where 0≤α≤1m
i 1max
=
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketWeighted average α=0.7
Construct large plant $200.000 ($180.000) 380α-180 86Construct small plant $100.000 ($20.000) 120α-20 64Do nothing $0 $0 $0 0
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 11
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
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Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
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Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWeighted averagea=0.7
Construct large plant$200,000($180,000)380a-18086
Construct small plant$100,000($20,000)120a-2064
Do nothing$0$0$00
Optimism-pessimism index (a)0.9MAX. W.A.=$0
DECISION :Do nothing
&A
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Minimax regret
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
REGRET VALUESMaximum in row
$0$180,000$180,000
$100,000$20,000$100,000
$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Monetary Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
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-200000-150000-100000
-500000
50000100000150000200000250000
0 0.2 0.4 0.6 0.8 1
α
Payo
ff
Point 1Small Plant
Large Plant Point 2
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Criterion of Realism
120α-20 = 0 ⇒ α = 0.1667380α-180 = 120α-20 ⇒ α = 0.6154
0 ≤ α ≤ 0.1667 ⇒ “Do nothing”0.1667 ≤ α ≤ 0.6154 ⇒ “Construct small plant”0.6154 ≤ α ≤ 1 ⇒ “Construct large plant”
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Equally LikelihoodLaplace argued that “knowing nothing at all about the
true state of nature” is equivalent to “all states having equal probability”
Choose the alternative with the maximum row average (expected value)
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketRow
averageConstruct large plant $200,000 ($180,000) $10,000Construct small plant $100,000 ($20,000) $40,000Do nothing $0 $0 $0
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 14
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$162,000
Construct small plant$100,000($20,000)$88,000
Do nothing$0$0$0
Optimism-pessimism index (a)0.9MAX. W.A.=$162,000
DECISION :Construct large plant
&A
Page &P
Minimax regret
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
REGRET VALUESMaximum in row
$0$180,000$180,000
$100,000$20,000$100,000
$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Monetary Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
MinimaxSavage defined the regret (opportunity loss) as the difference
between the value resulting from the best action given that state
of nature j is the true state and the value resulting from alternative i with respect to state
of nature jChoose the alternative with the minimum worst
(maximum) regretRegret Values STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketMaximum in
rowConstruct large plant $0 $180,000 $180,000Construct small plant $100,000 $20,000 $100,000Do nothing $200,000 $0 $200,000
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 15
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STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$54,0800.1666666667
Construct small plant$100,000($20,000)$53,9200.6153846154
Do nothing$0$0$0
Optimism-pessimism index (a)0.616MAX. W.A.=$54,080
DECISION :Construct large plant
&A
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Minimax regret
$200,000($180,000)
$100,000($20,000)
$0$0
Regret ValuesSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$0$180,000$180,000
Construct small plant$100,000$20,000$100,000
Do nothing$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Monetary Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
Results of the Example
METHOD DECISION
Maximax “Construct large plant”
Maximin “Do nothing”
Criterion of Realism depends on α
Equally likelihood “Construct small plant”
Minimax “Construct small plant”
The appropriate method is dependent on the personality and philosophy of the DM.
