King Saud University
College of Computer & Information Sciences
IS 466 Decision Support Systems
Lecture 3
Decision Analysis
Dr. Mourad YKHLEF
The slides content is derived and adopted from many references
Outline
•
•
•
•
•
•
•
•
Introduction to Decision Analysis
Decision making under Uncertainty
Decision making under Risk
Decision making with Perfect Information
Decision Making with Imperfect Information
Decision Tree
Decision Making and Utility
Decision making in light of competitive actions
(Game theory)
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Introduction to Decision Analysis
• Decision Analysis provides a framework for
making important decisions.
• Decision Analysis allows us to select a decision
from a set of possible decision alternatives when
uncertainties regarding the future exist.
• The goal is to optimize the resulting return
(payoff) in terms of a decision criterion.
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Decision Problem
• Decision tables
• Decision trees
Outcomes
States of Nature
(Event)
Alternatives
Decision Problem
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Decision table
Terms
• Alternative: choice
• State of nature (Event): an occurrence over which the
decision maker has no control
Symbols used in decision tree
A decision node from which one of several
alternatives may be selected
A state of nature node out of which one state of nature
will occur
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Decision Table
States of Nature
Alternatives
State 1
State 2
Alternative 1
Outcome 1
Outcome 2
Alternative 2
Outcome 3
Outcome 4
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Decision Tree
State 1
1
Outcome 1
State 2
Outcome 2
State 1
2
State 2
Outcome 3
Outcome 4
Decision
Node
State of Nature Node
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Products Decision Tree
Favorable market
A state of nature node
1
Unfavorable market
Favorable market
A decision node
Construct
small plant 2
Unfavorable market
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Decision Models
• The types of decision models:
– Decision making under certainty
• The future state of nature is assumed known.
– Decision making under uncertainty (no probabilities)
• There is no knowledge about the probability of the states of nature
occurring.
– Decision making under risk (with probabilities)
• There is (some) knowledge of the probability of the states of nature
occurring.
– Decision making with perfect information
• The future state of nature is assumed known with certain probability.
– Decision making with imperfect information (Bayesian Theory)
– Decision making in light of competitive actions (Game theory)
• All the actors (players) are seeking to maximize their return.
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Decision Models
• Under uncertainty we can not compute the
expected value and decisions are made based on
other criteria.
• One method for making decisions under risk is to
select the decision with the best expected value.
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Decision Making Under Uncertainty
• Maximax - Choose the alternative that
maximizes the maximum outcome for every
alternative (Optimistic criterion)
• Maximin - Choose the alternative that
maximizes the minimum outcome for every
alternative (Pessimistic criterion)
• Equally likely - chose the alternative with the
highest average outcome.
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Example - Decision Making Under Uncertainty
States of Nature
Alternatives Favorable Unfavorable Maximum
Construct
large plant
Construct
small plant
Do nothing
Minimum
Row
in Row Average
-180,000 10,000
Market
200,000
Market
-180,000
in Row
200,000
100,000
-20,000
100,000
-20,000
40,000
0
0
0
0
0
Maximax
Maximin
IS 466 – Decision Analysis - Dr. Mourad Ykhlef
Equally
likely
12
Outline
•
•
•
•
•
•
•
•
Introduction to Decision Analysis
Decision making under Uncertainty
Decision making under Risk
Decision making with Perfect Information
Decision Making with Imperfect Information
Decision Tree
Decision Making and Utility
Decision making in light of competitive actions
(Game theory)
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Decision under uncertainty
• Payoff Table (Decision Table)
-Payoff ‫العائد‬-
– Payoff table analysis can be applied when:
• There is a finite set of discrete decision alternatives.
• The outcome of a decision is a function of a single future event.
– In a Payoff table • The rows correspond to the possible decision alternatives.
• The columns correspond to the possible future events.
• States of nature (events) are mutually exclusive and
collectively exhaustive.
• The table entries are the payoffs.
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Example (1/3)
• Ahmed has inherited 1000
• He has to decide how to invest the money for one year.
• A broker has suggested five potential investments.
–
–
–
–
–
Gold
Company A
Company B
Company C
Company D
• The return on each investment depends on the (uncertain)
market behavior during the year.
• Ahmed would build a payoff table to help make the
investment decision
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Example (2/3)
Decision
States of Nature
Alternatives Large Rise Small Rise No Change Small Fall Large Fall
Gold
-100
100
200
300
0
Company A
250
200
150
-100
-150
Company B
500
250
100
-200
-600
Company C
60
60
60
60
60
Company D
200
150
150
-200
-150
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Example (3/3)
Decision
States of Nature
Alternatives Large Rise Small Rise No Change Small Fall Large Fall
Gold
-100
100
200
300
0
-100
200
150
Company A
250
-150
Company B
500
250
100
-200
-600
Company C
60
60
60
60
60
Company D
200
150
150
-200
-150
• The Company D alternative is dominated by the Company A alternative
• The Company D alternative can be dropped
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Decision making Criteria
• The decision criteria are based on the decision
maker’s attitude toward life.
