Transcript Slide 1
Chapter 12 Risk Attitude and Utility Theory
Utility
Theory Maximize Expected Utility
Motivation
Rescale all values to between 0 and 1 based on risk attitude
Certainty Equivalent
Manage
Risk – Utility Theory explains
Insurance
Partnership
Risk Aversion
assessment
Standard function with parameters
Direct assessment
Problems with assessment
Critique
Chapter 12
Utility Theory: Paradox and Prospect Theory
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Utility Theory - Overview
Incorporates
decision-maker’s risk attitude Value of
alternatives cannot be captured by E(X) because risk attitude
Certainty Equivalent (CE): The payoff amount we would accept
in lieu of under-going the uncertain situation, taking a gamble.
Risk Premium: A situation’s risk premium (RP) is the difference
between expected payoff (EP) and certainty equivalent (CE)
Risk Aversion: CE < E(X) Concave function
Profit – prefer sure profit that is less than E(X) of a profit gamble
Cost – prefer sure cost that is higher than E(Y) of a loss gamble
Risk
Prone: CE > E(X) Try for more money than average
With high guaranteed minimum, may be will to try for home run even if
offered a value somewhat higher than the expected value
Chapter 12
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TV show “Deal or No Deal” – Activate WEB GAME
Game Description and Risk Attitude
Chapter 12
Game starts with 26 suitcases, each with a dollar value ranging from $0.01 to $1M
Contestant picks one case as his to be opened at the end, and selects 6 cases to be
opened immediately. As they are opened, their stored dollar value is deleted from
the list.
After opening these six cases, the player is offered a sure amount or the option to
keep playing.
At each the step the number of boxes to open at this stage decreases until only
one at a time is opened each time.
For example, I played the game until there were 3 unopened boxes left ($25,
$50,000, and $1,000,000). One of these boxes I had selected as my box.
The offer was for $220,000.This is much less than the expected value
E(X) = (1/3)(25)+ (1/3)((50,000) + (1/3)(1,000,000) = (1/3) (1,000,025) =
$350,008
If you prefer to take less than the expected value risk averse
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Simple gamble (lottery) – Two Outcomes
Two boxes are left: $10,000 and $20,000.
You have a 50–50 chance of winning $10,000 or $20,000 depending upon what is
in your selected box. E(X) = $15,000
Would you accept an offer of $12,000 or continue Yes __ or No __
Would you accept an offer of $14,500 or continue Yes __ or No __
What is the minimum you would accept to stop the game? ___ this is Certainty
Equivalent (CE) of the gamble
If the CE is less than the expected value, you are a risk averse.
Risk Premium is the difference between your CE and the expected value
Assume you would accept $12,500
Risk Premium = E(X) – CE = $15,000 - $12,500 = $2,500
If the CE is more than the expected value, you are a risk prone.
Gamble is a set of uncertain outcomes with probabilities
Not necessarily just 2 outcomes
All outcomes may not have the same probability.
Chapter 12
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Expected Value is Inadequate
Generally Risk Averse
Extremely
Rare Catastrophic Events
Law of averages no help for individual
Motivates insurance industry
Companies may self-insure up to extreme costs
Can tolerate moderate risks that are not devastating
Deductible on insurance
Large
investments relative to size of company
Cannot afford the loss
Has need for certain profit more than high upside
potential sell development rights
Chapter 12
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Determine function U(X)
U(X)
function may be determined
Select from standard functional forms – exponential
or logarithmic and estimate parameters from
questions
Brute force graphing of the results of series of
Certainty Equivalent responses to gambles
Assume
exponential U(X) = 1-e(-x/15000)
U(20,000) = 1-e(-20000/15000) = .7364
Chapter 12
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Calculate Expected Utility E(U(X)) and Not E(X)
Assume exponential utility function for this example
(0.5)
$10,000
E(X) = 0.5($10000) + 0.5($20000) = $12,500
(0.5)
$20,000
Assume U(X) = 1-e(-x/15000)
U(10,000) = 1-e(-10000/15000) = 0.4866
U(20,000) = 1-e(-20000/15000) = 0.7364
E(U(X)) = 0.5(0.487) + 0.5(0.736) = 0.6115
invert exponential function
CE(0.6115) = $14,182
$14,250 would be preferred to gamble
$14,000 would not be preferred
Chapter 12
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(0.5)
0.4866
0.7364
(0.5)
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Invert function U(X) U-1(X)
0.6115
= 1-e(-x/15000)
e(-x/15000) =
ln
of both sides (-X/15000) = ln(0.3885) = -0.9455
X
= 15000 (0.9455) = 14,182
CE
of 50-50 gamble of 10,000 and 20,000 is $14,182
Risk
Chapter 12
1- 0.6115 = 0.3885
Premium = $15,000 - $14,182 = $818
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Risk Aversion
CONCAVE for Negative (Cost)
Prefer to PAY More than expected value to be sure Cost does not get
too large.