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Additional Example
Alternative
State of NatureGood
Market(TL)
Average Market
(TL)
Poor Market
(TL)Large Coffee House 75,000 25,000 -40,000
Small Coffee House 100,000 35,000 -60,000
Do nothing 0 0 0
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 17
PresenterPresentation NotesMaximax: 75, 100*, 0Maximin: -40, -60, 0*Hurwicz: (a=0.6) 29, 36*, 0Eq. Like.: 20, 25*, 0Regret: 40*, 60, 100
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Solutions with QM for Windows
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Decision Making Under RiskProbability
Objective Subjective
Expected (Monetary) ValueExpected Value of Perfect Information
Expected Opportunity Loss Utility Theory
Certainty EquivalenceRisk Premium
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Probability
A probability is a numerical statement about the likelihood that an event will occur
The probability, P, of any event occurring is greater than or equal to 0 and less than or equal to 1:
0 ≤ P(event) ≤ 1 The sum of the simple probabilities for all possible
outcomes of an activity must equal 1
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Objective Probability
Determined by experiment or observation: Probability of heads on coin flip Probability of spades on drawing card from deck
P(A): probability of occurrence of event An(A): number of times that event A occursn: number of independent and identical repetitions
of the experiments or observations
P(A) = n(A) / n∞→n
limit
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Subjective Probability Determined by an estimate based on the expert’s
personal belief, judgment, experience,
and existing knowledge of a situation
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PresenterPresentation Notes Subjective Probability is associated not with the system under observation, but with the observer of that systemDifferent DMs can assign different probabilities to the same event.
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Payoff Matrix with Probabilities
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketProbabilites 60% 40%Construct large plant $200,000 ($180,000)Construct small plant $100,000 ($20,000)Do nothing $0 $0
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 23
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Probabilites60%40%
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$162,000
Construct small plant$100,000($20,000)$88,000
Do nothing$0$0$0
Optimism-pessimism index (a)0.9MAX. W.A.=$162,000
DECISION :Construct large plant
&A
Page &P
Minimax regret
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
REGRET VALUESMaximum in row
$0$180,000$180,000
$100,000$20,000$100,000
$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Monetary Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketExpected
ValueConstruct large plant $200,000 ($180,000) $48,000Construct small plant $100,000 ($20,000) $52,000Do nothing $0 $0 $0PROBABILITIES 0.6 0.4
Expected (Monetary) ValueChoose the alternative with the maximum weighted row
averageEV(ai) = vij P(θj)∑
j
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 24
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$54,0800.1666666667
Construct small plant$100,000($20,000)$53,9200.6153846154
Do nothing$0$0$0
Optimism-pessimism index (a)0.616MAX. W.A.=$54,080
DECISION :Construct large plant
&A
Page &P
Minimax regret
$200,000($180,000)
$100,000($20,000)
$0$0
REGRET VALUESSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$0$180,000$180,000
Construct small plant$100,000$20,000$100,000
Do nothing$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
Sensitivity Analysis
EV (Large plant) = $200,000P – $180,000 (1 – P)
EV (Small plant) = $100,000P – $20,000(1 – P)
EV (Do nothing) = $0P + $0(1 – P)
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 25
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
Sensitivity Analysis
-200000-150000-100000
-500000
50000100000150000200000250000
0 0.2 0.4 0.6 0.8 1
P
EV
Point 1Small Plant
Large Plant Point 2
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 26
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
Expected Value of Perfect Information A consultant or a further analysis can aid the decision
maker by giving exact (perfect) information about the true state: the decision problem is no longer under risk; it will be under certainty.
Is it worthwhile for obtaining perfectly reliable information: is EVPI greater than the fee of the consultant (the cost of the analysis)?
