QUANTITATIVE BUSINESS ANALYSIS
Simple Decision Tree Analysis and Utility Theory
Oliver Yu, Ph.D.
Oliver Yu © 2017
DT&UT-1
OBJECTIVE AND BACKGROUND
Objective
In addition to AHP, we will present another approach for quantifying
values of a decision-maker by using mathematical axioms and
decision-tree analysis.
Background
This approach has a long history dating back to Daniel Bernoulli in the
18th century, but it was first formalized by the great mathematician,
John von Neumann, in the 1950s.
In this approach, utility is defined as a relative quantitative measure of
the decision-maker’s values. It is relative because, like temperature
with the Fahrenheit and Celsius measures and altitude with the
English and metric measures, the values of a decision-maker can be
represented by different but internally consistent measures of utilities.
Furthermore, the decision-tree analysis is both a way to estimate
utilities and a way to apply utilities for making choices.
Oliver Yu © 2017
DT&UT-2
DECISION TREE WITH UNCERTAINTY:
FRAMEWORK
A Decision Tree
A tree representation of the decision process over time as
• a series of decision time points (decision or choice nodes conventionally in
square shape) at which the decision-maker has full control, with available
choices as emanating branches
• interspersed by a series of uncertainty time points (probability or chance
nodes conventionally in round shape) at which the decision maker has no
control, with probable outcomes as emanating branches
Basic Elements of the Decision-Making Process
Value: Ultimate outcome of each series of branches
Choices: branches at each decision node
Relationships: The tree structure
Oliver Yu © 2017
DT&UT-3
DECISION TREE WITH UNCERTAINTY:
A SIMPLE GRAPHICAL EXAMPLE
Probability p
Outcome Value 1a
Probability
Node
Decision
Node
Outcome Value 1b
Choice 1
Probability 1-p
Outcome 2
Choice 2
Oliver Yu © 2017
DT&UT-4
DECISION TREE WITH UNCERTAINTY:
ANALYSIS
For a decision tree with uncertainty, we compare the expected value
of each choice and select the one with the greatest expected value.
Specifically, in the above example of a simple decision tree,
Expected value of Choice 1:
p (Outcome Value 1a) + (1-p) (Outcome Value 1b)
Expected value of Choice 2: Outcome Value 2
The best choice is the one with the greatest expected value or utility.
Note: This simple decision tree can also be used as a mechanism for
estimating the decision-maker’s relative degree of preference for a
choice that lies between the best and worst choices
Oliver Yu © 2017
DT&UT-5
UTILITY: DEFINITION
Utility is a measure of decision-maker’s relative degree of
preference, desirability, importance, benefit, or value of the outcome
of an Choice. In resource allocation, the degree of preference, or
utility, U(x), of the outcome from allocating resource x to an
investment is generally a nonlinear, often concave, function. A
concave utility function generally represents diminishing incremental
increases in the relative degree of preference, desirability,
importance, benefit, or value of the allocation.
If time factor is considered, then utility generally decreases with time.
This phenomenon is often called the time preference of utility. Such
a decrease in utility or preference is generally caused by the
perceived risk of future returns, which usually increases with time.
This decrease is often represented in a simplified manner as the
discount rate.
Oliver Yu © 2017
DT&UT-6
UTILITY: AXIOMS
1. Rankability and Completeness
Let U be the utility of an Choice, then for a choice between Choices
A and B, a decision-maker either prefers A to B, i.e., u(A)>u(B); or B
to A, i.e., u(A)<u(B); or indifferent between them, i.e., u(A)=u(B).
2. Transitivity
At a given time, if a decision-maker prefers A to B and B to C, then
the decision-maker must prefer A to C; i.e., u(A)>u(B)>u(C) implies
u(A)>u(C).
3. Computability
If a lottery L has probability p for A and 1-p for B, then the utility of L
will be
u(L) = (p) u(A) + (1-p) u(B).
4. Substitutability
If u(A)=u(B), then A and B are substitutable for the decision-maker.
5. Continuity and Certainty Equivalent
If a decision-maker prefers A to B and B to C, then there exists a
lottery L with probability p for A and 1-p for C such that the u(L) =
u(B), and B is the certainty equivalent of the lottery L.
Oliver Yu © 2017
DT&UT-7
UTILITY IS A RELATIVE MEASURE OF
PREFERENCE
Similar to temperature, utility is a relative measure, in the sense that
we can develop a complete utility measurement system by defining a
base value and a measuring unit. For example, there are two
common measuring systems for temperature, Celsius and Fahrenheit.
