Chapter 4
Making Hard
Decisions
Making Choices
R. T. Clemen, T. Reilly
Chapter 4 – Making Choices
Lecture Notes by: J.R. van Dorp and T.A. Mazzuchi
COPYRIGHT © 2006by GWU
Slide 1 of 53
Texaco Versus Pennzoil
In early 1984, Pennzoil and Getty Oil agreed to the
terms of a merger. But before any formal documents
could be signed, Texaco offered Getty a substantially
better price, and Gordon Getty, who controlled most of
the Getty Stock, reneged on the Pennzoil deal and sold to
Texaco. Naturally, Pennzoil felt as if it had been dealt
with unfairly and immediately filed a lawsuit against
Texaco alleging that Texaco had interfered illegally in the
Pennzoil-Getty negotiations. Pennzoil won the case: in
late 1985, it was awarded $11.1 billion, the largest
judgment ever in the United States. A Texas appeal court
reduced the judgement to $2 billion, but interest and
penalties drove the total back up to $10.3 billion. James
Kinnear, Texaco’s Chief executive officer, had said that
Texaco would file for bankruptcy if Pennzoil obtained
court permission to secure the judgment by filing liens
against Texaco’s assets.
Chapter 4 – Making Choices
Slide 2 of 53
Texaco Versus Pennzoil - Continued
Furthermore, Kinnear had promised to fight the case all
the way to the U.S. Supreme Court if necessary, arguing
in part that Pennzoil had not followed Security and
Exchange Commission regulations in its negotiations with
Getty. In April 1987, just before Pennzoil began to file
liens, Texaco offered to Penzoil $2 billion dollars to
settle the entire case. Hugh Liedtke, chairman of
Pennzoil, indicated that his advisors were telling him that
a settlement between $3 billion and $5 billion would be
fair.
What should Hugh Liedtke do?
1. Accept $2 Billion
2. Refuse $2 Billion and counter offer $5 Billion
Chapter 4 – Making Choices
Slide 3 of 53
Texaco Versus Pennzoil – Decision Tree
Max Settlement
Amount
($ Billion )
Accept $2 Billion
2
Texaco Accepts $5 Billion
5
High
Counteroffer
Texaco Refuses
$5 Billion
Counteroffer
Final Court
Decision
Medium
Low
High
Refuse
Final Court
Decision
Medium
Low
Texaco
Counter offers $3 Billion
Accept $3 Billion
Chapter 4 – Making Choices
10.3
5
0
10.3
5
0
3
Slide 4 of 53
Texaco Versus Pennzoil - Continued
•
Given tough negotiation positions of the two executives,
there could be an even chance (50%) that Texaco will
refuse to negotiate further.
•
Liedtke and advisor figure that it is twice as likely that
Texaco would counter offer $3 billion than accepting the
$5 billion. Hence, because there is a 50% chance of
refusal, there must be a 33% chance of a Texaco
counteroffer and a 17% chance of Texaco accepting
$5 billion.
Chapter 4 – Making Choices
Slide 5 of 53
Texaco Versus Pennzoil - Continued
What are the probabilities of the final court decision?
ƒ
ƒ
ƒ
Liedtke admitted that Pennzoil could lose the case. Thus there is
a significant possibility the outcome would be zero. It’s probability
is assessed at 30%.
Given the strength of the Pennzoil case it is also possible that the
court will upheld the judgment as it stands. It’s probability is
assessed at 20%.
Finally, the possibility exists that the judgment could be reduced
to $5 billion. Thus there must be a chance of 50% of this happening.