EVPI is the maximum amount a decision maker would pay for additional information
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 27
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
Expected Value of Perfect InformationEVPI is the increase in EMV that results from having
perfect informationEVPI = EV with PI – Maximum EV under risk
EV with Perfect Information: 200*.6+0*.4=120Maximum EV under risk: 52 EVPI = 120 – 52 = 68
STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketExpected
ValueConstruct large plant $200,000 ($180,000) $48,000Construct small plant $100,000 ($20,000) $52,000Do nothing $0 $0 $0PROBABILITIES 0.6 0.4
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 28
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$54,0800.1666666667
Construct small plant$100,000($20,000)$53,9200.6153846154
Do nothing$0$0$0
Optimism-pessimism index (a)0.616MAX. W.A.=$54,080
DECISION :Construct large plant
&A
Page &P
Minimax regret
$200,000($180,000)
$100,000($20,000)
$0$0
REGRET VALUESSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$0$180,000$180,000
Construct small plant$100,000$20,000$100,000
Do nothing$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
Regret Values STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketConstruct large plant $0 $180,000 $72,000Construct small plant $100,000 $20,000 $68,000Do nothing $200,000 $0 $120,000PROBABILITIES 0.6 0.4
Expected Opportunity
Loss
Expected Opportunity LossChoose the alternative with the minimum weighted row
average of the regret matrixEOL(ai) = rij P(θj)∑
j
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 29
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$54,0800.1666666667
Construct small plant$100,000($20,000)$53,9200.6153846154
Do nothing$0$0$0
Optimism-pessimism index (a)0.616MAX. W.A.=$54,080
DECISION :Construct large plant
&A
Page &P
Minimax regret
$200,000($180,000)
$100,000($20,000)
$0$0
REGRET VALUESSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$0$180,000$180,000
Construct small plant$100,000$20,000$100,000
Do nothing$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Value
Construct large plant$200,000($180,000)$48,000
Construct small plant$100,000($20,000)$52,000
Do nothing$0$0$0
PROBABILITIES0.60.4
MAX. EMV =$52,000
DECISION :Construct small plant
&A
Page &P
EOL
$200,000($180,000)
$100,000($20,000)
$0$0
Regret ValuesSTATES OF NATUREExpected Opportunity Loss
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$0$180,000$72,000
Construct small plant$100,000$20,000$68,000
Do nothing$200,000$0$120,000
PROBABILITIES0.60.4
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
Additional Example
Alternative
State of NatureGood
Market(TL)
Average Market
(TL)
Poor Market
(TL)Large Coffee House 75,000 25,000 -40,000
Small Coffee House 100,000 35,000 -60,000
Do nothing 0 0 0Probability 0.25 0.50 0.25
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 30
PresenterPresentation NotesEMV: 21.25, 27.5*, 0EVPI: 42.5 – 27.5 = 15EOL: 21.25, 15*, 42.5
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
«Deal or no Deal» TV Game Show The game revolves around the opening of a set of
numbered briefcases (or boxes), each of which contains a different cash.
The contents (i.e., the values) of all of the cases are known at the start of the game, but the specific location of any prize is unknown.
The contestant claims (or is assigned) a case to begin the game.
The case's value is not revealed until the conclusion of the game.
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 31
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
The contestant then begins choosing cases that are to be removed from play.
The amount inside each chosen case is immediately revealed
Throughout the game, after a predetermined number of cases have been opened, the banker offers the contestant an amount of money to quit the game
The offer based roughly on the amounts remaining in play and the contestant's demeanor, the bank tries to 'buy' the contestant's case for a lower price than what's inside the case.
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 32
«Deal or no Deal» TV Game Show
PresenterPresentation NotesDemeanor -> attitude, behaviour
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
The player then answers "Deal or no deal" question, choosing: "Deal", accepting the offer and ending the game "No Deal", rejecting the offer and continuing
This process of removing cases and receiving offers continues, until either the player accepts an offer to 'deal', or all offers have been rejected and player wins the cash
inside her/his case. To lengthen games, the banker's offer is usually less
than the expected value, particularly early in the game
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 33
«Deal or no Deal» TV Game Show
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
Utility Theory Utility assessment may assign the worst payoff a utility
of 0 and the best payoff a utility of 1. A standard gamble is used to determine utility values:
When DM is indifferent between two alternatives, the utility values of them are equal.