I can also developed a Yu system for measuring temperature, in
reference to the two common systems as follows:
Temperature (degree)
Celsius
Fahrenheit
Yu
Water Freezing
Water Boiling
Normal Body
0
100
37
32
212
98
-100
1000
307
Conversion formulas: F = 32+C(212-32)/100;
Y = -100+C(1000-100)/100.
Oliver Yu © 2017
DT&UT-8
A SIMPLE DECISION TREE FOR
UTILITY ESTIMATION
Probability p*
U(Best Outcome) = U(A)
Lottery L
U(Worst Outcome) = U(C)
Probability 1-p*
U(Outcome in Between) = U(B)
Certainty Equivalent
Oliver Yu © 2017
U(A) > U(B) = p*U(A) + (1-p*)U(C) = U(L) > U(C),
where is p* is the indifference probability between B and the Lottery.
DT&UT-9
UTILITY: ESTIMATION PROCESS
As a relative measure of value, a decision-maker’s utility can be
quantified through the simple decision tree analysis of two choices.
Using money as an example.
1. Assuming that utility as the measure for the value of money is a
non-decreasing function of the quantity of money for a decisionmaker, set an arbitrarily high utility, say 100, for a large amount of
money, say $1 million; and an arbitrarily low utility, say 0, for a small
amount of money, say $0.
2. Develop a simple decision tree with two choices:
• A lottery, L, with a probability p of getting the best outcome (A) of winning
$1 million and probability 1-p of getting the worst outcome (C) of $0.
• A certainty choice of getting the in-between outcome (B) of x amount of
money lying between $0 and $1 million with certainty.
Then by Utility Axioms 3 and 5, U(L) = pU(A) + (1-p)U(C).
Oliver Yu © 2017
DT&UT-10
UTILITY ESTIMATION - Concluded
3. For this artificial simple decision tree,
if p = 1, then clearly the decision-maker will choose the lottery L;
if p = 0, then clearly the decision-maker will choose the certainty
choice with outcome B.
By Utility Axiom 5, as we vary p continuously from 1 to 0, there exists
a probability value p* at which the decision-maker will choose L if the
probability p of getting the best outcome A is greater than p*, and
choose the certainty Choice if the probability p of getting the best
outcome A in the gamble is less than p*, and will be indifferent
between L and the certainty Choice if p = p*.
4. By Utility Axiom 4, if the probability of getting the best outcome A
in the lottery is p*, then the utility of B,
U(B) = U(L) = p*U(A) + (1-p*)U(C),
and B is the certainty equivalent of L with p = p*.
Oliver Yu © 2017
DT&UT-11
UTILITY: RISK ATTITUDES
This simple decision tree also can be used to determine the risk
attitudes of the decision-maker. Again using money as an example:
If U(x) is proportional to x, then the decision-maker is risk neutral as
the value of money is proportional to the amount of money.
If U(x) is more than proportional to x, then the decision-maker is
viewed as risk avoiding, as the utility of having x for sure is preferred
to the utility of a lottery with higher expected payoff.
On the other hand, if U(x) is less than proportional to x, then the
decision-maker is viewed as risk preferring, as the utility of having x
for sure is less than the utility of a lottery with lower expected payoff.
Risk preference is totally subjective, depending on the decisionmaker’s personality, the size of decision-maker’s assets, and the
amount at stake.
Oliver Yu © 2017
DT&UT-12
UTILITY: DIFFERENT RISK ATTITUDES
U
Risk Avoiding
Risk Neutral
Risk Preferring
$
Oliver Yu © 2017
DT&UT-13
UTILITY: A SIMPLE DETERMINATION
OF RISK ATTITUDE FOR MONEY
A simple way to determine whether a decision-maker’s risk attitude
for money is to compare the utility function with a risk neutral
person’s utility function that has the same utility for the maximum
amount of money, M, as well as the same utility for the minimum
amount of money, m (for example, both persons have utility 100 for
M=$1 million and 0 for m=$0).
Let x be an amount of money between M and m, then the riskneutral person’s utility for x, Un(x), is (x-m)/(M-m)[U(M)-U(m)] or
(x/1 million)(100) for the example.
Now, let U(x) be the utility of the decision-maker for x, then
If U(x) > Un(x), the decision-maker is risk avoiding;
If U(x) = Un(x), the decision-maker is risk neutral;
If U(x) < Un(x), the decision-maker is risk preferring.
