EMGT 269 - Elements of Problem Solving and Decision Making
Frequentist interpretation of probability:
Probability = Relative frequency of occurrence of an event
Example: Throwing a fair coin, Pr(Heads)=0.5
What about following events?

Core Meltdown of Nuclear Reactor

You being alive at the age of 50?

Unsure of the final outcome, though event has occurred

One basketball team beating the other in next day's match
Note: Without doubt the above events are uncertain and we talk about the probability of the above events. These probabilities are NOT Relative Frequencies of Occurrences.
What are they?
Subjectivist Interpretation of Probability:
Probability = Degree of belief in the occurrence of an event
By assessing a probability for the events above, one expresses one's DEGREE OF BELIEF . High probabilities coincide with high degree of belief. Low probabilities coincide with low degree of belief.
Why do we need it?

Frequentist interpretation not always applicable.

Model, and structure individualistic uncertainty through probability.
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 90

EMGT 269 - Elements of Problem Solving and Decision Making

Direct Methods: Directly asking for probability assessments.

Indirect Methods: Formulating questions in expert's domain of expertise and extract probability assessment through probability modeling, e.g. betting strategies and indifference lotteries
1. Betting Stategies
Event: Lakers winning the NBA title this season
STEP 1: Offer a person to choose between following the following bets, where X=100, Y=0.
Max Profit
Lakers Win
X
Bet for Lakers
Lakers Loose
Lakers Win
-Y
-X
Bet against Lakers
Lakers Loose
Y
STEP 2: Offer a person to choose between following the following bets, where X=0, Y=100. (Consistency Check)
STEP 3: Offer a person to choose between following the following bets, where X=100, Y=50.
STEP 4: Offer a person to choose between following the following bets, where X=50, Y=100. (Consistency Check)
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 91

EMGT 269 - Elements of Problem Solving and Decision Making
Continue until point of indifference has been reached.
Assumption:
When a person is indifferent between bets the expected payoffs from the bets must be the same.
Thus:
X

Pr(LW) - Y  Pr(LL)= -X

Pr(LW) + Y  Pr(LL) 

2

X

Pr(LW) - 2  Y  (1- Pr(LW) )=0 
Pr(LW) =
X
Y

Y
.
2
Example: X=100, Y=50  Pr(LW)=
3

 66.66%
2. Betting Stategies
Event: Lakers winning the NBA title this season
Choose two prices A and B, such that A>>B.
Lakers Win
Hawaiian Trip
Lottery 1
Lakers Loose
Beer
(p)
Hawaiian Trip
Lottery 2
(1-p)
Beer
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 92

EMGT 269 - Elements of Problem Solving and Decision Making
Lottery 2 is the REFERENCE LOTTERY. A probability mechanism is specified for lottery 2. For Example:

Throwing a fair coin

Ball in an urn

Throwing a die,

Wheel of fortune
Strategy:
1. Specify p
1
.
2. Ask which one do you prefer?
3. If Lottery 1 is preferred offer change p i
to p
1+i
 p i.
4. If Lottery 2 is preferred offer change p i
to p
1+i
 p i.
5. If indifference is reached STOP, else goto 2.
Assumption:
Indifference reached

Pr(LW) = p
Consistency Checking:
Subjective Probabilities must follow the laws of probability
Example:
If expert specifies Pr(A), Pr(B|A) and Pr(A
 then
Pr(B|A)  Pr(A)=Pr(A



Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 93

EMGT 269 - Elements of Problem Solving and Decision Making
1. Ask for distribution parameters (direct\parametric)
2. Ask for distribution quantities and solve for parameters
(direct\parametric)
3. Ask for shape of CDF (Non-parametric)
1. Assessing Quantiles
Definition: x p is the p-th quantile of random variable X 

F(x p
)=Pr(X

x p

=p
Terminology: quantile, fractile, percentile, quartile
Assessing CDF is often done by assessing a number of quantiles.
@ Method 2: Use quantile estimates to solve distribution parameters.
@ Method 3: Connect multiple quantile estimates by straight lines to approximate the CDF
Example:
Uncertain Event: Age of Movie Actress (e.g. ….)
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 94