Chapter 4 – Making Choices
Slide 6 of 53
Texaco Versus Pennzoil – Decision Tree
Max Settlement
Amount
($ Billion )
2
Accept $2 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)
High (0.20)
Refuse
Final Court
Decision
Medium (0.50)
Low (0.30)
Texaco
(0.33)
Counter offers $3 Billion
Accept $3 Billion
Chapter 4 – Making Choices
5
10.3
5
0
10.3
5
0
3
Slide 7 of 53
Decision Tree and Expected Monetary Value (EMV)
When objective is measured in dollars
First Suggestion:
Solve decision problem by choosing
that alternative that maximizes the EMV
Expected value of discrete random variable Y:
n
n
i =1
i =1
EY [Y ] = ∑ yi ∗ Pr(Y = yi ) = ∑ yi ∗ pi
Chapter 4 – Making Choices
Slide 8 of 53
A double-risk dillema
Max Profit
Win (0.20)
EMV= $4
$25
$24
y
$24.00
-$1.00
Pr(Y=y)
0.2
0.8
y*Pr(Y=y)
$4.80
-$0.80
$4.00
= EMV
y*Pr(Y=y)
$4.50
$0.00
$4.50
=EMV
Trade Ticket
EMV=
$4.5
-$1
EMV= $4.5
Lose (0.80)
$0
Win (0.45)
$10
-$1
$10
Keep Ticket
$0
Lose (0.55)
$0
$0
y
$10.00
$0.00
Pr(Y=y)
0.45
0.55
Interpretation EMV: Playing the same lottery a lot of times
will result over time in an average pay-off equal to the EMV
Chapter 4 – Making Choices
Slide 9 of 53
Texaco Versus Pennzoil – Decision Tree
Max Settlement
Amount
($ Billion )
Accept $2 Billion
2
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)
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
0
10.3
5
0
3
Solve tree using EMV by folding back the tree
Chapter 4 – Making Choices
Slide 10 of 53
Decision Tree and Expected Monetary Value (EMV)
Step 1: Calculate EMV of court decision uncertainty node
High (0.20)
EMV= $4.56
Final Court
Decision
Medium (0.50)
Low (0.30)
10.3
5
0
Step 1
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
=EMV
Chapter 4 – Making Choices
Slide 11 of 53
Decision Tree and Expected Monetary Value (EMV)
Step 2: Evaluate decision regarding Texaco’s counter offer
High (0.20)
EMV=
4.56
EMV=
4.56
Refuse
Final Court
Decision
Medium (0.50)
Low (0.30)
Accept $3 Billion
Chapter 4 – Making Choices
10.3
5
0
3
Slide 12 of 53
Decision Tree and Expected Monetary Value (EMV)
Step 3: Calculate EMV Texaco’s reaction uncertainty node
Accept $2 Billion
2
Texaco Accepts $5 Billion (0.17)
EMV=
4.63
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
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
5
EMV=
4.56
EMV=
4.56
Chapter 4 – Making Choices
Slide 13 of 53
Decision Tree and Expected Monetary Value (EMV)
Step 4: Evaluate the immediate decision
Max Result
Accept $2 Billion
EMV=
4.63
Counteroffer
2
EMV=
4.63
$5 Billion
Optimal decision: Counteroffer $5 Billion
Optimal decision strategy: Counteroffer $5 Billion
and if Texaco counteroffers $3 Billion, then
refuse this counteroffer.
Chapter 4 – Making Choices
Slide 14 of 53
Folding back the Decision Tree from right to left using EMV
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
5
Final Court
Decision
Medium (0.50)
Low (0.30)
EMV=
4.56
Refuse
Final Court
Decision
Medium (0.50)
Low (0.30)
Texaco
(0.33)
Counter offers $3 Billion
Accept $3 Billion
Chapter 4 – Making Choices
5
0
High (0.20)
EMV=
4.56
10.3
10.3
5
0
3
Slide 15 of 53
Decision Strategies Texaco-Pennzoil Case
How many decision strategies do we have in
the Texaco – Penzoil decision tree?