Oliver Yu © 2017
DT&UT-14
UTILITY: RISK PREMIUM
For a risk-averse decision-maker, if a lottery, L, with expected
monetary value E(L), then the monetary value of the certainty
equivalent, x, for the lottery is necessarily lower than E(L), because
of the uncertainty in the lottery. E(L) - x is then the risk premium of
the lottery. In other word, because of the risk involved in the lottery,
the decision-maker has discounted the monetary value of the lottery
to the amount represented by the risk premium.
In dealing with a lottery of potential loss, the reverse will be true; i.e.,
a risk-averse decision-maker would have a certainty equivalent x
higher than the expected loss E(L) from the lottery. Using insurance
as an example, the lottery for the insured has a expected loss E(L)
less than the insurance premium x charged by the insurer. In this
case, the risk premium is x-E(L), i.e., the portion of the premium that
the insured pays in excess of the expected payout by the insurer in
order to avoid the risk of a large loss without insurance coverage.
Oliver Yu © 2017
DT&UT-15
UTILITY: STYLIZED UTILITY FUNCTION
For analytical simplicity, utility functions for money are often
stylized. For example, the utility function for money of a riskavoiding person may be stylized to be proportional to the root
function of the amount of money, while that of a risk-preferring
person may be proportional to the power function of the
amount of money.
Example: The utility function for money U(x) of a risk-avoiding
person is stylized to be proportional to the square root of the
amount of money x, i.e., U(x) = c x0.5 . If the person sets
U($1 million) = 1,000 and U($0) = 0, then the proportionality
constant c can be determined to be 10, because U(1,000,000)
= c (1,000,000)0.5 = c (1,000) = 100.
Now, U(x) can be estimated for any x.
Oliver Yu © 2017
DT&UT-16
PRINCIPLE OF INSURANCE PRICING
Insurers generally set premiums based on the following principle:
The maximum insurance premium an insured person is willing to pay
is at an amount that the person is indifferent between having and not
having the insurance; i.e., when the two choices have same utility.
Application:
If a person has net monetary asset A, including an asset under risk
B, and a utility function for money U(x). During a year, assume that
B has a probability p of being totally destroyed in an accident, and
probability 1-p of being not harmed at all. Then the maximum
insurance premium IP the person would be willing to pay to fully
insure B can be obtained by equating the utility of the asset after
subtracting IP, i.e., U(A-IP), and the expected utility of the assets
without insurance, i.e., p U(A-B) + (1-p) U(A).
Oliver Yu © 2017
DT&UT-17
INSURANCE PRICING EXAMPLE
A person has a net asset A=$40,000, including a car valued at B=$7,600.
Through demographic profiling, the utility function for money of this person
U(x) for x amount of money is stylized to be proportional to the square root
of money; i.e., U(x) = c x0.5. By setting U($10000) = 1000 and U(0) = 0, we
obtain c = 10, and can then estimate U(x) for any amount of money.
Accident statistics for the person’s demographic segment has shown that
the car has a probability p= 0.05 of being totaled during a year. The insurer
can estimate the maximum insurance premium, IP, the person would be
willing to pay to have the car fully insured by equating the utilities of the
person for having and not having the insurance as follows:
Utility of full insurance = (40000-IP) = 10(40000-IP)0.5
= Utility of no insurance = 0.05 U(car is totaled) + 0.95 U(car not damaged)
= 0.05U(40000-7600)+0.95U(40000)=0.05(1800)+0.95(2000) = 1990
or 40000-IP = (199)2 = 39601 or IP = $399.
Risk premium = IP – Insurance expected payout
= 399 – [0.05 (7600) + 0.95 (0)] = $19, which is the insurer’s gross profit.
Oliver Yu © 2017
DT&UT-18
RECONCILIATION WITH ANALYTIC
HIERARCHY PROCESS
For computers A, B, and C, assume that a decision-maker prefers A the
most and C the least. Using AHP, we can estimate the values or degrees of
preference, VA, VB, and VC of the three computers respectively. Clearly
VA>VC>VC.
By setting U(A)=VA and U(C)=VC, we can then apply the utility decision
tree to estimate, U(B), the utility of B. Since utility also measures the
relative preference, U(B) should equal VB.
What if U(B) does not equal VB, how can we reconcile the difference?
