AP / ADMS 3300

DECISION ANALYSIS

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Week 8
Chapter 14

Risk Attitudes

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 Session 8- Objective
This session covers the problems associated with risk
and return trade-offs.
The preference side of decision analysis
How can we model a decision maker’s preferences?
Problems associated with risk and return tradeoffs
Introduce an approach called utility theory that allows
us to incorporate riskiness
Develop the basic tools of utility theory for risky
decision making
How to deal with some decision making biases

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Risk
Basing decision on expected monetary values
(EMV’s) is convenient, but it can lead to decisions
that may not seem intuitively appealing.
Game 1 Win $30 with probability 0.5
Lose $1 with probability 0.5
Game 2 Win $2000 with probability 0.5
Lose $1900 with probability 0.5
Which one would you prefer to play?

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 Risk
For example, how would you decide on which gamble to
take?

Game 1:
Win $30 with probability 0.5
Lose $1 with probability 0.5
Game 2:
Win $2,000 with probability 0.5
Lose $1,900 with probability 0.5

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Game 1 has expected value of $14.5. Game 2, on the
other hand, has an expected value of $50.00.
Most of us, however, would consider Game 2 to be
riskier than Game 1.
Many of the examples and problems that we have
considered so far have been analyzed in terms of
expected monetary value.
EMV, however, does not capture risk attitude.

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Utility Function
Individuals who are afraid of risk or are
sensitive to risk are called risk-averse.
Utility function represents a way to translate
dollars into “utility units”.
If we take some dollar amount (X), we can
locate that amount on the horizontal axis.
A utility function can be specified in terms of
graph, table or mathematical expression.

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 Risk
Risk-averse: afraid of or sensitive to risk
Risk-aversion can be
understood as a utility
function, which can be
presented as an
upward sloping,
concave curve
(opening downward).

On the graph, $2.75

has a utility value of
$3.75. So losing $2.75
would feel like losing
$3.75.
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Risk Attitude
A typical utility curve is upward sloping and
concave
( the curve opens downward).
An upward sloping utility curve makes fine
sense: it means that more wealth is better than
less wealth, everything else being equal.
Concavity in a utility curve implies that an
individual is risk-averse.
Generally, if you would trade a gamble for a
sure amount that is less than the expected value
of the gamble, you are risk-averse. Purchasing
insurance is an example of risk-averse behavior.

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Risk Attitudes
A convex (opening upward) utility curve indicates risk
seeking behavior.
The risk seeker might be eager to enter into a gamble.
An individual can also be risk-neutral. Risk neutrality
is reflected by a utility curve that is simply a straight
line.

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 Risk

Attitudes
Risk-averse (concave curve) decision maker
Shies away from gambling
Doesn’t like to lose—overestimates value of losing
Risk-seeking (convex curve) decision maker
Eager to enter a gamble
Overestimates value of winning
Risk-neutral (linear) decision maker
Ignores risk
Maximizing EMV equals maximizing expected utility

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 Risk Attitudes
The three types of risk attitude utility functions:

Convex

Concav
e

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 Investing in the Stock Market
If we can translate a utility function from dollars to
utility, how should we use it?
A utility function should help us choose from among
alternatives that have uncertain payoffs.
Instead of maximizing expected value, the decision
maker should maximize expected utility.
The best choice then should be the action with the
highest expected utility.

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 Stock Market Example
Recall from Chapters 5 and 12 the problem of the investor who has funds
that he wishes to invest. He has three choices:
 High-risk stock
 Low-risk stock
 Savings account that would pay $500
If he invests in the stocks, he must pay a $200 brokerage fee.
 If the market goes up, he will earn $1,700 from the high-risk stock and
$1,200 from the low risk stock.
 If the market stays at the same level, his payoffs for the high- and low-
risk stocks will be $300 and $400, respectively.
 If the stock market goes down, he will lose $800 with the high-risk
stock but still earn $100 from the low-risk stock.
The probabilities that the market will go up, stay the same, or go down are
0.5, 0.3, and 0.2, respectively.