First strategy: “Accept $2 billion”
Accept $2 Billion
Chapter 4 – Making Choices
2
Slide 16 of 53
Decision Strategies Texaco-Pennzoil Case
Second strategy: “Counter $5 billion and if Texaco counter
offers $3 billion refuse this counteroffer of $3 Billion”
Texaco Accepts $5 Billion (0.17)
5
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
Final Court
Decision
Medium (0.50)
Low (0.30)
Final Court
Decision
Medium (0.50)
Low (0.30)
Texaco
(0.33)
Counter offers $3 Billion
Chapter 4 – Making Choices
5
0
High (0.20)
Refuse
10.3
10.3
5
0
Slide 17 of 53
Decision Strategies Texaco-Pennzoil Case
Third strategy: “Counter $5 billion and if Texaco counter
offers $3 billion accept this counteroffer of $3 Billion”
Texaco Accepts $5 Billion (0.17)
5
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
Final Court
Decision
Low (0.30)
(0.33)
Texaco
Counter offers $3 Billion
Medium (0.50)
Accept $3 Billion
Chapter 4 – Making Choices
10.3
5
0
3
Slide 18 of 53
Risk Profiles and Cumulative Risk Profiles
RISK PROFILES = Graph that shows probabilities for each of
the possible outcomes given a particular decision strategy.
Note: Risk Profile is a probability mass function for the discrete random
variable Y representing the outcomes for the given decision strategy.
CUMMULATIVE RISK PROFILES = Graphs that shows cumulative
probabilities associated with a risk profile
Note: Cumulative risk profile is a cumulative
distribution function for the discrete random variable Y representing
the outcomes for the given decision strategy.
Chapter 4 – Making Choices
Slide 19 of 53
Risk Profiles
First strategy: “Accept $2 billion”
Accept $2 Billion
Pr(Outcome|D)
1
Risk Profile
D="Accept $2 Billion"
1
Pr(Outcome|D)
Outcome x ($Billion)
2
2
1
0.8
0.6
0.4
0.2
0
-1
2
5
8
11
Outcome ($Billion)
Chapter 4 – Making Choices
Slide 20 of 53
Risk Profiles
Second strategy: “Counter $5 billion and if Texaco counter
offers $3 billion refuse this counteroffer of $3 Billion”
Calculation
Prob
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
Medium (0.50) 5
0.33*0.50
0.165
Low (0.30)
0.33*0.30
0.099
Texaco Accepts $5 Billion (0.17)
High (0.20)
Counteroffer
Texaco Refuses (0.50)
$5 Billion
Counteroffer
Texaco
Counter - (0.33)
offers $3 Billion
10.3
Final
Medium (0.50) 5
Court
Decision
Low (0.30)
0
High (0.20)
Refuse
5
Final
Court
Decision
10.3
0
Total
Chapter 4 – Making Choices
1.000
Slide 21 of 53
Risk Profiles
Second strategy: “Counter $5 billion and if Texaco counter
offers $3 billion refuse this counteroffer of $3 Billion”
Outcome x ($Billion)
0
5
10.3
Calculation
0.150+0.099
0.170+0.250+0.165
0.100+0.066
Pr(Outcome| D)
0.249
0.585
0.166
1.000
Risk Profile D="Counter $5 Billion, refuse counter
offer of $3 Billion if given"
Pr(Outcome|D)
1
0.8
0.585
0.6
0.4
0.249
