1. QUANTITATIVE BUSINESS ANALYSIS
Simple Decision Tree Analysis and Utility Theory
Oliver Yu, Ph.D.

2. OBJECTIVE AND BACKGROUND
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.

3. 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

4. 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

5. 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

6. 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.

7. 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.

8. 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
Conversion formulas: F = 32+C(212-32)/100;
Y = -100+C( )/100.

9. 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
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.

10. 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.
Assuming that utility as the measure for the value of money is a non-decreasing function of the quantity of money for a decision-maker, 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.
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).

11. 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*.

12. 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 decision-maker’s personality, the size of decision-maker’s assets, and the amount at stake.

13. UTILITY: DIFFERENT RISK ATTITUDES
Risk Avoiding
Risk Neutral
Risk Preferring U $

14. 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 risk-neutral 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.

15. 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.

16. UTILITY: STYLIZED UTILITY FUNCTION
For analytical simplicity, utility functions for money are often stylized. For example, the utility function for money of a risk-avoiding 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.

17. 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).

18. 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) U(car not damaged)
= 0.05U( )+0.95U(40000)=0.05(1800)+0.95(2000) = 1990
or IP = (199)2 = or IP = $399.
Risk premium = IP – Insurance expected payout
= 399 – [0.05 (7600) (0)] = $19, which is the insurer’s gross profit.

19. 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.

20. 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?

21. 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 entry-level position for a year, she believes that she has a 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

22. 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 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.

23. 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?

24. 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).