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A utility function in tabular form
Wealth ($) Utility Value
1500 1.00
1000 0.86
500 0.65
200 0.52
100 0.46
-100 0.33
-1000 0.00

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 Stock Market Example
In the decision tree (or influence-diagram) payoff table,
the net dollar payoffs are replaced by the corresponding
utility values and the analysis is performed using those
values. What’s the best choice based on EMV? EU?

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 Stock Market Example
The problem of interpretation:
What does it mean to say “EU equals 0.652”?
There is no ready interpretation of a unit of utility.
Even more problematic is trying to interpret the
difference between EUs.
We saw that the EU of the savings account was a “close”
second at 0.650, only 0.002 utility units less than the
low-risk stock. But what does a difference of 0.002
mean? Is it significant?

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 Expected Utility, Certainty Equivalents, and Risk
Premiums

One very common solution to interpreting utility is to

translate EU into dollars.
Certainty equivalent (CE): amount of money that is
equivalent in your mind to a given situation that
involves uncertainty
E.g., the least you would sell your position for in a
gamble. Below that dollar amount, “No deal.”

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 Expected Utility, Certainty Equivalents, and Risk
Premiums

Risk premium: how much you are willing to “give up”

to avoid the risk inherent in an uncertain situation.
Related to certainty equivalent as:
Risk Premium = EMV – CE
EMV = Risk Premium + CE

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 Expected Utility, Certainty Equivalents, and Risk
Premiums
For example, the EMV in the stock investment case is
$580. If you would sell the opportunity for $250, then that
is your CE. What is your risk premium, that is, how much
would you give up to avoid the risk of investing in the
stock market?
Risk Premium = $580 – $250

Risk Premium = $330

You are definitely risk-averse!

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 Expected Utility, Certainty Equivalents, and Risk
Premiums

Note that EU = U(CE). The

decision maker should be
indifferent in choosing
between them.

© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Risk Attitudes
For risk-neutral person, maximizing EMV is the
same as maximizing expected utility. This makes
sense; someone who is risk-neutral does not care
about risk and can ignore risk aspects of the
alternatives that he or she faces. Thus, EMV is a
fine criterion for choosing among alternatives,
because it also ignores risk.
Although most of us are not risk-neutral, it is
often reasonable for a decision maker to assume
that his or her utility curve is nearly linear in the
range of dollar amounts for a particular decision.
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Certainty Equivalent
Certainty equivalent is the amount of money that
is equivalent in your mind to a given situation that
involves uncertainty.
suppose you face the following gamble:
Win $2000 with probability 0.50
Lose $20 with probability 0.50
Now imagine that one of your friends is interested in
taking your place.

What will you do?

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After thought and discussion, you conclude
that the least you would sell your position for is
$300. If your friend cannot pay that much, then
you would rather keep the gamble.
Your certainty equivalent for the gamble is
$300. This is sure thing; no risk is involved.
We can rank the investments by their certainty
equivalents.
Risk Premium:
The risk premium is defined as the difference
between the
EMV and the certainty equivalent.
Risk Premium = EMV - Certainty Equivalent

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Consider the gamble between winning $2000 and
losing $20, each with probability 0.50. The EMV of
this gamble is $990. On reflection, you assessed
your certainty equivalent to be $300, and so your
risk premium is
Risk Premium= $990 - $300
= $690
Because you were willing to trade the gamble for
$300, you were willing to “give up” $690 in expected
value in order to avoid the risk inherent in the
gamble.

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Now, we can pull all of pieces together in graph.
Imagine a gamble has expected utility Y. The value
Y is in utility units, so we must first locate Y on the
vertical axis. Trace a horizontal line from the
expected utility point until the line intersects the
utility curve. Now drop down to the horizontal axis
to find the certainty equivalent. The difference
between the expected value and the certainty
equivalent is the risk premium.
In any given situation, the certainty equivalent,
expected value, and risk premium all depend on
two factors: the decision maker’s utility function
and the probability distribution for the payoffs.

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Utility Function Assessment
There are two utility-assessment approaches based on
the idea of certainty equivalents.
Assessment Using Certainty Equivalents:
The first assessment method requires the decision maker
to assess several certainty equivalent.
Now imagine that you have the opportunity to play the
following lottery,
Win $100 with probability 0.5
Win $10 with probability 0.5
What is the minimum amount for which you would be
willing to sell your opportunity to play this game.