Choose the alternative with the maximum expected utility
EU(ai) = u(ai) = u(vij) P(θj)∑j
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 34
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
Standard Gamble for Utility AssessmentBest payoff (v*)u(v*) = 1
Worst payoff (v–)u(v–) = 0
Certain payoff (v)u(v) = 1*p+0*(1–p)
(p)
(1–p)
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 35
http://www.ilkertopcu.info/mailto:kabak@itu.edu.tr
Utility Assessment (1st approach)v*u(v*) = 1
x1u(x1) = 0.5
x2u(x2) = 0.75
(0.5)
(0.5)
Best payoff (v*)u(v*) = 1
Worst payoff (v–)u(v–) = 0
Certain payoff (x1)u(x1) = 0.5
(0.5)
(0.5)
x1u(x1) = 0.5
v–u(v–) = 0
x3u(x3) = 0.25
(0.5)
(0.5)
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 200Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 36
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trSheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Getiri
Fayda E?risi
-180.0
5.0
100.0
175.0
200.0
0.0
0.25
0.5
0.75
1.0
Expected Utility (Example 1)v ij u(v ij ) 200 1175 0.75100 0.55 0.250
-20-180 0
v ij u(v ij ) 5 0.250 0.2432
-20 0.2162-180 0
Utilities STATES OF NATURE
ALTERNATIVESFavorable
marketUnfavorable
marketExpected
UtilityConstruct large plant 1 0 0.6Construct small plant 0.5 0.2162 0.3865Do nothing 0.2432 0.2432 0.2432PROBABILITIES 0.6 0.4
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 37
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trSheet1
fayda değerleriOLAYLAR
SEÇENEKLERYüksek TalepDüşük TalepBeklenen
Olasılıklar%60%40Fayda
Büyük fabrika kurma100.60200-180
Küçük fabrika kurma0.50.21620.3865100-20
Yatırım yapmama0.24320.24320.243200
viju(vij)
2001
1750.75
1000.5
50.25
0
-20
-1800
viju(vij)
50.25
00.24320.0270270270.027027
-200.21620.13513513510.135135
-1800
Sheet1
fayda değerleriOLAYLAR
SEÇENEKLERYüksek TalepDüşük TalepBeklenen
Olasılıklar%60%40Fayda
Büyük fabrika kurma100.60200-180
Küçük fabrika kurma0.50.21620.3865100-20
Yatırım yapmama0.24320.24320.243200
viju(vij)
2001
1750.75
1000.5
50.25
0
-20
-1800
viju(vij)
50.25
00.24320.0270270270.0272
-200.21620.13513513510.1352
-1800
Input
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$54,0800.1666666667
Construct small plant$100,000($20,000)$53,9200.6153846154
Do nothing$0$0$0
Optimism-pessimism index (a)0.616MAX. W.A.=$54,080
DECISION :Construct large plant
&A
Page &P
Minimax regret
$200,000($180,000)
$100,000($20,000)
$0$0
REGRET VALUESSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$0$180,000$180,000
Construct small plant$100,000$20,000$100,000
Do nothing$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
UtilitiesSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Utility
Construct large plant100.6
Construct small plant0.50.21620.3865
Do nothing0.24320.24320.2432
PROBABILITIES0.60.4
MAX. EMV =$0
DECISION :Construct large plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
Utility Assessment (2nd approach)
Best payoff (v*)u(v*) = 1
Worst payoff (v–)u(v–) = 0
Certain payoff (vij)u(vij) = p
(p)
(1–p)
In the example:u(-180) = 0 and u(200) = 1For vij=–20, p=%70 ⇒ u(–20) = 0.7For vij=0, p=%75 ⇒ u(0) = 0.75For 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
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 38
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trSheet: Sheet1
Sheet: Sheet2
Sheet: Sheet3
Getiri
Fayda E?risi
-180.0
-20.0
0.0
100.0
200.0
0.0
0.7
0.75
0.9
1.0
Expected Utility (Example 2)
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
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 39
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trInput
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
&A
Page &P
Maximin
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMinimum in row
Construct large plant$200,000($180,000)($180,000)
Construct small plant$100,000($20,000)($20,000)