• The criteria include the
– Maximin Criterion - pessimistic or conservative
approach.
– Minimax Regret Criterion - pessimistic or conservative
approach.
– Maximax Criterion - optimistic or aggressive approach.
– Principle of Insufficient Reasoning – no information
about the likelihood of the various states of nature.
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A- The Maximin Criterion
• This criterion is based on the worst-case scenario.
– It fits both a pessimistic and a conservative decision
maker’s styles.
– A pessimistic decision maker believes that the worst
possible result will always occur.
– A conservative decision maker wishes to ensure a
guaranteed minimum possible payoff.
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A- The Maximin Criterion
• To find an optimal decision
– Record the minimum payoff across all states of nature
for each decision.
– Identify the decision with the maximum payoff.
The Maximin Criterion
Decisions
Gold
Company A
Company B
Company C
Large Rise Small rise
-100
250
500
60
100
200
250
60
Minimum
No Change Small Fall
200
150
100
60
300
-100
-200
60
Large Fall Payoff
0
-150
-600
60
-100
-150
-600
60
Optimal Decision
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B- The Minimax Regret Criterion
• The Minimax Regret Criterion
– This criterion fits both a pessimistic and a conservative decision
maker approach.
– The payoff table is based on “lost opportunity,” or “regret.”
– The decision maker incurs regret by failing to choose the “best”
decision.
• To find an optimal decision
– For each state of nature:
• Determine the best payoff over all decisions.
• Calculate the regret for each decision alternative as the
difference between its payoff value and this best payoff value.
– For each decision find the maximum regret over all states of
nature.
– Select the decision alternative that has the minimum of these
“maximum regrets.”
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B- The Minimax Regret Criterion
The Payoff Table
Decision
Gold
Company A
Company B
Company C
Decision
Gold
Company A
Company B
Company C
Large rise Small rise No change Small fall Large fall
-100
250
500
60
100
200
250
60
200
150
100
60
300
-100
-200
60
0
-150
-600
60
The Regret Table
Large rise Small rise No change Small fall Large fall
600
250
0
440
150
0
0
Let50
us build the50Regret Table
400
0
100
500
190
140
240
IS 466 – Decision Analysis - Dr. Mourad Ykhlef
60
210
660
0
22
B- The Minimax Regret Criterion
The Payoff Table
Decision
Gold
Company A
Company B
Company C
Large rise Small rise No change Small fall Large fall
-100
250
500
60
100
200
300 a regret
0
Investing in
gold generates
200of 600 when
150 the market
-100 exhibits
-150
250
100
-200
-600
a large rise
60
60
60
60
500 – (-100) = 600
The Regret Table
Maximum
Decision Large rise Small rise No change Small fall Large fall Regret
Gold
Company A
Company B
Company C
600
250
0
440
150
50
0
190
0
50
100
140
0
400
500
240
60
210
660
0
IS 466 – Decision Analysis - Dr. Mourad Ykhlef
600
400
660
440
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C- The Maximax Criterion
• This criterion is based on the best possible scenario.
It fits both an optimistic and an aggressive decision maker.
• An optimistic decision maker believes that the best
possible outcome will always take place regardless of the
decision made.
• An aggressive decision maker looks for the decision with
the highest payoff (when payoff is profit).
• To find an optimal decision.
– Find the maximum payoff for each decision alternative.
– Select the decision alternative that has the maximum of the
“maximum” payoff.
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C- The Maximax Criterion
Decision
Gold
Company A
Company B
Company C
The Maximax Criterion
Maximum
Large rise Small rise No change Small fall Large fall Payoff
-100
250
500
60
100
200
250
60
200
150
100
60
300
-100
-200
60
0
-150
-600
60
300
250
500
60
Optimal Decision
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D- Insufficient Reason
• This criterion might appeal to a decision maker
who is neither pessimistic nor optimistic.
– It assumes all the states of nature are equally likely to
occur.
– The procedure to find an optimal decision.
• For each decision add all the payoffs.
• Select the decision with the largest sum (for profits).
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D- Insufficient Reason
• Sum of Payoffs
–
–
–
–
Gold
Company A
Company B
Company C
500
350
50
300
• Based on this criterion the optimal decision
alternative is to invest in gold.