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Your job is to find the certainty equivalent (CE) so that you are
Indifferent to options A and B.
(0.5)
$100

(0.5)
$10
A

B
CE

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 Finding your certainty equivalent is where your subjective
judgment comes into play. The CE undoubtedly will vary from
person to person. Suppose that for this reference gamble your
certainty equivalent is $30.

 Because you are indifferent between $30 and the risky gamble, the
utility of $30 must equal the expected utility of the gamble. We can
get the first two points of your utility function by arbitrarily setting
U(100)=1 and U(10)=0

 Because you are indifferent between $30 and the risky gamble, the
utility of $30 must equal the expected utility of the gamble.
U(30) = 0.5U(100) +0.5 U(10)
=0.5(1) + 0.5(0)
=0.5
We have found a third point on your utility curve.

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To find another, take a different reference
lottery:
Win $100 with probability 0.5
Win $30 with probability 0.5
Again, the certainty equivalent will vary from
person to person, but suppose that you settle on
$50. We can do exactly what we did before, but
with the new gamble.
U (50) = 0.5 U(100) +0.5 U(30)
= 0.5(1) + 0.5 (0.5)
=0.75
This is the fourth point on your utility curve.
We now have four points on utility curve, and we
can graph and draw a curve through them.

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 1.00

0.75
Utility
0.50

0.25

0 20 40 60 100
Wealth
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Assessment Using Probabilities
This approach involves setting the sure amount in
Alterative B and adjusting the probability in the
reference gamble to achieve indifference. This is
called the probability-equivalent (PE) assessment
technique.
Consider the reference lottery:
Win $100 with probability p
Win $10 with probability (1-P)

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A reference gamble for assessing the utility of $65 using the
Probability-equivalent method.
(p)
$100

(1-p)
$10
C

D
$65

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To find utility value for $65, adjust p until you are
indifferent between the sure $65 and the reference
gamble.
U (65) = p U(100) + (1-p) U(10)
= p (1) + (1-p) (0)
=p
The probability that makes you indifferent just
happens to be your utility value for $65. For
example, if you choose p=0.87 to achieve
indifference, the U(65)=0.87.

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Risk Tolerance and the Exponential Utility Function
An alternative approach is to base the assessment
on a particular mathematical function, such as
exponential utility function.

This utility function is based on the constant

e=2.71828 ….,the base of natural logarithms. This
function is concave and thus can be used to
represent risk-averse preference. As x becomes
large, U(x) approaches 1. The utility of zero, U(0), is
equal to 0, and the utility for negative x (being in
debt) is negative.

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 Example
Start with:

Assuming R = 900, solve the exponential

equation:

Solve for EU:

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 Example
Assuming R = 900, solve the exponential equation:

Continuing solve for x, which is CE:

You’re indifferent between taking the gamble

or taking $1114.71 as a sure thing.

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 Some Caveats
1. Utilities do not add up.
U(A + B) ≠ U(A) + U(B)
Inherent in the nature of nonlinear utility functions
You must calculate net payoffs or net contributions at
the endpoints of the decision tree before transforming
to utility values.

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 Some Caveats
2. Utility differences do not express strength of
preferences.
Utility only provides a numerical scale for ordering or
ranking preferences.
Does not measure strength
Whether one agrees or not, it is necessary to interpret
utility carefully in this regard.

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 Some Caveats
3. Utilities are not comparable from person to person
A utility function is strictly a subjective personal
statement of a single individual’s preferences.
It does not provide a basis for comparing utilities
among individuals.
Nor can it be transferred from one individual to another.