50-50 COST Gamble $90,000 or $10,000 => prefer sure $60,000 cost
Basis for Insurance Industry
CONCAVE for Maximization (Profit)
Prefer to EARN less than expected value to be sure of earning at least
that profit.
50-50 PROFIT Gamble $90,000 or $10,000
===> would prefer sure $40,000 profit to the gamble.
Basis for Selling off bank Debts such as mortgages
Chapter 12
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Risk Aversion – Concave Curve
Profit
Cost
Risk Aversion - Cost
1.2
1.2
1
1
0.8
0.8
0.6
Series1
Utility
Utility
Risk Aversion - Profit
0.6
0.4
0.4
0.2
0.2
0
0
20000
40000
60000
80000
0
100000
-100000
Profit
-80000
-60000
-40000
-20000
0
Cost
0.5 for $10,000 or 0.5 for $90,000
E(X) = $50,000
Accept $40,000 = CE U(40000)=.5
Risk Premium $10,000
Chapter 12
Utiltiy
0.5 for -$10,000 or 0.5 for -$90,000
E(X) = -$50,000
Accept -$60,000 = CE U(60000)=.5 Risk
Premium $10,000
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Activity – Risk Aversion
Pay to avoid loss
___________________________________________
___________________________________________
Accept less profit but be sure (or more certain) of profit
___________________________________________
___________________________________________
Other Examples
___________________________________________
___________________________________________
___________________________________________
___________________________________________
Chapter 12
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Utility Function Assessment
Brute Force
Series
of 3 lotteries
Obtain Certainty Equivalent of utility values of .25, .5,
& .75
Extrapolate over the range of 0 to 1.
Create your own utility function for $0M to $10M
set U(min) = 0 & U(max) = 1
set U(0) = 0 & U(10) = 1
Chapter 12
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Decision tree Boss Controls automation investment
Need utility function for four values (0.8, 1.9, 8.5, 10)
40.0%
9.9
Take Rate
5.86
60.0%
50% Take
16.5
0
1.9
40.0%
13.8
Take Rate
6.32
60.0%
50% Take
23
0.4
0.8
30% Take
FALSE
Low
Automation Investment
-8
0
8.5
How Much
6.32
30% Take
High
Chapter 12
TRUE
-13
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10
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Certainty equivalents
(0.5)
XL
≈ CE50
(0.5)
XH
(0.5)
XL
≈ CE25
(0.5)
(0.5)
CE50
XH
≈ CE75
(0.5)
Chapter 12
CE50
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BC automation investment certainty
equivalent for 0.50, 0.25, and 0.75 utility
(0.5)
0
(0.5)
(0.5)
(0.5)
(0.5)
(0.5)
Chapter 12
10
≈ CE50 = 3.6
0
≈ CE25 = 1.6
3.6
10
≈ CE75 = 6.2
3.6
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Utility assessment for BC automation
investment example
1
Utility Score
0.8
0.6
0.4
0.2
0
0
2
4
Dollars
6
8
10
Value
0
0.8
1.6
1.9
3.6
6.2
8.5
10
Utility
0
0.13
0.25
0.29
0.5
0.75
0.90
1
Table 12-4: Utility scores for BC automation investment
Chapter 12
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Decision tree for BC automation investment using
utility function
FALSE
40%
0
0.13
60%
0
1.0
40%
0.29
0.4
0.29
60%
0.9
0.6
0.90
30% Take Rate 0.13
Take Rate
0.652
High
50% Take Rate 1.0
Automation Investment
Decision
0.657
30% Take Rate
TRUE
Take Rate
0.657
Low
50% Take Rate
CE: $5.142 (0.652) vs. $5.196 million (0.657)
Chapter 12