Do nothing$0$0$0
MAXIMIN =$0
DECISION :Do nothing
&A
Page &P
Maximax
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$200,000($180,000)$200,000
Construct small plant$100,000($20,000)$100,000
Do nothing$0$0$0
MAXIMAX =$200,000
DECISION :Construct large plant
&A
Page &P
Hurwicz criterion
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketWEIGHTED AVERAGE
Construct large plant$200,000($180,000)$54,0800.1666666667
Construct small plant$100,000($20,000)$53,9200.6153846154
Do nothing$0$0$0
Optimism-pessimism index (a)0.616MAX. W.A.=$54,080
DECISION :Construct large plant
&A
Page &P
Minimax regret
$200,000($180,000)
$100,000($20,000)
$0$0
REGRET VALUESSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketMaximum in row
Construct large plant$0$180,000$180,000
Construct small plant$100,000$20,000$100,000
Do nothing$200,000$0$200,000
MINIMAX =$100,000
DECISION :Construct small plant
&A
Page &P
Laplace's
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketRow average
Construct large plant$200,000($180,000)$10,000
Construct small plant$100,000($20,000)$40,000
Do nothing$0$0$0
MAX. R.A. =$40,000
DECISION :Construct small plant
&A
Page &P
EMV
UtilitiesSTATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable marketExpected Utility
Construct large plant100.6
Construct small plant0.90.70.82
Do nothing0.750.750.75
PROBABILITIES0.60.4
MAX. EMV =$0
DECISION :Construct large plant
&A
Page &P
EOL
STATES OF NATURE
ALTERNATIVESFavorable marketUnfavorable market
Construct large plant$200,000($180,000)
Construct small plant$100,000($20,000)
Do nothing$0$0
PROBABILITIES0.60.4
REGRET VALUESExpected Opportunity Loss
$0$180,000$72,000
$100,000$20,000$68,000
$200,000$0$120,000
MIN. EOL =$68,000
DECISION :Construct small plant
&A
Page &P
Preferences for Risk
Monetary outcome
Util
ity
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Certainty Equivalence If a DM is indifferent between accepting a lottery
ticket and accepting a sum of certain money, the monetary sum is the Certainty Equivalent (CE) of the lottery
Z is CE of the lottery (Y>Z>X).
X
Y
Z
p
1–p
Dr. Y. İlker Topcu (www.ilkertopcu.n ) & Dr. Özgür Kabak (kabak@itu.edu.tr)et 41
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Risk Premium Risk Premium (RP) of a lottery is the difference between
the EV of the lottery and the CE of the lottery If the DM is risk averse (avoids risk), RP > 0
S/he prefers to receive a sum of money equal to expected value of a lottery than to enter the lottery itself
If the DM is risk prone (seeks risk), RP < 0S/he prefers to enter a lottery than to receive a sum of money equal to its expected value
If the DM is risk neutral, RP = 0S/he is indifferent between entering any lottery and receiving a sum of money equal to its expected value
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Decision Analysis�(Decision Tables, Utility)IntroductionComponents of Decision AnalysisBasic Steps in Decision Analysis�Decision-Making EnvironmentsDecision Making Under CertaintyDecision Making Under UncertaintyDecision Table / Payoff MatrixMaximaxMaximinCriterion of RealismSlide Number 12Criterion of RealismEqually LikelihoodMinimaxResults of the ExampleAdditional ExampleSolutions with QM for WindowsDecision Making Under RiskProbabilityObjective ProbabilitySubjective ProbabilityPayoff Matrix with ProbabilitiesExpected (Monetary) ValueSensitivity AnalysisSensitivity AnalysisExpected Value of Perfect InformationExpected Value of Perfect InformationExpected Opportunity LossAdditional Example«Deal or no Deal» TV Game Show«Deal or no Deal» TV Game Show«Deal or no Deal» TV Game ShowUtility TheoryStandard Gamble for Utility AssessmentUtility Assessment (1st approach)Expected Utility (Example 1)Utility Assessment (2nd approach)Expected Utility (Example 2)Preferences for RiskCertainty EquivalenceRisk Premium