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D- Insufficient Reason
Payoff Table
Gold
Company A
Company B
Company C
Large Rise
-100
250
500
60
RESULTS
Criteria
Decision
Maximin
Company C
Minimax Regret
Company A
Maximax
Company B
Insufficient Reason
Gold
Small Rise No Change Small Fall Large Fall
100
200
300
0
200
150
-100
-150
250
100
-200
-600
60
60
60
60
Payoff
60
400
500
100
• Decision under uncertainty can present a problem if the attitude toward life
(optimistic, pessimistic, or somewhere in between) change rapidly
• Solution: obtain probability estimates for the states of nature and implement
decision making under risk.
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Outline
•
•
•
•
•
•
•
•
Introduction to Decision Analysis
Decision making under Uncertainty
Decision making under Risk
Decision making with Perfect Information
Decision Making with Imperfect Information
Decision Tree
Decision Making and Utility
Decision making in light of competitive actions
(Game theory)
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Decision making under Risk
• Probabilistic decision situation
• States of nature have probabilities of occurrence
• The probability estimate for the occurrence of
each state of nature (if available) can be
incorporated in the search for the optimal
decision.
• For each decision calculate its expected payoff.
Expected Payoff = Σ(Probability)(Payoff)
• Select the decision with the best expected payoff
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The Expected Value Criterion
Optimal Decision
Decision
Gold
Company A
Company B
Company C
Prior Prob.
Expected
The Expected Value Criterion
Value
Large rise Small rise No change Small fall Large fall
-100
250
500
60
0.2
100
200
250
60
0.3
200
150
100
60
0.3
300
-100
-200
60
0.1
0
-150
-600
60
0.1
100
130
125
60
EV(Company A) = (0.2)(250) + (0.3)(200) + (0.3)(150) + (0.1)(-100) + (0.1)(-150) = 130
Expected Return of The EV (EREV) = max (100, 130, 125, 60) = 130
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The expected value Criterion
• The expected value criterion is useful generally in
the case where the decision maker is risk neutral.
• This criterion does not take into account the
decision maker’s attitude toward possible losses.
We will see that utility theory offers an alternative
to the expected value approach.
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Outline
•
•
•
•
•
•
•
•
Introduction to Decision Analysis
Decision making under Uncertainty
Decision making under Risk
Decision making with Perfect Information
Decision Making with Imperfect Information
Decision Tree
Decision Making and Utility
Decision making in light of competitive actions
(Game theory)
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Expected Value of Perfect Information
• The gain in expected return obtained from
knowing with certainty the future state of nature
is called: Expected Value of Perfect Information
(EVPI)
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Expected Value of Perfect Information
The Expected Value of Perfect Information
Decision
Large rise
Small rise
-100
250
500
60
0.2
Gold
Company A
Company B
Company C
Prob.
No change
100
200
250
60
0.3
Small fall
200
150
100
60
0.3
Large fall
300
-100
-200
60
0.1
0
-150
-600
60
0.1
• If we know with certainty that the market were going to “Large Rise” (resp.
small fall) the optimal decision would be to invest in Company B (Gold).
• Expected Return with Perfect information ERPI
= (Probability of 1st state of nature )*(best outcome of 1st state of nature )
+…+ (Probability of 5th state of nature )*(best outcome of 5th state of nature )
= 0.2(500)+0.3(250)+0.3(200)+0.1(300)+0.1(60) = 271
EREV Expected Return of the EV criterion = 130
EVPI = ERPI - EREV = 271 - 130 = 141
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Expected Value of Perfect Information
If Ahmed knew in advance
the Market would undergo
His optimal decision
With a gain of Payoff
(with respect to risk
case) of
a large rise
Company B
500-250=250
a small rise
Company B
250-200=50
no c hange
Gold
200-150=50
a small fall
Gold
300-(-100)=400
a large fall
Company C
60-(-150)=210
EVPI =.2(250)+.3(50)+.3(50)+.1(400)+.1(210) = 141
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Outline
•
•
•
•
•
•
•
•
Introduction to Decision Analysis
Decision making under Uncertainty
Decision making under Risk
Decision making with Perfect Information
Decision Making with Imperfect Information
Decision Tree
Decision Making and Utility
Decision making in light of competitive actions
(Game theory)
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Decision Making with Imperfect Information
Bayesian Analysis
• Some statisticians argue that is unnecessary to
practice decision making under uncertainty coz
one always has at least some probabilistic info
related to the states of nature.
• Bayesian Statistics play a role in assessing
additional information obtained from various
sources.
• This additional information may assist in refining
original probability estimates, and help improve
decision making.
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Example (1/6)
• Ahmed can purchase econometric forecast results for 50 SR.
• The forecast predicts “negative” or “positive” econometric
growth.
