EMGT 269 - Elements of Problem Solving and Decision Making
4. EVALUATING DECISION TREES
Objective measured in Dollars
Solve Decision Problem using
Expected Monetary Value (EMV)
Expected Value of discrete random variable Y:
n
n
i 1
i 1
EY [Y ]   yi  Pr(Y  yi )   yi  pi
Max Profit
Trade Ticket
EMV=
$4.5
EMV=
$4
Win (0.20)
$25
$24
y
$24.00
-$1.00
Pr(Y=y)
0.2
0.8
-$1
EMV=
$4.5
Lose (0.80)
$0
Win (0.45)
$10
=EMV
y*Pr(Y=y)
$4.50
$0.00
$4.50
=EMV
-$1
$10
y
$10.00
$0.00
Keep Ticket
$0
Lose (0.55)
$0
y*Pr(Y=y)
$4.80
-$0.80
$4.00
Pr(Y=y)
0.45
0.55
$0
Interpretation:
Playing the lottery a lot of times will result in
an average payoff equal to the EMV.
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 52
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Texaco Pennzoil Case:
Max Result
Accept $2 Billion
2
Texaco Accepts $5 Billion (0.17)
5
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
10.3
Medium (0.50)
Final Court
Decision
Low (0.30)
0
High (0.20)
Refuse
10.3
Medium (0.50)
Final Court
Decision
Low (0.30)
Texaco
(0.33)
Counter offers $3 Billion
5
5
0
Accept $3 Billion
3
Solve tree using EMV by folding back the Tree
High (0.20)
Final Court
Decision
Medium (0.50)
Low (0.30)
Step 1
10.3
5
0
y
10.300
5.000
0.000
Pr(Y=y)
0.2
0.5
0.3
y*Pr(Y=y)
$2.06
$2.50
$0.00
$4.56
1.
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 53
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
=EMV
EMGT 269 - Elements of Problem Solving and Decision Making
High (0.20)
EMV=
4.56
Refuse
Medium (0.50)
Final Court
Decision
EMV=
4.56
10.3
Low (0.30)
5
0
Accept $3 Billion
3
2.
Max Result
Accept $2 Billion
2
Texaco Accepts $5 Billion (0.17)
3.
EMV=
4.63
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
5
EMV=
4.56
Texaco
Counter offers $3 Billion
(0.33)
y
5.000
4.560
4.560
Pr(Y=y)
0.17
0.5
0.33
y*Pr(Y=y)
$0.85
$2.28
$1.50
$4.63
EMV=
4.56
Max Result
Accept $2 Billion
2
EMV=
4.63
Counteroffer
$5 Billion
4.
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 54
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
=EMV
EMGT 269 - Elements of Problem Solving and Decision Making
Definition Decision Path:
A path starting add the left most node up to the values at the end
of a branch by selecting one alternative from a decision node or
by following one outcome from uncertainty nodes.
Definition Decision Strategy:
The collection of decision paths connected to a branch of the left
most node by selecting one alternative from each decision node
along these paths.
Optimal Decision Strategy:
That Decision Strategy which results in the highest EMV if we
maximize profit and the lowest EMV if we minimize cost.
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 55
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Max Result
Accept $2 Billion
2
EMV=
4.63
Texaco Accepts $5 Billion (0.17)
EMV=
4.63
High (0.20)
EMV=
4.56
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
Final Court
Decision
Low (0.30)
High (0.20)
EMV=
4.56
EMV=
4.56
Medium (0.50)
Refuse
Final Court
Decision
Medium (0.50)
Low (0.30)
Texaco
(0.33)
Counter offers $3 Billion
Accept $3 Billion
5
10.3
5
0
10.3
5
0
3
OPTIMAL DECISION STRATEGY:
Counteroffer $5 Billion. Next, if Texaco
Counteroffers
$3 Billion, refuse the counteroffer.
Number of Decision Strategies in Decision Tree above: 3
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 56
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
1. D =“Accept 2 Billion”.
Accept $2 Billion
2
2. D =”Counter 5 Billion, Refuse Counter of $3 Billion”.
Texaco Accepts $5 Billion (0.17)
5
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
Final Court
Decision
10.3
Medium (0.50)
5
Low (0.30)
0
High (0.20)
Refuse
Final Court
Decision
10.3
Medium (0.50)
5
Low (0.30)
Texaco
(0.33)
Counter offers $3 Billion
0
3. D =”Counter 5 Billion, Accept $3 Billion”.
Texaco Accepts $5 Billion (0.17)
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
Final Court
Decision
Medium (0.50)
Low (0.30)
5
10.3
5
0
(0.33)
Texaco
Counter offers $3 Billion
Accept $3 Billion
3
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 57
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
RISK PROFILES = Graph that shows probabilities
for each possible outcome given a decision strategy.