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Utility Function: Simplify Interview
Goal: simplify
interview process One question for each
parameter
Exponential U(x) = 1 – exp(-x/R) Constant Risk Aversion
Implies that the magnitude of cash on hand does not affect
attitude towards Risk
R = Risk Tolerance
R = Sum of Money about which you are indifferent between
a 50-50 chance of gaining the whole sum, R and losing the
half sum, R/2
Chapter 12
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Constant Risk Aversion
Risk Premium is Constant
R=35
Chapter 12
U(X) = 1 – exp(-x/35)
50-50 Gamble
Expected
Certainty
Risk
Between ($)
Value
Equivalent
Premium
10,40
25
21.88
3.12
20,50
30,60
40,70
35
45
55
31.88
41.88
51.88
3.12
3.12
3.12
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Determine R = Risk Tolerance
50- 50 gamble to determine R
Personal
Chapter 12
Win
Lose
$20
$200
$2,000
$20,000
$10
$100
$1,000
$10,000
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Would take
Gamble Y/N
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Determine R = Risk Tolerance
50- 50 gamble to determine R
Corporate
Chapter 12
Win
Lose
$20,000
$200,000
$2,000,000
$20,000,000
$10,000
$100,000
$1,000,000
$10,000,000
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Would Corp.
Gamble Y/N
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ENCO Project Selection
An energy company will select one of two projects. If the
company chooses Project A, it will undertake the development
of a new power plant in one developing country. The company
estimates that the investment cost will be $50 million and
total revenue for five-year after the operation cost will be $80,
$90, or $110 million. There is a 20% chance that the local
government will take over the plant once it is finished due to
some legal issues and just repay the original investment cost
($50 million). Project B also requires a $50 million investment.
Total revenue for five-year after the operation cost will be $66,
$80, or $90 million. In this second country, there is no chance
that the government will take over this project when it is
completed.
Chapter 12
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Maximize expected value
Decision tree for ENCO project selection
0.2
20.0%
Yes
0
50
TRUE
Project A
Government Take Project?
34.4
-50
30.0%
Low
80
43
0
40.0%
Medium
90
30.0%
High
110
Project
30
Revenue
80.0%
No
0.24
0.32
40
0.24
60
Project Decision
34.4
Low
Project B
FALSE
0
66
16
40.0%
0
80
30
30.0%
0
90
40
Revenue
28.8
-50
Medium
High
Chapter 12
30.0%
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Figure 12.11: Cumulative risk profile for ENCO project
selection Redraw diagram without student version
1
Cumulative Probability
0.8
0.6
Project A
Project B
0.4
0.2
Chapter 12
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60
50
40
30
20
10
0
-10
0
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A utility score was calculated for each value.
R= $30 million
EU(Project A) = 0*20% + (0.632*30%+0.736*40%+0.865*30%)*80% = 0.595
EU(Project B) = 0.413*30%+0.632*40%+0.736*30% = 0.598
Chapter 12
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Decision tree for ENCO: using expected utility
Yes
Project A
FALSE
-50
20.0%
0
0.000
50
Government Take Project?