• Statistics regarding the forecast are:
The Forecast
predicted
Positive econ. growth
Negative econ. growth
When the Company showed a...
Large Rise
Small Rise
No Change
Small Fall
Large Fall
80%
20%
70%
30%
50%
50%
40%
60%
0%
100%
When the Company B showed a large rise the
Forecast predicted a “positive growth” 80% of the time.
P(forecast predicts “positive ” | small rise in market) = .7
P(forecast predicts “negative” | small rise in market) = .3
• Should Ahmed purchase the forecast?
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Example (2/6)
• To find Expected payoff with forecast Ahmed should
determine what to do when:
– The forecast is “positive growth”,
– The forecast is “negative growth”.
• Ahmed needs to know the following probabilities
–
–
–
–
–
P(Large rise | The forecast predicted “Positive”)
P(Small rise | The forecast predicted “Positive”)
P(No change | The forecast predicted “Positive ”)
P(Small fall | The forecast predicted “Positive”)
P(Large Fall | The forecast predicted “Positive”)
–
–
–
–
–
P(Large rise | The forecast predicted “Negative ”)
P(Small rise | The forecast predicted “Negative”)
P(No change | The forecast predicted “Negative”)
P(Small fall | The forecast predicted “Negative”)
P(Large Fall) | The forecast predicted “Negative”)
• Bayes’ Theorem provides a procedure to calculate these probabilities
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Bayes’ Theorem (1/4)
P( A | B) = P( B | A)P( A)
P( B)
Proof : P(A|B)=P(A and B)/P(B)
(1)
P(B|A)=P(A and B)/P(A)
P(A and B) =P(B|A)*P(A)
(1) P(A|B)=P(B|A)*P(A) / P(B)
Bayes, Thomas (1763) An essay towards solving a problem in
the doctrine of chances. Philosophical Transactions of the
Royal Society of London, 53:370-418
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Bayes’ Theorem (2/4)
• Often we begin probability analysis with initial or prior
probabilities.
• Then, from a sample, special report, or a product test we
obtain some additional information.
• Given this information, we calculate revised or posterior
probabilities.
• Bayes’ theorem provides the means for revising the prior
probabilities.
Prior
Probabilities
New
Information
Application
of Bayes’
Theorem
IS 466 – Decision Analysis - Dr. Mourad Ykhlef
Posterior
Probabilities
42
Bayes’ Theorem (3/4)
• Knowledge of sample (survey) information can be used
to revise the probability estimates for the states of nature.
• Prior to obtaining this information, the probability
estimates for the states of nature are called prior
probabilities.
• With knowledge of conditional probabilities for the
indicators of the sample or survey information, these prior
probabilities can be revised by employing Bayes' Theorem.
• The outcomes of this analysis are called posterior
probabilities or branch probabilities for decision trees.
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Bayes’ Theorem (4/4)
• A1, A2, …, An are mutually exclusive and collectively
exhaustive (e.g., events)
P(Ai|B) =
P(B|Ai)P(Ai)
P(B|A1)P(A1)+ P(B|A2)P(A2)+…+ P(B|An)P(An)
Posterior Probabilities
Probabilities determined
after the additional info
becomes available.
Prior probabilities
Probability estimates
determined based on
current info, before the
new info becomes available.
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Example (3/6)
• The tabular approach to calculating posterior
probabilities for “positive” economical forecast
Prior
Prob.
States of
Nature
Large Rise
Small Rise
No Change
Small Fall
Large Fall
0.2
0.3
0.3
0.1
0.1
Prob.
(Positive|State)
X
0.8
0.7
0.5
0.4
0
Joint
Prob.
= 0.16
0.21
0.15
0.04
0
Ai:Large Rise
B: forecast positive
P(B|Ai) P(Ai)
P(forecast = positive | large rise) P(large rise)
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Example (4/6)
• The tabular approach to calculating posterior
probabilities for “positive” economical forecast
States of
Nature
Large Rise
Small Rise
No Change
Small Fall
Large Fall
Case 1: Forecast predicts "Positive"
Prior
Prob.
Joint
Posterior
Prob.
(Positive|State)
Prob.
Prob.
0.2
0.3
0.3
0.1
0.1
X
0.8
=
0.7
0.5
0.4
0
Sum =
0.16
0.21
0.15
0.04
0
0.56
0.286
0.375
0.268
0.071
0.000
0.16
0.56
Revision of
state of nature
probability
P(large rise | Forecast = positive)
Probability(Forecast = positive) = .16 + .21 + .15 + .04 +.0 = .56
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Example (5/6)
• The tabular approach to calculating posterior
probabilities for “negative” economical forecast
States of
Nature
Large Rise
Small Rise
No Change
Small Fall
Large Fall
Case 2: Forecast predicts "Negative"
Prior
Prob.