In this case, because of the hierarchical structure and comparison
precision, AHP generally produces more accurate and consistent reflections
of the decision-maker’s preferences. Furthermore, since, VB is between VA
and VC, U(B) can always be made equal to VB by setting p for getting A in
the gamble to be (VB-VC)/(VA-VC).
However, if U(B) obtained directly from the Utility Theory approach is very
different from VB, then the decision-maker should review the AHP process
to gain a deeper insight about the difference.
Oliver Yu © 2017
DT&UT-19
HOMEWORK 5
5a. (10 points) Set your utility for $0 to be 0 and $1 million to be
100. Apply the simple decision tree used in determining your utility
for money, where the lottery has a prize of $1 million if you win and
$0 if you lose, to determine your utility for $500,000 and $200,000.
Use these 4 utilities to draw your utility curve for money.
5b. (10 points) Go to your HW3, and use the highest and lowest total
scores of the best and worst computers as the their respective
utilities. Again use the simple decision tree, where the lottery will
give you the best computer if you win and the worst computer if you
lose, to determine the total score of the middle computer. Check
whether this total score for the middle computer is close to the one
you obtained in HW3. Which total score for the middle computer you
will feel more confident and why?
Oliver Yu © 2017
DT&UT-20
HOMEWORK 6
A college senior must choose between two Choices: going for an MBA
or taking a full-time entry-level-level position right after graduation.
She thinks that she has 0.65 probability of completing the MBA in a
year. If she completes the MBA, she believes that she has 0.3
probability of getting a manager position; otherwise, she will get a
senior staff position. Should she fails the MBA, she will have to take
the entry job but with less seniority than what she would have if she
had gone to work right after graduation. Once started at the entrylevel position for a year, she believes that she has a 50-50 chance of
moving up to a junior staff position versus staying at the entry-level
position. Her preferences for the possible outcomes of her choice at
the end of two years are listed in decreasing order below:
(1) Completing the MBA and getting a management position
(2) Completing the MBA and getting a senior staff position
(3) Moving to junior staff without going to MBA and thus more seniority
(4) Moving to junior staff after failing the MBA
(5) Staying at entry level without going to MBA and thus more seniority
(6) Staying at entry level after failing the MBA
Oliver Yu © 2017
DT&UT-21
HOMEWORK 6 - concluded
Using the simple decision tree for utility estimation, she has found
that she would be indifferent between:
Outcome (2) and a lottery with a 50-50 chance of yielding the
best outcome (1) and the worst outcome (6).
Outcome (3) and the lottery if the lottery has a 0.4 probability
yielding (1) and a 0.6 probability of yielding (6).
Outcome (4) and the lottery if the lottery has a 0.25 probability
yielding (1) and a 0.75 probability of yielding (6).
Outcome (5) and the lottery if the lottery has a 0.1 probability
yielding (1) and a 0.9 probability of yielding (6).
6a (10 points) By assigning a utility 0 to (6) and 100 to (1), find the
utility for each of the four outcomes between (1) and (6).
6b (10 points) Draw a decision tree for her career decision and find
her best Choice for the two-year period.
Oliver Yu © 2017
DT&UT-22
HOMEWORK 7
A person has a net asset of $1 million, including a $300,000
net equity of a house (market value of the house – mortgage).
Specifically, the house has a market value of $600,000
including $400,000 for the structure and $200,000 for the land,
and a mortgage of $300,000. The person plans to buy
$400,000 fire insurance for full coverage of the house. For
simplicity, assume that each year the house has a 1%
probability of being totally destroyed by fire and a 99%
probability of no damage occurring to the house. The person’s
utility for money is approximately proportional to the quartic
root of money with U($100,000,000)=1,000 and U($0)=0.
7a.(5 points) Draw the decision tree for the person’s decision
of buying or not buying the insurance.
7b.(10 points) Determine the maximum insurance premium IP
the person would be willing to pay.
7c. (5 points) What is the risk premium at the maximum IP?
Oliver Yu © 2017
DT&UT-23
BONUS PROBLEM
(20 points) Determine the maximum insurance
premium the person would be willing to pay for a
$300,000 insurance just to cover the mortgage.
(Hint: in this case, the house is under-insured. In
other words, with the $300,000 insurance, if the
house is totally destroyed by fire, the person will
suffer a loss in the net asset because the insurance
covers only the mortgage not the full net equity of the
house, and the maximum insurance premium the
person would be willing to pay will need to be
determined through numerical iterations).
Oliver Yu © 2017
DT&UT-24