0.2
0.166
0
-1
2
5
8
11
Outcome ($Billion)
Chapter 4 – Making Choices
Slide 22 of 53
Risk Profiles
Third strategy: “Counter $5 billion and if Texaco counter
offers $3 billion accept this counteroffer of $3 Billion”
Texaco Accepts $5 Billion (0.17)
5
High (0.20)
Counteroffer
$5 Billion
Texaco Refuses (0.50)
Counteroffer
Texaco
(0.33)
Counter offers $3 Billion
10.3
Final
Medium (0.50) 5
Court
Decision
Low (0.30)
0
Accept $3 Billion
3
Calculation
Prob
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
Total
Chapter 4 – Making Choices
1.000
Slide 23 of 53
Risk Profiles
Third strategy: “Counter $5 billion and if Texaco counter
offers $3 billion accept this counteroffer of $3 Billion”
Outcome x ($Billion)
0
3
5
10.3
Calculation
0.15
0.33
0.170+0.250
0.1
Pr(Outcome| D)
0.15
0.33
0.42
0.1
1.000
Risk Profile D="Counter $5 Billion, Accept Counter
Offer of $3 Billion if given"
Pr(Outcome|D)
1
0.8
0.6
0.4
0.42
0.33
0.2
0.15
0.10
0
-1
2
5
8
11
Outcome ($Billion)
Chapter 4 – Making Choices
Slide 24 of 53
Cumulative Risk Profiles
First strategy: “Accept $2 billion”
Pr(Outcome|D)
1
1
Pr(Outcome|D)
Outcome x ($Billion)
2
Risk Profile
D="Accept $2 Billion"
1
0.8
0.6
0.4
0.2
0
-1
2
5
8
11
Outcome ($Billion)
Pr(Outcome ≤ x|D)
1
Cumulative Risk Profile
D="Accept $2 Billion"
Pr(Outcome ≤ x|D)
Outcome x ($Billion)
2
1
1
0.8
0.6
0.4
0.2
0
0
-1
1
3
5
7
9
11
Outcome ($Billion)
Chapter 4 – Making Choices
Slide 25 of 53
Cumulative Risk Profiles
Outcome x ($Billion)
0
5
10.3
Pr(Outcome|D)
0.249
0.585
0.166
Risk Profile D="Counter $5 Billion, refuse counter
offer of $3 Billion if given"
1
Pr(Outcome|D)
Second strategy: “Counter $5 billion
and if Texaco counter offers $3 billion
refuse this counteroffer of $3 Billion”
0.8
0.585
0.6
0.4
0.249
0.2
0.166
0
-1
2
5
8
11
Outcome ($Billion)
Outcome x ($Billion)
0
5
10.3
Pr(Outcome ≤ x|D)
0.249
0.249 + 0.585 = 0.834
0.834 + 0.166 = 1
Pr(Outcome ≤ x|D)
Cumulative Risk Profile D="Counter $5 Billion,
refuse counter offer of $3 Billion if given"
1
1
0.834
0.8
0.834
0.6
0.4
0.249
0.249
0.2
0
0
-1
2
5
8
11
Outcome ($Billion)
Chapter 4 – Making Choices
Slide 26 of 53
Cumulative Risk Profiles
Outcome x ($Billion)
0
3
5
10.3
Pr(Outcome|D)
0.15
0.33
0.42
0.1
Risk Profile D="Counter $5 Billion, Accept Counter
Offer of $3 Billion if given"
1
Pr(Outcome|D)
Third strategy: “Counter $5 billion
and if Texaco counter offers $3 billion
accept this counteroffer of $3 Billion”
0.8
0.6
0.4
0.15
0.2
0.42
0.33
0.10
0
-1
2
5
8
11
Outcome ($Billion)
Cumulative Risk Profile D="Counter $5 Billion,
accept counter offer of $3 Billion if given"
Pr(Outcome ≤ x|D)
0.15
0.15 + 0.33 = 0.48
0.48 + 0.42 = 0.90
0.90 + 0.10 = 1
1
Pr(Outcome ≤ x|D)
Outcome x ($Billion)
0
3
5
10.3
1
0.90
0.8
0.6
0.90
0.48
0.4
0.15
0.2
0.15
0
-1
0.48
2
5
8
11