Joint
Posterior
Prob.
(negative|State) Probab.
Probab.
0.2
0.3
0.3
0.1
0.1
0.2
0.3
0.5
0.6
1
Sum =
0.04
0.09
0.15
0.06
0.1
0.44
0.091
0.205
0.341
0.136
0.227
Probability(Forecast = negative) = .44
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Example (6/6)
Bayesian Analysis
Indicator 1
Indicator 2
States
Prior
Conditional
Joint
Posterior
States
Prior
Conditional
Joint
Posterior
of Nature Probabilities Probabilities Probabilities Probabilites of Nature Probabilities Probabilities Probabilities Probabilites
Large Rise
0.2
0.8
0.16
0.286
Large Rise
0.2
0.2
0.04
0.091
Small Rise
0.3
0.7
0.21
0.375
Small Rise
0.3
0.3
0.09
0.205
No Change
0.3
0.5
0.15
0.268
No Change
0.3
0.5
0.15
0.341
Small Fall
0.1
0.4
0.04
0.071
Small Fall
0.1
0.6
0.06
0.136
Large Fall
0.1
0
0
0.000
Large Fall
0.1
1
0.1
0.227
s6
0
0
0.000
s6
0
0
0.000
s7
0
0
0.000
s7
0
0
0.000
s8
0
0
0.000
s8
0
0
0.000
P(Indicator 1)
0.56
P(Indicator 2)
0.44
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Expected Value of Sample Information EVSI
• This is the expected gain from making decisions
based on Sample Information.
• Revise the expected return for each decision using
the posterior probabilities as follows:
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Conditional Expected Values
The revised probabilities payoff table
Decision
Gold
Company A
Company B
Company C
P(State|Positive)
P(State|Negative)
Revision of
state of nature
probability
Large rise Small rise No change Small fall Large fall
-100
250
500
60
0.286
0.091
100
200
250
60
0.375
0.205
200
150
100
60
0.268
0.341
300
-100
-200
60
0.071
0.136
0
-150
-600
60
0
0.227
EV(Invest in…….
GOLD|“Positive” forecast) =
=.286(-100)+.375(100 )+.268( 200)+.071( 300)+0( 0 ) = 84
EV(Invest in …….
GOLD | “Negative” forecast) =
=.091(-100 )+.205( 100 )+.341( 200 )+.136( 300 )+.227( 0 ) = 120
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Conditional Expected Values
• The revised expected values for each decision:
Positive forecast
Negative forecast
EV(Gold|Positive) = 84
EV(C A|Positive) = 180
EV(C B|Positive) = 250
EV(C C|Positive) = 60
EV(Gold|Negative) = 120
EV(C A|Negative) = 65
EV(C B|Negative) = -37
EV(C C|Negative) = 60
If the forecast is “Positive”
Invest in Company B.
If the forecast is “Negative”
Invest in Gold.
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Conditional Expected Values
• Ahmed can now calculate the expected return
from purchasing the forecast (ERSI).
• ERSI Expected Return of Sample Information
• ERSI = P(Forecast is positive) * EV(Company B|Forecast is positive)
+ P(Forecast is negative”) * EV(Gold|Forecast is negative)
= (.56)(250) + (.44)(120) = 192.5
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Expected Value of Sampling Information EVSI
• The expected gain from buying the forecast is:
EVSI = ERSI – EREV = 192.5 – 130 = 62.5
• Ahmed should purchase the forecast., his
expected gain is greater than the forecast cost
(50 SR < 62.5 SR).
• Efficiency = EVSI / EVPI = 63 / 141 = 0.45
– Efficiency is between 0 and 1
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Example (Summary)
Payoff Table
Gold
Company A
Company B
Company C
d5
d6
d7
d8
Prior Prob.
Ind. 1 Prob.
Ind 2. Prob.
Ind. 3 Prob.
Ind 4 Prob.
Large Rise Small Rise No Change Small Fall Large Fall
-100
100
200
300
0
250
200
150
-100
-150
500
250
100
-200
-600
60
60
60
60
60
0.2
0.286
0.091
0.3
0.375
0.205
0.3
0.268
0.341
0.1
0.071
0.136
0.1
0.000
0.227
Ind. 2
120.45
Gold
Ind. 3
0.00
Ind. 4
0.00
s6
s7
s8 EV(prior) EV(ind. 1) EV(ind. 2)
100
83.93
120.45
130
179.46
67.05
125
249.11
-32.95
60
60.00
60.00
#### ### ##
#### ### ##
0.56
0.44
RESULTS
Prior
Ind. 1
130.00
249.11
optimal payoff
optimal decision Company A Company B
EVSI =
EVPI =
Efficiency=
62.5
141
0.44
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Summary of some laws
• EVPI = ERPI – EREV
• EVSI = ERSI – EREV
• Efficiency = EVSI / EVPI
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Outline
•
•
•
•
•
•
•
•
Introduction to Decision Analysis
Decision making under Uncertainty
Decision making under Risk
Decision making with Perfect Information
Decision Making with Imperfect Information
Decision Tree
Decision Making and Utility
Decision making in light of competitive actions
(Game theory)
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Decision Tree
• The Payoff Table (Decision Table) approach is
useful for a non-sequential or single stage.