0.595
No
Project
80.0%
0
Revenue
0.744
40.0%
Medium
90
30.0%
High
110
0
0.632
0
0.736
0
0.865
Project Decision
0.598
Project B
Low
TRUE
-50
Revenue
0.598
Medium
High
Chapter 12
30.0%
80
Low
30.0%
66
0.3
0.413
40.0%
80
30.0%
90
0.4
0.632
0.3
0.736
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Impact of risk tolerance on certainty
equivalent
Certainty Equivalent
35.00
30.00
25.00
20.00
15.00
Project A
10.00
Project B
5.00
0.00
1
10
20
30
40
50
60
Risk Tolerance
Chapter 12
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Risk Sharing Strategies
Buy
insurance
Partnerships
Example: Transmission plant by GM and Ford
Joint ventures with foreign companies
Diversification
Diversification with independent investments
Example: Dual sourcing, investment in different stocks
Diversification with dependent investments (Hedging)
Chapter 12
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Risk Sharing through Buy Insurance:
Project Selection
Chapter 12
Buy insurance against a government takeover of project A.
$4 million premium the company will receive a $10
million payment if the government takes the project.
Add new decision branch to tree as part of Project A. (top
of next tree)
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ENCO: insurance reporting expected value
20.0%
Yes
FALSE
Buy Insurance
-4
0
6
60
Government Take Project?
32.400
30.0%
80
Revenue
39.000
Medium 40.0%
90
Low
80.0%
0
No
TRUE
Project A
-50
High
Insurance Decicion
34.400
20.0%
Yes
TRUE
Do Not Buy
0
50
Government Take Project?
34.400
30.0%
110
80.0%
0
Project Insurance
30.0%
80
Revenue
43.000
Medium40.0%
90
0.24
30
30.0%
110
0.24
60
High
Project Decision
34.400
Low
Project B
FALSE
-50
Revenue
28.800
Medium
High
Chapter 12
30.0%
66
0
16
40.0%
80
30.0%
0
30
0
40
Chelst & Canbolat
90
Value Added Decision Making
0
36
0
56
0.2
0
Low
No
0
26
03/06/12
0.32
40
30
ENCO : insurance option reporting CE
TRUE
Buy Insurance -4
Insurance Decision
27.630
Project A
TRUE
-50
FALSE
Do Not Buy
0
Project Decision
27.630
20.0%
0.2
60
6
Government Take Project?
27.630
30.0%
Low
80
80.0%
Revenue
No
0
36.830
Medium 40%
90
Yes
High 30.0%
110
20.0%
0
50
0
Government Take Project?
27.107
30.0%
Low
80
80.0%
Revenue
No
0
40.83040.0%
Medium
90
High
Project B
Chapter 12
FALSE
-50
0.32
36
0.24
56
Yes
Project Insurance
30.0%
66
Revenue
27.322
40.0%
Medium
80
30.0%
High Chelst & Canbolat
90
Value Added Decision Making
Low
0.24
26
0
16
0
30
0
40
30.0%
110
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0
30
0
40
0
60
31
Decision Tree for Insurance company
Tree Not in Textbook
Yes
TRUE
Sell
20.0%
0.2
-10
-6
Government Take Project?
4
2
80.0%
No
0
Insurance Company
0.8
4
Decision
2
Do Not Sell
Chapter 12
FALSE
0
0
0
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Share Risk through Partnership
Concept: Commit
to only a percentage of the cost
(liability) and accrue the equivalent percentage of
revenue
Common amongst insurance companies to reinsure and
share risk of catastrophic event – Lloyd’s of London
Decision Tree Method – reduce costs and revenues
proportionate to share and calculate new utility scores.
Chapter 12
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Share Risk through Partnership:
Project Selection
Chapter 12
Outside investor shares 50% of cost and benefit of each
project.
By sharing the investment, the company can now be
involved in more projects. Let’s evaluate a 50% share. As a
result the company can invest in BOTH A and B.
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ENCO with 50% partnership reporting expected profit
Government takes away project A?
TRUE
34.400
-50 +
Project Decision
34.400
FALSE
Revenue
Project B
+
28.800
-50
30.0%
Low
33
20.0%
Revenue
from
B
Yes
14.400
25
40.0%
Medium
40
30.0%
High
45
FALSE
Government
takes
away
project
A?