Outcome ($Billion)
Chapter 4 – Making Choices
Slide 27 of 53
Deterministic Dominance
Original Tree
Accept $2 Billion
Penzoill-Texaco
FALSE
0.000
2.000
2.000
Decision
4.635
Texaco Accept $5 Biilion
Counteroffer $5 Billion TRUE
Chance
0.000
4.635
17%
0.169
5.0
5.000
20%
0.100
10.3
10.300
Medium Award
50%
0.249
5.0
5.000
Low Award
30%
0.150
0.0
0.000
High Award
Texaco Refuses Counteroffer
50%
Chance
0.000
4.6
20%
0.066
10.3
10.300
Medium Award
50%
0.166
5.0
5.000
Low Award
30%
0.100
0.0
0.000
High Award
branch
Texaco Counteroffers $3 Billion
33%
Decision
0.000
4.560
branch
Chapter 4 – Making Choices
TRUE
Chance
0.000
4.6
FALSE
0.000
3.0
3.000
Slide 28 of 53
Deterministic Dominance
Modified Tree
Accept $2 Billion
Penzoill-Texaco
FALSE
0.000
2.000
2.000
Decision
5.257
Texaco Accept $5 Biilion
Counteroffer $5 Billion TRUE
0.000
17%
0.169
5.0
5.000
Chance
5.257
20%
0.100
10.3
10.300
Medium Award
50%
0.249
5.0
5.000
Low Award
30%
0.150
2.5
2.500
High Award
Texaco Refuses Counteroffer
50%
Chance
0.000
5.3
20%
0.066
10.3
10.300
Medium Award
50%
0.166
5.0
5.000
Low Award
30%
0.100
2.5
2.500
High Award
branch
Texaco Counteroffers $3 Billion
33%
Decision
0.000
5.310
branch
Chapter 4 – Making Choices
TRUE
Chance
0.000
5.3
FALSE
0.000
3.0
3.000
Slide 29 of 53
Deterministic Dominance
Based on EMV analysis we still choose
the alternative “Counteroffer $5 Billion”
Max Result
Accept $2 Billion
EMV=
5.26
Counteroffer
2
EMV=
5.26
$5 Billion
Could we have made a decision
here without an EMV analysis ?
Chapter 4 – Making Choices
Slide 30 of 53
Deterministic Dominance
Formal Definition: Deterministic Dominance:
If the worst outcome of Alternative B is at least as good
as that of the best outcome of Alternative A, then
Alternative B deterministically dominates Alternative A.
•
Deterministic dominance may also be concluded by
drawing cumulative risk profiles and using the
definition:
Definition: Range of a Cumulative Risk Profile = [L,U],
where L= Smallest 0% point in Cumulative Risk Profile
and U= Largest 100% point in Cumulative Risk Profile
Chapter 4 – Making Choices
Slide 31 of 53
Deterministic Dominance
Deterministic dominance via cumulative risk
profiles:
Step 1: Draw cumulative risk profiles in one graph
Step 2: Determine range for each risk profile
Step 3: If ranges are disjoint or their intersections contain a single point
then deterministic
dominance
is present
Cumulative
Risk Profiles
Range 1: {2}
Revised Texaco-Penzoil Case
ƒ
ƒ
ƒ
1
Pr(Outcome ≤ x)
•
Range 2: [2.5,10.3]
0.8
0.6
0.4
2.5
10.3
0.2
0
-1
2
5
8
Outcome ($Billion)
Accept $2 Billion
Counteroffer $5 Billion and Refuse $3 Billion
11
Chapter 4 – Making Choices
Ranges 1 and 2
are disjoint. The
Objective is
Max Result,
hence Green CRP
deterministically
dominates the Red
one.