• Decision Tree is useful in analyzing multi-stage
decision problems consisting of a sequence of
dependent decisions.
• A Decision Tree is a chronological representation
of the decision process.
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Decision Tree
• The tree is composed of nodes and branches.
Chance
node
Decision
node
P(S2)
P(S2)
A branch emanating from a
decision node corresponds to a
decision alternative. It includes a
cost or benefit value.
A branch emanating from a
state of nature (chance) node
corresponds to a particular state
of nature, and includes the
probability of this state of
nature.
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Example
•
•
Company A (Co. A) plans to do a commercial development on a property.
Relevant data
–
–
–
–
Asking price for the property is 300,000.
Construction cost is 500,000.
Selling price is approximated at 950,000.
Variance application costs 30,000 in fees and expenses
• There is only 40% chance that the variance will be approved.
• If Co. A purchases the property and the variance is denied, the property can be
sold for a net return of 260,000.
• A three month option on the property costs 20,000, which will allow Co. A to
apply for the variance.
– A consultant can be hired for 5000.
– The consultant will provide an opinion about the approval of the variance
application
• P (Consultant predicts approval | approval granted) = 0.70
• P (Consultant predicts denial | approval denied) = 0.80
• Co. A wishes to determine the optimal strategy
– Hire/not hire the consultant now,
– Other decisions that follow sequentially.
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Bayes’ calculation
• P(variance approved) = .4
• P(variance denied) = .6
• P(consultant predicts approved | variance approved) = .7
• P(consultant predicts denial
| variance approved) = 1 - .7 = .3
• P(consultant predicts approved | variance denied) = 1 - .8 = .2
• P(consultant predicts denial
| variance denied) =.8
• Wants to compute
P(variance approved | consultant predicts approved)
P(variance denied | consultant predicts approved)
P(variance approved | consultant predicts denial)
P(variance denied | consultant predicts denial)
P(consultant predicts approved)
P(consultant predicts denial)
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Bayes’ calculation
States of
Nature
variance approved
variance denied
States of
Nature
variance approved
variance denied
Consultant predicts variance approval
Prior
Prob.
Joint
Posterior
Prob.
(positive|State) Probab.
Probab.
0.4
0.6
0.7
0.2
Sum =
0.28
0.12
0.4
.28/.4 = .70
.12/.4 =.30
Consultant predicts variance denial
Prior
Prob.
Joint
Posterior
Prob. (negative|State) Probab.
Probab.
0.4
0.6
0.3
0.8
Sum =
0.12
0.48
0.6
IS 466 – Decision Analysis - Dr. Mourad Ykhlef
.12/.6 = .20
.48/.6 =.80
62
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The case “Do not hire consultant”
• The expected value associated with buying land & apply for variance
6000 = .4(120 000) + .6(-70 000)
• The expected value associated with buying option & apply for
variance
10 000 = .4(100 000) + .6(-50 000)
• Thus if company A decides not to hire consultant the
corresponding expected values that results if it does
nothing, buy the land or purchases the option are 0, 6000,
10 000 respectively
– Purchasing the option is the optimal decision
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Outline
•
•
•
•
•
•
•
•
Introduction to Decision Analysis
Decision making under Uncertainty
Decision making under Risk
Decision making with Perfect Information
Decision Making with Imperfect Information
Decision Tree
Decision Making and Utility
Decision making in light of competitive actions
(Game theory)
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Decision Making and Utility
Decision
No accident
Buy insurance
Do not buy insurance
Prob.
Accident
Expected Value
500
500
500
0
10000
80
0.992
0.008
• Decision should be do not purchase insurance, but people
almost always do purchase insurance.
• The expected value criterion may not be appropriate if the
decision is a one-time opportunity with substantial risks.
• Decision makers do not always choose decisions based on
the expected value criterion.
– Insurance policies cost more than the present value of the expected
loss the insurance company pays to cover insured losses.
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Decision Making and Utility
• In Utility approach, it is assumed that a decision
maker can rank decisions in a coherent manner.