50% of Both Projects
31.600
-50
Project A
Project - Diversification
0
8
0
15
0
20
30.0%
33
Revenue from B
29.400
40.0%
Medium
40
30.0%
High
45
Low
Low
No
Chapter 12
80.0%
0
30.0%
40
0
23
0
30
0
35
Revenue from A
35.900
40.0% Revenue from B
Medium
34.400
45 +
Revenue
from B
30.0%
High
44.400
55 +
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ENCO decision tree with partnership option
reporting certainty equivalent
Project A
Project - Diversification
FALSE
-50
Government Take Project?
+
27.107
FALSE
-50
+
Project Decision
29.450
Project B
Revenue
27.322
0.06
8
30.0%
33
Revenue from B
20.0%
Yes
14.032
25
Medium 40.0%
40
30.0%
High
45
Government Take Project A?
29.450
Low
50% of Both Projects
TRUE
-50
0.08
15
0.06
20
30.0%
33
Revenue from B
29.032
40.0%
Medium
40
30.0%
High
45
Low
Low
No
Chapter 12
80.0%
0
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30.0%
40
Revenue from A
34.966
Medium 40.0%
+
45
30.0%
High
+
55
0.072
23
0.096
30
0.072
35
Revenue from B
34.032
Revenue from B
44.032
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Optimize PERCENT Share of Project
Can optimize based on risk attitude
Percent share of option A
Percent share of option B
Different companies with different risk attitudes
will have different optimum preferences
Chapter 12
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Phillips Petroleum and Onshore US Oil Exploration
Prospect Ranking: R equal to $25 million
Expected Value Basis
Prospect
Rank EV 100% share
Certainty Equivalent Basis
Rank
Optimal
Share
Certainty
Equivalent
South Louisiana
1
18.6
8
12.5%
0.6
Norphlet
2
16.5
6
12.5%
0.8
Wilcox
3
11.8
5
25%
0.8
Frio
4
10.8
7
12.5%
0.7
Vicksburg
5
4.0
4
75%
1.0
Yegua Deep
6
3.0
3
100%
1.0
Smackover
7
2.5
1
100%
1.8
Yegua Shallow
8
2.2
2
100%
1.1
Chapter 12
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Following two slides not in the textbook
Impacts
of the target setting on the risk attitude
Consistent with prospect theory
Chapter 12
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Risk Attitude and Targets
Component Cost Target $46 – Goal: Minimize Cost
You are near completion on a design and are almost certain that you will
be able to reach the variable cost target of $46. A suggestion comes
along that by changing materials the component should be easier to
manufacture with the cost dropping to $40 per part. However, time is
short and there is a concern that without proper testing of Design for
Manufacture the change could, in fact, increase the cost to $47 per
component and miss the target. Would you make the change assuming
the two outcomes were equally likely?
.5
$46
OR
.5
Chapter 12
$47
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Value Added Decision Making
$40
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Risk Attitude and Targets
Component Cost Target $40
You are near completion on a design and are almost certain that you will
NOT be able to reach the variable cost target of $40 and that the part
will cost $42. A suggestion comes along that by changing materials the
part should be easier to manufacture with the cost dropping to $40 per
part. However, time is short and there is a concern that without proper
testing of Design for Manufacture the change could, in fact, increase the
cost to $47 per part and miss the target. Would you make the change
assuming the two outcomes were equally likely?
.5
$42
OR
.5
Chapter 12
$47
Chelst & Canbolat
Value Added Decision Making
$40
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Problems With Carrying Out Risk Aversion
Assessment in Real-World
Managers
are uncomfortable with this abstract concept
Managers are inconsistent -- Prefer ranges for CE and not
specific values
People are inconsistent
risk prone for small values ( i.e. will buy lottery tickets)
risk averse for large values (i.e. buy insurance)
May be risk prone if values are negative but risk averse if all
values are positive and vice versa
Attitudes towards risk are influenced by artificial targets
Target secure – take no gamble to improve
Target at risk – take extreme gambles to reach target
Chapter 12
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Risk Assessment In Practice
(Done Less than 10% of the time)
Howard Rule of Thumb for R in major decisions:
R = 6.4% of sales or 1.25 Net Income – Your company value?