© 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
Week 8
Chapter 15

Utility Axioms,
Paradoxes and
Implications
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
Axioms for Expected Utility
Our first step is to look at the behavioral assumptions that form
the basis of expected utility. These assumptions, or axioms, relate
to the consistency with which an individual expresses preferences
from among a series of risky prospects. Instead of axioms, we
might call these rules for clear thinking.
Ordering and Transitivity:
Ordering: A decision maker can rank order (establish
preference or indifference) any two consequences, and the
ordering is transitive.
Transitivity: If A1 is preferred to A2 and A2 is preferred to A3,
then A1 is preferred to A3.
For example, if a person prefers Amsterdam to London and
London to Paris, then he would prefer Amsterdam to Paris.
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 Axioms for Expected Utility
2. Reduction of compound uncertain events

Reduction axiom: A decision maker is indifferent

between a compound uncertain event (a complicated
mixture of gambles or lotteries) and a simple uncertain
event as determined by reduction using standard
probability manipulations.

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 Axioms for Expected Utility

Decision trees of the reduction axiom: compound uncertain events

can be reduced to simple uncertain events without changing the
decision maker’s preferences.
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 Axioms for Expected Utility
3. Continuity
Given three consequences A1, A, and A2 where the
decision maker’s stated preferences are A1 > A > A2,
then we can always find a probability value p, 0 < p <1,
such that the decision maker is indifferent between
receiving A for sure and a lottery with probability p of
receiving A1 and probability 1 – p of receiving A2.
We can always construct a reference gamble with some
probability p, 0 < p <1, for which the decision maker
will be indifferent between the reference gamble and
A.
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Axioms for Expected Utility
Suppose that you find yourself as the plaintiff
in a court case. You believe that court will award
you either $5000 or nothing. Now, imagine that
the defendant offer to pay you $1500 to drop the
charges. According to the continuity axiom,
there must be some probability p of winning
$5000(and the corresponding 1-p probability of
winning nothing) for which you would be
indifferent between taking or rejecting the
settlement offer. Of course, if your subjective
probability of winning happens to be lower
than p, then you would accept the proposal.
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Axioms for Expected Utility
4. Substitutability: A decision maker is indifferent
between any original uncertain event that includes
outcome A and one formed by substituting for A an
uncertain event that is judged to be its equivalent.
For example, you are interested in playing the
lottery, and you are just barely willing to pay 50
cents for a ticket. If I owe you 50 cents, then you
should be just as willing to accept a lottery ticket as
the 50 cents in cash.

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 Axioms for Expected Utility

Two decision trees that show how substitutability works: If A is

equivalent to a lottery with a p chance at C and 1 – p chance at D,
then Decision Tree I is equivalent to Decision Tree II.
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 Axioms for Expected Utility
5. Monotonicity: Given two reference gambles with the
same possible outcome, a decision maker prefers the one
with the higher probability of winning the preferred
outcome.
6. Invariance: All that is needed to determine a decision
maker’s preferences among uncertain events are the payoffs
( or consequences) and the associated probabilities.
7. Finiteness: No consequences are considered infinitely bad
or infinitely good.
The finiteness axiom assures us that expected utility will never
be infinite, and so we always will be able to make meaningful
comparisons

49
Paradoxes
Even though the axioms of expected utility
theory appear to be compelling when we
discuss them, people do not necessarily make
choices in accordance with them.
Framing effects are among the most pervasive
paradoxes in choice behavior.
An individual’s risk attitude can change
depending on the way the decision problem is
posted---that is , on the “frame” in which a
problem is presented.
The difficulty is that the same decision problem
usually can be expressed in different frames.
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Paradoxes
Example:
The United States is preparing for an outbreak
of an unusual Asian strain of influenza. Experts
expect 600 people to die from the disease. Two
programs are available that could be used to
combat the disease, but because of limited
resources only one can be implemented.
Program A (Tried and True) 400 people will be saved
Program B ( Experimental) There is an 80% chance
that 600 people will be saved and a 20% chance that no
one will be saved.
Which of these two programs do you prefer?

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Paradoxes
Now consider the following two programs:
Program C 200 people will die.
Program D There is a 20% chance that 600 people
will die and an 80% chance that no one will die.
Would you prefer C or D ?
You may have noticed that program A is same as C
and that B is the same as D. It all depends on
whether you think in terms of deaths or lives
saved. Many people prefer A on one hand, but D
on the other.
The reason for the inconsistent choices appear to
be that different points of reference are used to
frame the problem in two different ways.
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