• Utility values, U(V), reflect the decision maker’s
perspective and attitude toward risk. Instead of
maximizing expected value, the decision maker
should maximize expected utility.
• Each payoff is assigned a utility value.
– Higher payoffs get larger utility value.
• The optimal decision is the one that maximizes
the expected utility.
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Determining the Utility Values of Payoffs
• List every possible payoff in the payoff table in
ascending order.
• Assign a utility of 0 to the lowest value and a
value of 1 to the highest value.
• For all other possible payoffs (Rij) ask the decision
maker the following question:
– Suppose you are given the option to select one of the
following two alternatives:
• Receive Rij (one of the payoff values) for sure,
• Play a game of chance where you receive either
– The highest payoff of Rmax with probability p, or
– The lowest payoff of Rmin with probability 1- p.
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Determining the Utility Values of Payoffs
p
1-p
Rij
Rmax
Rmin
• What value of p would make you indifferent between the two
situations?
• The answer to this question is the indifference probability for
the payoff Rij and is used as the utility values of Rij.
U(Rij) = p (simple version)
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Determining the Utility Values of Payoffs
d1
d2
Alternative 1
A sure event
100
s1
s2
150
100
-50
140
•
For p = 1.0, you
Alternative 2
prefer alternative 2.
(Game-of-chance)
• For p = 0.0, you’ll
prefer alternative 1.
• Thus, for some p
between 0.0 and 1.0
150
you’ll be indifferent
between the alternatives.
1-p
p
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-50
70
Determining the Utility Values of Payoffs
d1
d2
s1
s2
150
100
-50
140
Alternative 1 • Let’s assume the
A sure event probability of
indifference is p = .7
100
U(100)=.7U(150)+.3U(-50)
= .7(1) + .3(0) = .7
Alternative 2
(Game-of-chance)
150
1-p
p
-50
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Example 1 (1/2)
•
Data
– The highest payoff was 500. Lowest payoff was –600, which
Ahmed would incur if he he invested in Company B and there
were a large fall in the market.
– The second highest payoff was 300. The case of Gold and a large
fall in the market
– We ask Ahmed: “if you could receive 300 for sure or you could
receive 500 with probability p and lose 600 with probability 1-p,
what value of p would make you indifference between these two
choices?”
– The indifference probabilities provided by Ahmed are
Payoff
Prob.
-600 -200 -150 -100
0
0.25
0.3
0.36
0
60
100
150
200
250
300
500
0.5
0.6
0.65
0.7
0.75
0.85
0.9
1
– Ahmed wishes to determine his optimal investment decision.
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Example 1 (2/2)
Utility Analysis
Gold
Company A
Company B
Company C
d5
d6
d7
d8
Probability
RESULTS
Criteria
Exp. Utility
Large Rise Small Rise No Change Small Fall Large Fall
0.36
0.65
0.75
0.9
0.5
0.85
0.75
0.7
0.36
0.3
1
0.85
0.65
0.25
0
0.6
0.6
0.6
0.6
0.6
0.2
0.3
Decision
Company B
Value
0.675
0.3
0.1
0.1
EU
0.632
0.671
0.675
0.6
0
0
0
0
Certain Payoff
-600
-200
-150
-100
0
60
100
150
200
250
300
500
Utility
0
0.25
0.3
0.36
0.5
0.6
0.65
0.7
0.75
0.85
0.9
1
EU(Gold)=0.36x0.2 + 0.65x0.3 + 0.75x0.3 + 0,9x0.1 + 0.5x0.1 = 0.632
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Three types of Decision Makers
• Risk Averse -Prefers a certain outcome (payoff) to
a chance outcome (utility) having the same
expected value.
• Risk Taking - Prefers a chance outcome (utility) to
a certain outcome (payoff) having the same
expected value.
• Risk Neutral - Is indifferent between a chance
outcome and a certain outcome having the same
expected value.
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Three Types of Utility Curves
Payoff
Risk Averter (Concave
curve): Utility rises
slower than payoff
Payoff
Risk Seeker
(Convex curve):
Utility rises faster
than payoff
Payoff
Risk-Neutral:
Maximizes
Expected payoff
and ignores risk
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Example 2 (1/10)
Consider the following three-state, four-decision problem
with the following payoff table in Riyals:
d1
d2
d3
d4
s1
+100,000
+50,000
+20,000
+40,000
s2
+40,000
+20,000
+20,000
+20,000
s3
-60,000
-30,000
-10,000
-60,000
The probabilities for the three states of nature are:
P(s1) = .1, P(s2) = .3, and P(s3) = .6.
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Example 2 (2/10)
• Risk-Neutral Decision Maker
If the decision maker is risk neutral the expected value
approach is applicable.