Slide 32 of 53
Stochastic Dominance: Example 1
Firm A: Original Tree
Accept $2 Billion
Penzoill-Texaco
FALSE
0.000
2.000
2.000
Decision
4.635
Texaco Accept $5 Biilion
Counteroffer $5 Billion TRUE
Chance
0.000
4.635
17%
0.169
5.0
5.000
20%
0.100
10.3
10.300
Medium Award
50%
0.249
5.0
5.000
Low Award
30%
0.150
0.0
0.000
High Award
Texaco Refuses Counteroffer
50%
Chance
0.000
4.6
20%
0.066
10.3
10.300
Medium Award
50%
0.166
5.0
5.000
Low Award
30%
0.100
0.0
0.000
High Award
branch
Texaco Counteroffers $3 Billion
33%
Decision
0.000
4.560
branch
Chapter 4 – Making Choices
TRUE
Chance
0.000
4.6
FALSE
0.000
3.0
3.000
Slide 33 of 53
Stochastic Dominance: Example 1
Firm B: Modified Tree
Accept $2 Billion
Penzoill-Texaco
FALSE
0.000
2.000
2.000
Decision
4.718
Texaco Accept $5 Biilion
Counteroffer $5 Billion TRUE
Chance
0.000
4.718
17%
0.169
5.0
5.000
20%
0.100
10.3
10.300
Medium Award
50%
0.249
5.2
5.200
Low Award
30%
0.150
0.0
0.000
High Award
Texaco Refuses Counteroffer
50%
Chance
0.000
4.7
20%
0.066
10.3
10.300
Medium Award
50%
0.166
5.2
5.200
Low Award
30%
0.100
0.0
0.000
High Award
branch
Texaco Counteroffers $3 Billion
33%
Decision
0.000
4.660
branch
Chapter 4 – Making Choices
TRUE
Chance
0.000
4.7
FALSE
0.000
3.0
3.000
Slide 34 of 53
Stochastic Dominance: Example 1
Based on EMV analysis we still choose
the alternative “Firm B”
Max Result
EMV=
4.63
Firm A
EMV=
4.72
EMV=
4.72
Firm B
Could we have made a decision
here without an EMV analysis ?
Chapter 4 – Making Choices
Slide 35 of 53
Stochastic Dominance: Example 1
Optimal Cumulative risk profiles in
“Firm A” Tree and “Firm B” Tree
Cumulative Risk Profiles: Firm A and Firm B
Pr(Outcome ≤ x)
1
0.8
0.6
0.4
0.2
0
-1
2
5
8
11
Outcome ($Billion)
Firm A
Firm B
Chapter 4 – Making Choices
Slide 36 of 53
Stochastic Dominance: Example 1
Pr(Outcome ≤ x| Firm B) ≤
Pr(Outcome ≤ x| Firm A)
or equivalently:
Pr(Outcome ≥ x| Firm B) ≥
Pr(Outcome ≥ x| Firm A)
1
Pr(Outcome ≤ x)
Note that for all
possible values of x:
Cumulative Risk Profiles: Firm A and Firm B
0.8
0.6
0.4
0.2
0
-1
2
5
8
11
Outcome ($Billion)
Firm A
Firm B
Hence the chances of winning with Firm B
are always better than that of Firm A.
Conclusion: Firm B stochastically dominates Firm A
Chapter 4 – Making Choices
Slide 37 of 53
Stochastic Dominance: Example 2
Firm A: Original Tree
Accept $2 Billion
Penzoill-Texaco
FALSE
0.000
2.000
2.000
Decision
4.635
Texaco Accept $5 Biilion
Counteroffer $5 Billion TRUE
Chance
0.000
4.635
17%
0.169
5.0
5.000
20%
0.100
10.3
10.300
Medium Award
50%
0.249
5.0
5.000
Low Award
30%
0.150
0.0
0.000
High Award
Texaco Refuses Counteroffer
50%
Chance
0.000
4.6
20%
0.066
10.3
10.300
Medium Award
50%
0.166
5.0
5.000
Low Award
30%
0.100
0.0
0.000
High Award
branch
Texaco Counteroffers $3 Billion
33%
Decision
0.000
4.560
branch
Chapter 4 – Making Choices
TRUE
Chance
0.000
4.6
FALSE
0.000
3.0
3.000
Slide 38 of 53
Stochastic Dominance: Example 2
Firm C: Modified Tree
Accept $2 Billion
Penzoill-Texaco
FALSE
0.000
2.000
2.000
Decision
6.005
Texaco Accept $5 Biilion
Counteroffer $5 Billion TRUE
Chance
0.000
6.005
17%
0.169
5.0
5.000
30%
0.150
10.3
10.300
Medium Award
60%
0.299
5.2
5.200
Low Award
10%
0.050
0.0
0.000
High Award
Texaco Refuses Counteroffer
50%
Chance
0.000
6.2
30%
0.100
10.3
10.300
Medium Award
60%
0.199
5.2
5.200
Low Award
10%
0.033
0.0
0.000
High Award
branch
Texaco Counteroffers $3 Billion
33%
Decision
0.000
6.210
branch
Chapter 4 – Making Choices
TRUE
Chance
0.000
6.2
FALSE
0.000
3.0
3.000
Slide 39 of 53
Stochastic Dominance: Example 2
Based on EMV analysis we still choose
the alternative “Firm C”
Max Result
EMV=
4.63
Firm A
EMV=
6.00
EMV=
6.00
Firm C
Could we have made a decision
here without an EMV analysis ?