Note:
 Risk Profile is a probability density function for the discrete
random variable Y representing the outcomes for the given
decision strategy.
Accept $2 Billion
2
Outc om e x ($Billion)
Pr(Outc ome| D ))
2
1
Pr(Outcome| D )
1
1
0 .8
0 .6
0 .4
0 .2
0
-1
0
1
2
3
4
5
6
7
O u tc o m e
1.
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 58
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
8
9
10
11
EMGT 269 - Elements of Problem Solving and Decision Making
C a lc u la t io n
Texaco Accepts $5 Billion (0.17)
0.17
0.170
0.50*0.20
0.100
0.50*0.50
0.250
0.50*0.30
0.150
0.33*0.20
0.066
0.33*0.50
0.165
0.33*0.30
0.099
5
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
10.3
Final
Medium (0.50) 5
Court
Decision
Low (0.30)
0
High (0.20)
Refuse
Final
Court
Decision
Texaco
Counter - (0.33)
offers $3 Billion
Calc ulati on
0.150+0.0 99
0.170+0.250+0.1 65
0.100+0.0 66
10.3
Medium (0.50) 5
Low (0.30)
0
Total
Pr (Ou tc ome| D)
0.2 49
0.5 85
0.1 66
1.0 00
1.000
1
P r(Ou tco m e| D )
Outc o me x ($Billion)
0
5
10.3
Prob
0 .8
0 .6
0 .5 8 5
0 .4
0 .2 4 9
0 .2
0 .1 6 6
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
O u tc o m e
2.
C a lc u la t io n
Texaco Accepts $5 Billion (0.17)
Counteroffer
$5 Billion
Texaco Refuses (0.50)
Counteroffer
Texaco
(0.33)
Counter offers $3 Billion
0.17
0.170
0.50*0.20
0.100
0.50*0.50
0.250
0.50*0.30
0.150
0.33
0.330
5
High (0.20)
10.3
Final
Medium
(0.50)
Court
5
Decision
Low (0.30)
0
Accept $3 Billion
3
Prob
Total
1.000
Calc ulation
0.15
0.33
0.170+0.250
0.1
Pr(Outc ome| D)
0.15
0.33
0.42
0.1
1.000
P r(Ou tco m e| D )
1
Outc ome x ($Billion)
0
3
5
10.3
0 .8
0 .6
0 .3 3
0 .2
0 .1 5
0 .1
0
-1
3.
0 .4 2
0 .4
0
1
2
3
4
5
6
7
O u tc o m e
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 59
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
8
9
10
11
EMGT 269 - Elements of Problem Solving and Decision Making
CUMMULATIVE RISK PROFILES = Graphs that shows
cumulative probabilities associated with a risk profie
Note:
 Cumulative Risk Profile is a cumulative distribution function for
the discrete random variable Y representing the outcomes for
the given decision strategy.
Risk Profiles
Cumulative Risk Profiles
1 .2
1 .2
1
P r(Y<=y | D )
Pr(Outcome| D )
1
0 .8
0 .6
0 .4
1
1
0 .6
0 .4
0 .2
0 .2
0
0
-1
1
0 .8
0
1
2
3
4
5
6
7
8
9
10
0
-1
11
0
0
1
2
3
4
5
6
7
8
9
10
11
y
O u tc o m e
1.
1 .2
1 .2
1
P r(Y<=y| D )
P r(Outcome| D )
1
0 .8
0 .6
0 .5 8 5
0 .4
0 .2 4 9
0 .2
0 .4
0 .2 4 9
0 .2
0 .1 6 6
0
0
1
2
3
4
5
6
7
8
9
10
-1
11
1
0 .8 3 4
0 .6
0
-1
1
0 .8 3 4
0 .8
0 .2 4 9
0
0
1
2
3
4
5
6
7
8
9
10
11
y
O u tc o m e
1
1 .2
0.8
1
P r(Y<=y | D )
Pr(Outcome| D)
2.
0.6
0.42
0.4
0.33
0.2
0.15
0.1
0 .8
0 .6
0.4 8
0 .4
0 .2
0
-1
0.1 5
0
0
0
1
2
3
4
5
Outcome
6
7
8
9
10
11
1 1
0 .9
0 .9
-1
0
1
0.4 8
0.1 5
2
3
4
5
6
7
y
3.