EV(d1) = .1(100,000) + .3(40,000) + .6(-60,000) = -14,000
EV(d2) = .1( 50,000) + .3(20,000) + .6(-30,000) = - 7,000
EV(d3) = .1( 20,000) + .3(20,000) + .6(-10,000) = + 2,000
Note the EV for d4 need not be calculated as decision d4 is
dominated by decision d2.
The optimal decision is d3.
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Example 2 (3/10)
• Decision Makers with Different Utilities
Suppose two decision makers have the following
utility values:
Utility
Utility
Amount Decision Maker I Decision Maker II
100,000
100
100
50,000
94
58
40,000
90
50
20,000
80
35
-10,000
60
18
-30,000
40
10
-60,000
0
0
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Example 2 (4/10)
• Graph of the Two Decision Makers’ Utility Curves
Utility
100
Decision Maker I (Risk avoider)
80
60
40
Decision Maker II (Risk taker)
20
-60
-40
-20
0
20
40
60
Monetary Value (in 1000
1000’s)
’s)
80
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100
79
Example 2 (5/10)
• Expected Utility: Decision Maker I
Optimal
decision
is d3
s1
d1
100
d2
94
d3
80
Probability .1
s2
90
80
80
.3
Expected
s3 Utility
0
37.0
40
57.4
Largest
60
68.0
expected
.6
utility
Note: d4 is dominated by d2 and hence is not considered
Decision Maker I should make decision d3.
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Example 2 (6/10)
• Expected Utility: Decision Maker II
Optimal
decision
is d1
s1
d1
100
d2
58
d3
35
Probability .1
s2
50
35
35
.3
Expected
s3 Utility
0
25.0
Largest
10
22.3
expected
18
24.8
utility
.6
Note: d4 is dominated by d2 and hence is not considered
Decision Maker II should make decision d1.
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Example 2 (7/10)
• Suppose the probabilities for the three states of
nature in this example were changed to:
P(s1) = .5, P(s2) = .3, and P(s3) = .2.
– What is the optimal decision for a risk-neutral decision
maker?
– What is the optimal decision for Decision Maker I?
– . . . for Decision Maker II?
– What is the value of this decision problem to Decision
Maker I? . . . to Decision Maker II?
– What conclusion can you draw?
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Example 2 (8/10)
• Risk-Neutral Decision Maker
EV(d1) = .5(100,000) + .3(40,000) + .2(-60,000) = 50,000
EV(d2) = .5( 50,000) + .3(20,000) + .2(-30,000) = 25,000
EV(d3) = .5( 20,000) + .3(20,000) + .2(-10,000) = 14,000
The risk-neutral optimal decision is d1.
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Example 2 (9/10)
• Expected Utility: Decision Maker I
EU(d1) = .5(100) + .3(90) + .2( 0) = 77.0
EU(d2) = .5( 94) + .3(80) + .2(40) = 79.0
EU(d3) = .5( 80) + .3(80) + .2(60) = 76.0
Decision Maker I’s optimal decision is d2.
• Expected Utility: Decision Maker II
EU(d1) = .5(100) + .3(50) + .2( 0) = 65.0
EU(d2) = .5( 58) + .3(35) + .2(10) = 41.5
EU(d3) = .5( 35) + .3(35) + .2(18) = 31.6
Decision Maker II’s optimal decision is d1.
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Example 2 (10/10)
• Value of the Decision Problem: Decision Maker I
– Decision Maker I’s optimal expected utility is 79.
– He assigned a utility of 80 to +20,000, and a utility of 60 to -10,000.
– Linearly interpolating in this range 1 point is worth
30,000/20 = 1,500.
– Thus a utility of 79 is worth about 20,000 - 1,500 = 18,500.
• Value of the Decision Problem: Decision Maker II
–
–
–
–
Decision Maker II’s optimal expected utility is 65.
He assigned a utility of 100 to 100,000, and a utility of 58 to 50,000.
In this range, 1 point is worth 50,000/42 = 1190.
Thus a utility of 65 is worth about 50,000 + 7(1190) = 58,330.
The decision problem is worth more to Decision Maker II (since
58,330 > 18,500).
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Decision making in light
of competitive actions (Game theory)
• Game theory can be used to determine optimal
decisions in face of other decision making players.
• All the players are seeking to maximize their
return.
• The payoff is based on the actions taken by all the
decision making players.
• Details are beyond this lecture.
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Conclusion :
The Six Steps in Decision Theory
•
•
•
•
Clearly define the problem at hand
List the possible alternatives
Identify the possible outcomes
List the payoff or profit of each combination of
alternatives and outcomes
• Select one of the mathematical decision theory
models
• Apply the model and make your decision
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