Chapter 4 – Making Choices
Slide 40 of 53
Stochastic Dominance: Example 2
Optimal Cumulative risk profiles in
“Firm A” Tree and “Firm C” Tree
Cumulative Risk Profiles: Firm A and Firm C
Pr(Outcome ≤ x)
1
0.8
0.6
0.4
0.2
0
-1
2
5
8
11
Outcome ($Billion)
Firm A
Firm C
Chapter 4 – Making Choices
Slide 41 of 53
Stochastic Dominance: Example 2
Pr(Outcome ≤ x| Firm C) ≤
Pr(Outcome ≤ x| Firm A)
or equivalently:
Pr(Outcome ≥ x| Firm C) ≥
Pr(Outcome ≥ x| Firm A)
1
Pr(Outcome ≤ x)
Note that for all
possible values of x:
Cumulative Risk Profiles: Firm A and Firm C
0.8
0.6
0.4
0.2
0
-1
2
5
8
11
Outcome ($Billion)
Firm A
Firm C
Hence the chances of winning with Firm C
are always better than that of Firm A.
Conclusion: Firm C stochastically dominates Firm A
Chapter 4 – Making Choices
Slide 42 of 53
Stochastic Dominance: Examples 1 & 2
Cumulative Risk Profiles: Firm A and Firm B
Commonality CRP plots:
• Cumulative risk profiles in
both plots do not cross
Pr(Outcome ≤ x)
1
0.8
0.6
0.4
0.2
0
-1
• The CRP that is toward the
“right and below”
stochastically dominates
2
5
8
11
Outcome ($Billion)
Firm A
Firm B
Cumulative Risk Profiles: Firm A and Firm C
• The objective in both plots
is to Maximize the Result
• What if the objective is to
Minimize the Result?
Pr(Outcome ≤ x)
1
0.8
0.6
0.4
0.2
0
-1
2
5
8
11
Outcome ($Billion)
Firm A
Chapter 4 – Making Choices
Firm C
Slide 43 of 53
Making Decisions with Multiple Objectives
•Two Objectives:
Amount of
Fun
Making Money
(Measured in $)
Fun
Having Fun
(Measured on
Constructed attribute
scale):
Best(5), Good(4),
Middle(3), Bad(2),
Worst (1)
Job
Decision
Overall
Satisfaction
Salary
Amount
of Work
Chapter 4 – Making Choices
Slide 44 of 53
Making Decisions with Multiple Objectives
Consequences
Fun
level
Forest Job
Salary
Fun Level
5 (0.10)
$2600.00
5
4 (0.25)
$2600.00
4
$2600.00
3
$2600.00
2
$2600.00
1
$2730.00
3
$2320.50
3
$2047.50
3
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)
Chapter 4 – Making Choices
Slide 45 of 53
Analysis Salary Objective
Salary*Prob
1.00
$2,600.00
E[Salary]=
Conclusion:
• Forest Job preferred
Over In-Town job
• CRP’s cross. Hence,
No Stochastic
Dominance
In-Town Job
Prob
0.15
0.50
$2,600.00
Salary*Prob
$307.13
$1,160.25
0.35
E[Salary]=
$955.50
$2,422.88
$2,400
$2,800
1
Pr(Outcome ≤ x)
Salary
$2,047.50
$2,320.50
$2,600.00
$2,730.00
Forest Job
Prob
0.8
0.6
0.4
0.2
0
$2,000
$2,200
$2,600
$3,000
Salary
Forest Job
Chapter 4 – Making Choices
In-Town Job
Slide 46 of 53