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 60
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
8
9
10
11
EMGT 269 - Elements of Problem Solving and Decision Making
Deterministic Dominance
Max Result
Accept $2 Billion
2
Texaco Accepts $5 Billion (0.17)
High (0.20)
Counteroffer
$5 Billion
Texaco Refuses (0.50)
Counteroffer
Final Court
Decision
Medium (0.50)
Low (0.30)
High (0.20)
Refuse
Final Court
Decision
Medium (0.50)
Low (0.30)
Texaco
(0.33)
Counter offers $3 Billion
Accept $3 Billion
5
10.3
5
2.5
10.3
5
2.5
3
Definition:
A = largest 0% point in Cumulative Risk Profile
B = smallest 100% point in Cumulative Risk Profile
Range of a Cumulative Risk Profile = [A,B]
Deterministic Dominance:
Step 1: Draw cumulative risk profiles in one graph
Step 2: Determine range for each risk profile
Step 3: Ranges are disjoint
Deterministic Dominance
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 61
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
1 .2
Accept 2 Billion
1
0 .8
Counteroffer 5 Billion, Refuse
Texaco 3 Billion Counteroffer
0 .6
0 .4
0 .2
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
y
Maximize Return
"Counteroffer" deterministically
dominates "Accept"
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 62
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Stochastic Dominance
Example 1:
Max Result
Texaco Accepts $5 Billion (0.17)
5
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
10.5
Medium (0.50)
Final
Court
Decision
5.2
Low (0.30)
0
High (0.20)
10.5
Firm A
Refuse
Texaco
Counter - (0.33)
offers $3 Billion
Medium (0.50)
Final
Court
Decision
5.2
Low (0.30)
0
Accept $3 Billion
3
Texaco Accepts $5 Billion (0.17)
5
Firm B
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
Final
Court
Decision
Medium (0.50)
Low (0.30)
High (0.20)
Refuse
Texaco
Counter - (0.33)
offers $3 Billion
Final
Court
Decision
Medium (0.50)
Low (0.30)
10.3
5
0
10.3
5
0
Accept $3 Billion
3
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 63
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Stochastic Dominance:
Step 1: Draw cumulative risk profiles in one graph
Step 2: CRP do not cross
Stochastics Dominance
1
0 .8
0 .6
Firm B
Firm A
0 .4
0 .2
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
y
Max Return
Firm A stochastically dominates Firm B
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 64
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Example 2:
Max Result
Texaco Accepts $5 Billion (0.17)
5
High (0.30)
Counteroffer
$5 Billion
Texaco Refuses (0.50)
Counteroffer
10.3
Medium (0.60)
Final
Court
Decision
5.0
Low (0.10)
0
High (0.30)
10.5
Firm C
Refuse
Texaco
Counter - (0.33)
offers $3 Billion
Medium (0.60)
Final
Court
Decision
5.0
Low (0.10)
0
Accept $3 Billion
3
Texaco Accepts $5 Billion (0.17)
5
Firm D
High (0.20)
Counteroffer
$5 Billion
Texaco Refuses (0.50)
Counteroffer
Final
Court
Decision
Medium (0.50)
Low (0.30)
High (0.20)
Refuse
Texaco
Counter - (0.33)
offers $3 Billion
Final
Court
Decision
Medium (0.50)
Low (0.30)
10.3
5
0
10.3
5
0
Accept $3 Billion
3
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 65
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
1
0 .8
Firm D
0 .6
Firm C
0 .4
0 .2
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
y
Max Return
Firm C stochastically dominates Firm D
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 66
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Making Decisions with Multiple Objectives
Influence Diagram
Amount of
Fun
Fun
Job
Decision
Overall
Satisfaction
Salary
Amount
of Work
Two Objectives:
 Making Money (Measured in $)
 Having Fun (Constructed attribute scale, see page 128)
Best(5), Good(4), Middle(3), Bad(2), Worst (1)
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 67
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Decision Tree
Consequences
Salary
Fun
level
Forest Job
5 (0.10)