Example: R = $1 Billion dollar investment or purchase decision for
a company with over $15 Billion in sales.
First
carry out analysis without Risk Aversion
Display Comparative Risk Profiles and discuss
Insert exponential function & determine whether or not
the optimal solution is Sensitive to Risk Tolerance Value
within a realistic range for your company.
If optimal decision can change within a reasonable range
of R then assess actual utility function.
Chapter 12
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Assumptions of Utility Theory
Expectation: U(x1,p1;…;xn,pn)
= p1u(x1)+…+pnu(xn)
Asset Integration: The domain of the utility function is final
states (which include one’s asset position) rather than gains
or losses.
Risk Aversion: u is concave (u’’< 0)
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Critique of Utility Theory
Expectation Assumption
- Allais paradox
Problem 1: Choose between
A: $1 Million with certainty
Problem 2: Choose between
X: $1 Million with probability 0.11
0 with probability
0.89
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B: $1 Million with probability 0.89
$5 Million with probability 0.10
0 with probability
0.01
Y: $5 Million with probability 0.10
0 with probability
0.90
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Critique of Utility Theory- Actual preferences
Expectation Assumption
- Allais paradox
Problem 1: 82% choose A
A: $1 Million with certainty
B: $1 Million with probability 0.89
$5 Million with probability 0.10
0 with probability
0.01
Problem 2: 83% Choose Y
X: $1 Million with probability 0.11
0 with probability
0.89
Y: $5 Million with probability 0.10
0 with probability
0.90
Conclusion: People overweight “certain” outcomes more than merely probable
outcomes.
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Utility Function Stability and Accuracy
Affected
by mood
Quality of life QOL
Assessment of QOL living with serious disease.
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Prospect Theory (Kahneman and Tversky)
Example not in textbook
Value Function
◦ Concave for gains and Convex for losses;
◦ Steeper for losses than gains (Loss Aversion)
v(x)
1200
800
400
0
-1200
-800
-400-400 0
v(x)
400
800
1200
-800
-1200
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Prospect Theory (Kahneman and Tversky)
Example not in textbook
Weighting Function (The impact of events on desirability
of prospects, not merely the likelihood of events)
◦ Overweight certainty and small probabilities;
◦ Underweight moderate and high probabilities;
1
0.8
0.6
w+(p)
w-(p)
0.4
0.2
0
0
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0.2
0.4
0.6
0.8
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Prospect Theory (Kahneman and Tversky)
Example not in textbook
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Fourfold risk attitude:
Small
Probabilities
Moderate or high
probabilities
Gains
risk seeking
risk averse
Losses
risk averse
risk seeking
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Remember The Goal of Analysis
Update Intuition of Decision Maker
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Where are we headed next?
Analytic Tools
completed! YEAH! No more new software!
Data input: forecasting biases Structured expert interview
Decision making biases awareness
Ethics Decisions
Negotiations
Strategic Decisions & Scenario Planning
Final Projects
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TREES-A Decision-Maker's Lament
I think that I shall never see
A decision as complex as that treeA tree with roots in ancient days
(At least as old as Reverend Bayes);
A tree with flowers in its tresses
(Each flower made of blooming
guesses);
A tree with utiles at its tips
(Values gleaned from puzzled lips);
A tree with trunk all gnarled and twisted
With axioms by Savage listed;
A tree with stems so deeply nested
Intuition's completely bested;
A tree with branches sprouting branches
And nodes declaring what the chance is; A tree with branches in a tangle
Impenetrable from any angle;
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TREES-A Decision-Maker's Lament
A tree that tried to tell us "should"
Although its essence was but "would";
A tree that did decision hold back
'Til calculation had it rolled back.
Decisions are reached by fools like me,
But it took a consultant to make that tree.
Michael H. Rothkopf
with apologies to Joyce Kilmer and to competent,
conscientious decision analysts everywhere
Operations Research Vol. 28, No. 1,
January-February 1980
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