Fun Level Objective
Conclusion:
• Forest Job preferred
Over In-Town job
• CRP’s cross. Hence,
No Stochastic
Dominance
Fun Level*Prob
10.0%
22.5%
24.0%
5.0%
0.0%
61.5%
In-Town Job
Prob
Fun Level*Prob
1.00
60.00%
E[Fun Level]=
60.00%
1
Pr(Outcome ≤ x)
Outcome
5 -BEST
4 -GOOD
3 - MIDDLE
2 - BAD
1 - WORST
Forest Job
Fun Level
Prob
100.00%
0.10
90.00%
0.25
60.00%
0.40
25.00%
0.20
0.00%
0.05
E[Fun Level]=
0.8
0.6
0.4
0.2
0
0%
20%
40%
60%
80%
100%
120%
Fun Level
Forest Job
Chapter 4 – Making Choices
In-Town Job
Slide 47 of 53
Multiple Objective Analysis
• It is clear from both objective analyses that the Forest-Job
is the strongly preferred, although neither Stochastic
nor Deterministic Dominance can be observed in them.
• Before trade-off analysis can be conducted both objectives
have to be measured on a “comparable” scale.
Chapter 4 – Making Choices
Slide 48 of 53
Multiple Objective Analysis: Construct 0-1 Scale
• Having Fun Objective already has a 0-1 scale:
Transformed 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%)
• Construct 0-1 scale for Salary Objective:
$2730.00=100%, $2047.50=0%
Intermediate dollar amount X:
X − $2047.50
⋅100%
$2730 − $2047.50
Chapter 4 – Making Choices
Slide 49 of 53
Multiple Objective Analysis: Assess Trade-Off
ks = weight for salary
ks + k f = 1
k f = weight for fun
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. Hence: k s = 1.5 ⋅ k f
1
2

1.5 ⋅ k f + k f = 1  k f = 2.5 = 5
 ks + k f = 1
⇔
⇔

3 3 3
 k s = 1.5 ⋅ k f
k s = 1.5 ⋅ k f
k s = ⋅ =
2 5 5

Chapter 4 – Making Choices
Slide 50 of 53
Multiple Objective Analysis: Convert Scales
Consequences
Salary (0.6)
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 (0.4)
40 hours (0.35)
34 hours (0.50)
30 hours (0.15)
Chapter 4 – Making Choices
Slide 51 of 53
Multiple Objective Analysis: Combine Objectives
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%
Chapter 4 – Making Choices
Slide 52 of 53
Analysis Overall Satisfaction
Forest Job
Overall Satisfaction
Prob
OS*Prob
88.57%
0.10
8.9%
84.57%
0.25
21.1%
72.57%
0.40
29.0%
58.57%
0.20
11.7%
48.57%
0.05
2.4%
E[OS]=
73.2%
• CRP’s do not cross.
Hence, Stochastic
Dominance present.
1
Pr(Outcome ≤ x)
Conclusion:
• Forest Job preferred
Over In-Town job
In-Town Job
Overall Satisfaction
Prob
OS*Prob
84.00%
0.35
29.40%
48.00%
0.50
24.00%
24.00%
0.15
3.60%
E[OS]=
57.00%
0.8
0.6
0.4
0.2
0
0%
20%
40%
60%
80%
100%
Overall Satisfaction
Forest Job
Chapter 4 – Making Choices
In-Town Job
Slide 53 of 53