EMGT 269 - Elements of Problem Solving and Decision Making
STEP 1: You know age of actress is between 30 and 65
STEP 2: Consider Reference Lottery
Age  46
Hawaiian Trip
Lottery 1
Age  46
Beer
(p)
Hawaiian Trip
Lottery 2
(1-p)
Beer
You decide you are indifferent for p=0.5

Pr(Age  46)=0.5
STEP 3: Consider Reference Lottery
Age  50
Hawaiian Trip
Lottery 1
Age  50
Beer
(p)
Hawaiian Trip
Lottery 2
(1-p)
Beer
You decide you are indifferent for p=0.8

Pr(Age  50)=0.8
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 95

EMGT 269 - Elements of Problem Solving and Decision Making
STEP 4: Consider Reference Lottery
Age  40
Hawaiian Trip
Lottery 1
Age  40
Beer
(p)
Hawaiian Trip
Lottery 2
(1-p)
Beer
You decide you are indifferent for p=0.05

Pr(Age  40)=0.05
STEP 5. Approximate CDF
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0 10 20 30
Years
40 50 60 70
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 96

EMGT 269 - Elements of Problem Solving and Decision Making
Use of REFERENCE LOTTERIES to assess quantiles:
1. Fix Horizontal Axes:
Age  46
Hawaiian Trip
Lottery 1
Age  46
Beer
(p)
Hawaiian Trip
Lottery 2
(1-p)
Beer
Strategy: Adjust p until indifference point has been reached by using a probability mechanism.
2. Fix Vertical Axes:
Age  x
Hawaiian Trip
Lottery 1
Age  x
Beer
(0.35)
Hawaiian Trip
Lottery 2
(0.65)
Beer
Strategy: Adjust x until indifference point has been reached.
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 97

EMGT 269 - Elements of Problem Solving and Decision Making
Overall Strategy to assess CDF:
STEP 1: Ask for the Median (50% Quantile).
STEP 2: Ask for Extreme Values (0% quantile, 100% quantile).
STEP 3: Ask for High/Low Values (5% Quantile, 95%
Quantile).
STEP 4: Ask for 1 st and 3 rd Quartile (25% Quantile, 75%
Quantile).
STEP 5 A: Approximate CDF through straight line technique
STEP 5 B: Model CDF between assessed points
STEP 5 C: Calculate “Best Fit” in a family of CDF’s.
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 98

EMGT 269 - Elements of Problem Solving and Decision Making

Advanced Approaches:
1. Fitting Distributions
2. Numerical Integrations
3. Monte Carlo Simulation

Simple aproach: Use Discrete Approximation
Extended Pearson Tukey-Method:
A Fan is replaced by a three branch uncertainy node
95% Quantile
Median
5% Quantile
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0 10 20 30
Years
40 50 60 70
40 46 61
Age=30 Age= 40 (0.185)
Age = 46 (0.630)
Age = 65 Age = 61 (0.185)
Age
40
46
61
Pr(Age) Age*Pr(Age)
0.185
7.4
0.63
0.185
E[Age]
29.0
11.3
47.7
Instructor: Dr. J. Rene van Dorp
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
Session 6 - Page 99

EMGT 269 - Elements of Problem Solving and Decision Making
Bracket Median Method:
A Fan is replaced by a five branch uncertainy node
STEP 1: Divide total range in five equally likely intervals
STEP 2: Determine bracket median in each interval
STEP 3: Assing equal probabilitities in to all bracket medians
0.25
0.25
0.25
0.25
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
0.125
10 20 30 40
Years
41 44 48
50
53
60 70
Age=30
1 2 3 4
Age= 41 (0.25)
Age= 44 (0.25)
Age = 65
Age= 48 (0.25)
Age = 53 (0.22)
Age
41
44
48
Pr(Age) Age*Pr(Age)
0.25
0.25
0.25
53 0.25
E[Age]
10.3
11.0
12.0
13.3
46.5
Instructor: Dr. J. Rene van Dorp Session 6 - Page 100
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen

EMGT 269 - Elements of Problem Solving and Decision Making
When eliciting expert judgment, expert use primitive congnitive techniques, e.g. intuition, to make assessments.
These techniques are in general simple and intuitively appealing, however they may result in a number of biases.

Representative Bias:
Probability estimate is made on the basis of similarities with a group. One tend to ignore relevant information such as incidence/base rate.
Example:

X is the event that a person is dresses sloppy

Managers (M) are well dressed: Pr(X|M)=0.1.