$2600.00
5
4 (0.25)
$2600.00
4
$2600.00
3
$2600.00
2
$2600.00
1
3 (0.40)
2 (0.20)
1 (0.05)
# hours
per week
In-Town Job
Fun Level
40 hours (0.35)
34 hours (0.50)
30 hours (0.15)
$2730.00
3
$2320.50
3
$2047.50
3
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 68
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Analysis based on Salary Objective
Salary
$2,047.50
$2,320.50
$2,600.00
$2,730.00
Forest Job
Prob
Prob*Salary
1.00
E[Salary]=
In-Town Job
Prob
Prob*Salary
0.15
$307.13
0.50
$1,160.25
$2,600.00
$2,600.00
0.35
E[Salary]=
1.00
0.80
$955.50
$2,422.88
1.00
0.60
0.40
0.50
0.35
0.20
0.15
0.00
$2,000.00 $2,200.00
$2,400.00
$2,600.00
$2,800.00
Salary
Forest Job
In-Town Job
Optimal Decision Strategy: Forest-Job
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 69
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Analysis based on Fun Level Objective
Forest Job
In-Town Job
Fun Level
Prob
Fun Level*Prob
Prob
Fun Level*Prob
0.00%
0.05
0.0%
25.00%
0.20
5.0%
60.00%
0.40
24.0%
1.00
60.00%
90.00%
0.25
22.5%
100.00%
0.10
10.0%
E[Fun Level]=
61.5%
E[Fun Level]=
60.00%
1.00
1.00
0.80
0.60
0.80
0.40
0.60
1.00
0.20
0.40
0.00
0.20
0.00%
0.40
20.00%
40.00% 0.20
60.00%
80.00%
0.05
0.00
0
1
2
3
0.25
100.00% 120.00%
0.10
4
5
6
Summer Fun
Forest Job
In-Town Job
Optimal Decision Strategy: Forest-Job
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 70
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Analysis based on Two Objectives:
1. Before overall score can be calculated the scales of each
alternative need to be the same i.e.
 Transform to 0-1 scale or 0%-100% scale
 Set Best=100%, Worst=0%
 Determine intermediate values
Having Fun objective:
Best(100%), Good(90%), Middle(60%), Bad(25%), Worst (0%)
Making Money Objective:
 $2730.00=100%, $2047.50=0%
X  $2047.50
 100%
 Intermediate dollar amount X:
$2730  $2047.50
2. Assess Trade-off weights
k s = Weight for Salary
k f = Weight for Fun Level
ks  k f  1
Using Expert Judgment:
Going from worst to best in salary objective is
1.5 times more important than going from
worst to best in having fun objective
 k s  1.5  k f . With k s  k f  1 follows that:
k f  0.40; k s  0.60.
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 71
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Consequences
Salary (0.4)
Fun
level
Forest Job
5 (0.10)
81%
100%
4 (0.25)
81%
90%
81%
60%
81%
25%
81%
0%
100%
60%
40%
60%
0%
60%
3 (0.40)
2 (0.20)
1 (0.05)
# hours
per week
In-Town Job
Fun Level (0.6)
40 hours (0.35)
34 hours (0.50)
30 hours (0.15)
Total Score
Fun
level
Forest Job
5 (0.10)
88.6%
4 (0.25)
84.6%
3 (0.40)
2 (0.20)
1 (0.05)
# hours
per week
In-Town Job
40 hours (0.35)
34 hours (0.50)
30 hours (0.15)
72.6.%
58.6%
48.6%
84.0%
48.0%
24.0%
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 72
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
Combined Score
48.6%
58.6%
72.6%
84.6%
88.6%
Combined Score
24.0%
48.0%
84.0%
Forest Job
Prob
0.05
0.20
0.40
0.25
0.10
E[Comb Score]=
Combined Score*Prob
2.4%
11.7%
29.0%
21.1%
8.9%
73.2%
In-Town Job
Prob
Combined Score*Prob
0.15
3.6%
0.50
24.0%
0.35
29.4%
E[Comb Score]=
57.0%
Optimal Decision Strategy: Forest-Job
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 73
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen
EMGT 269 - Elements of Problem Solving and Decision Making
1 .0 0
0 .8 0
0 .6 0
0. 50
0 .4 0
0 .2 0
0 .0 0
0 . 0%
0. 40
0. 20
0. 15
0. 05
2 0 .0 %
4 0 .0 %
6 0 .0 %
0.35
0. 25
0.10
8 0 .0 %
1 0 0. 0%
C om bin e d S c ore
F ore st Jo b
In- T ow n Jo b
1.00
0.80
0.60
0.40
0.20
0.00
0. 0%
20. 0%
40.0%
60. 0%
80.0%
100. 0%
C o m bin ed Sco r e
F or est J ob
I n- Town J ob
"Forest-Job" stochastically dominates
"In-Town Job" alternative in terms of
the overall score
Instructor: Dr. J. Rene van Dorp
Session 4 - Page 74
Source: Making Hard Decisions, An Introduction to Decision Analysis by R.T. Clemen