Computer Scientist are badly dressed. Pr(X|C)=0.8.
At a conference with 90% attendence of managers and 10% attendance of computer scientist you observe a person and notices that he dresses sloppy.

What do think is more likely?:
"The person is a computer scientist" or
"The persion is a manager"
Instructor: Dr. J. Rene van Dorp Session 6 - Page 101
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen

EMGT 269 - Elements of Problem Solving and Decision Making
Note that:
Pr( X | C ) Pr( C )
Pr( C
Pr( M
| X )
| X )

Pr( X
Pr( X )
| M ) Pr( M )

Pr(
Pr( X
X | C ) Pr( C )
| M ) Pr( M )

0 .
8

0 .
1

1
0 .
1 * 0 .
9
Pr( X )
In other words: It is more likely that this person is a manager than a computer scientist.

Availability Bias:
Probability estimate is made according to the ease with which we can retrieve similar events

Anchoring Bias:
One makes first assessment (anchor) and make subsequent assessments relative to this anchor.

Motivational Bias:
Incentives are always present, such that people do not really say what they believe.

Break down problem into finer detail using probability laws until you have reached a point at which experts are comfortable in making the assessment in a meaning full manner. Next aggregate the detail assessment to obtain probability estimates at a lower level of detail.
Instructor: Dr. J. Rene van Dorp Session 6 - Page 102
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen

EMGT 269 - Elements of Problem Solving and Decision Making

Stock Market Example
Stock
Price
Market
Market Up
Stock Price Up
Stock Price Down
Market Down

Nuclear Regulatory Example
Stock
Price
Stock Price Up
Stock Price Down
Stock Price Up
Stock Price Down
Electrical Power
Failure?
Control
System Failure?
Control
System Failure?
Accident?
Instructor: Dr. J. Rene van Dorp Session 6 - Page 103
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen

EMGT 269 - Elements of Problem Solving and Decision Making
Which probability estimates do we need to calculate
The probability of an accident?:
A = Accident, L = Cooling System Failure, N = Control
System Failure, E = Electrical System Failure.
STEP 1: Assess the four conditional probabilities:
Pr( A | L , N ), Pr( A | L , N ), Pr( A | L , N ), Pr( A | L , N )
STEP 2: Assess the two conditional probabilities:
Pr( L | E ), Pr( L | E )
STEP 3: Assess the two conditional probabilities
Pr( N | E ), Pr( N | E )
STEP 4: Assess the probability:
Pr( E )
STEP 5: Apply Law of Total Probability
Pr( A )

Pr( A | L , N )

Pr( L , N )

Pr( A | L , N )

Pr( L , N )

Pr( A | L , N )

Pr( L , N )

Pr( A | L , N )

Pr( L , N )
STEP 6: Apply Law of Total Probability
Pr( L , N )

Pr( L , N | E )

Pr( E )

Pr( L , N | E )

Pr( E )
Same with:
Pr( L , N ), Pr( L , N ), Pr( L , N )
STEP 7: Apply Conditional Independence assumption
Pr( L , N | E )

Pr( L | E )

Pr( N | E )
Instructor: Dr. J. Rene van Dorp Session 6 - Page 104
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen

EMGT 269 - Elements of Problem Solving and Decision Making

Subjective probabilities must follow the laws of probability.

If they do not, the person assessing the probabilities is incoherent.
Incoherence

Possibility of a Dutch Book
Dutch Book:
A series of bets in which the opponent is guaranteed to loose and you win.
Example:
Basketball Game between Lakers and Celtics and your friend says that Pr(LW)=40% and Pr(CW)=50%. You note that the probabilities do not add up to 1, but your friend stubbornly refuses to change his initial estimates. You think,
"GREAT!" let's set up a series of bets!
BET 1
Lakers Win (0.4)
Max Profit
$60
BET 2
Celtics Loose (0.5)
Max Profit
-$50
Lakers Loose (0.6) Celtics Win (0.5)
$50
-$40
Note that both EMV of bets are 0 according to his probability assessments, so he should be willing to engage in both.
Lakers Win: Bet 1 - You win $60, Bet 2: You Loose $50
Lakers Loose: Bet 1 - You loose $40, Bet 2: You win $50
Instructor: Dr. J. Rene van Dorp Session 6 - Page 105
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen