1.
11
Ancient Notes on Decision Analysis
1.1
Introduction to Decision Analysis
Almost everyone makes a great many decisions every day, usually without any form of detailed
analysis. We decide what to eat for breakfast, how much time to spend on studying, which friends to visit,
at what time to go to bed, and  less frequently  some weightier matters also. This course is directed to
students of management and is concerned with the kinds of decisions that are made within business
enterprises and similar organizations. These are like the familiar personal decisions of daily life in one
very significant respect: the vast majority of business decisions, like the bulk of purely personal decisions,
must be and are made on the basis of very little data, calculation, and thought. The busy executive cannot
afford the time for prolonged deliberation as to what letters he will write Monday morning (or even as to
what he will say in most of them) any more than in his nonbusiness life can he afford to ponder over his
choice of breakfast cereal.
However many decisions are of sufficient importance and complexity to merit more extensive
analysis. In such cases the decision maker is apt to obtain some data, perform some calculations or
analysis, and put together a report, recommendation, or other document intended to justify the selection of
one course of action in preference to others. The amount of time and effort needed to prepare and present
this analysis and conclusions will depend mainly on how much is at stake in the decision (as the executive
sees it) and on how difficult it is to see and prove which course of action is the “right” one.
Let us consider the following example, which is simplified for illustrative purposes.
Example:
“War Stars”.
Mr. Fox, manager of Mount Para Productions, has recently received a script for a movie called
“War Stars”. Responsible for all movie productions Mr. Fox has to decide whether or not to produce
“War Stars”. Advisors told him it is typically an “all or nothing” (i.e., it will be either a “success” or a
“failure”) script with a fiftyfifty chance of success. To his best estimate Mr. Fox believes the movie will
result in a $6 million dollar net profit in the case that it is a success. However, Mount Para has to face an
estimated net loss of $4 million dollars if the movie is not well received.
Question:
Should Mr. Fox produce “War Stars”?
Decision Tree.
In “solving” the problem we will make use of a decision tree (also called decision diagram)
which provides a symbolic representation of a sequential decision process. A decision tree shows, at one
glance, when decisions can be made, what the possible consequences are, and what the resultant payoffs
will be. Another advantage of a decision tree is that the results of the computations are depicted directly
on the tree, thus simplifying the analysis.
Question:
What does the “War Stars” decision tree look like?
First a decision has to be made whether or not to produce the movie:
Produce movie
Do not produce movie
2.
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The above is called a fork, in this case an act fork. It is always represented by a small box with
several branches emanating from it. The individual branches of the fork represent all the options which
the decision maker wishes to consider in making his choice. It does not necessarily represent all possible
options. (For example, it may be possible to use the script for a play, but Mr. Fox does not wish to
consider this option as he is only concerned with movies).
Thus an act fork represents a decision point at which the decision maker has a choice.
If it is decided to produce the movie, two outcomes may occur, each with equal chance. (Recall:
fiftyfifty chance of “success”):
“success”
(50%)
“failure”
(50%)
The above fork is called an event fork, for which we will always use a circle followed by several
branches. The branches represent the relevant consequences of the preceding decision (act). An event fork
indicates a chance event whose outcome is not known to the decision maker at the time the decision is to
be made. For instance, in the War Stars example we “only” know there is a fiftyfifty chance for success
or failure. The decision maker has no choice, the outcome of the event is out of his/her control.
If it is decided not to produce “War Stars”, no relevant event or act (decision) is anticipated. We
are now ready to create the first decision tree using the previous forks:
“success”
(50%)
“failure”
(50%)
Produce movie
Do not produce movie
Even though the above diagram clarifies Mr. Fox’s decision process, it is possible to include more
information. In the above tree there are clearly three end positions and each position actually refers to a
sequence of branches from the beginning to this end position. And to each end position we can assign a
net payoff value, which we will call an end point. The three sequences and their end points are:
1) “produce movie” and “success”;
2) “produce movie” plus “failure”;
3) “do not produce movie”;
Endpoint:
Endpoint:
Endpoint:
$6 million.
$4 million.
$0.
The tree will be redrawn to include the above information:
Produce movie
Do not produce movie
“success”
(.5)
$6 million
“failure”
(.5)
$4 million
Notice that the fiftyfifty chance is changed into the equivalent fractions of one. Also, instead of chance,
we will often refer to such a fraction as the probability or likelihood. For example, the probability, that the
movie will be a “success” is .5.
3.
13
Before discussing the optimal strategy for Mr. Fox’s decision problem, we will indicate some
difficulties that may arise in more complicated and realistic situations. Also, some definitions are given.
Requirements about Forks.
A fork is either an event fork or an act fork; any situation which might appear to be a mixture of
chance and choice should be represented by two or more forks, including at least one act fork and one
event fork.
Regardless of whether a fork is an event fork or an act fork, the events or acts represented by its
branches should be of sufficient number and so labelled that they:
1)
2)
Include all possibilities under consideration, and
Include each possibility only once.
The term collectively exhaustive is a technical term used to mean that all possibilities under
consideration are included.
The following event fork represents the residential area of a randomlyselected full time U.B.C.
student:
Vancouver
“Lower Mainland” excluding Vancouver
British Columbia excluding “Lower Mainland”
The branches of this fork do not exhaust all the possibilities. The student may live somewhere else, for
instance in Bellingham (U.S.A.). One way of making the fork collectively exhaustive would be to add a
branch “Outside British Columbia”.
The technical term mutually exclusive means that each possibility is included only once. In other words,
the descriptions of each of the branches do not overlap; the selection of one excludes all the others. For
example, the following event fork  again representing the residential area of a randomlyselected full
time U.B.C. student  does not have mutually exclusive branches:
Vancouver
“Lower Mainland”
British Columbia
Outside British Columbia
Figure 1.1. A fork whose branches are not mutually exclusive.
Living in the “Lower Mainland” does not exclude living in Vancouver. Moreover, living in British
Columbia does not exclude both living in Vancouver and living in the “Lower Mainland”
4.
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Questions:
1.1.a.
Are the branches of the fork in Figure 1.1 collectively exhaustive?
1.1.b.
A world traveller is deciding where to go on her next vacation and drew the decision diagram
below. Are the branches mutually exclusive?
Travel to Europe first
Travel to Italy first
Travel to Amsterdam first
1.1.c.
A student wishes to buy a typewriter and draws the diagram given below. Are the branches
mutually exclusive?
Buy a secondhand typewriter
Buy an electric typewriter
Do not buy a typewriter
1.1.d.
Can you say whether the branches of the fork in problem 1.1.c are collectively exhaustive? if
not, make assumptions and modify (if necessary) the fork into one whose branches are
collectively exhaustive.
Forks with a Large Number of Branches
There are many situations in which one would like to have a fork with a large number of branches.
For instance, a production manager, in deciding on what quantity of a certain item to produce, may wish
to consider a whole range of possibilities. Or there may be uncertainty about the demand for an item in a
given time period, and this demand might have a great range of possible values. In such a case, it is
impractical to include every branch on a decision diagram individually. Instead, such a fork is represented
schematically by a “fan” which indicates that there is a great range of possibilities and shows a few
typical branches. For example, if the abovementioned production manager wishes to consider production
runs lying between 50 and 275 items, a fan representing his act fork would look like that shown below.
50
27.5
The analysis of diagrams involving fans is conceptually no more difficult than analysis of diagrams
involving simpler forks. However, because a greater number of possibilities are implied by a fan, the
amount of computation involved is often greater.
In the “War Stars” example a fan would appear if it had not been an “all or nothing” script, but
instead several levels of “success” and “failure” had been recognized, each having its own end point.
5.
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Assignment of Probabilities
In many real life situations the probabilities for the different outcomes are not easy to obtain. It can
be difficult and sometimes even impossible to assign the “right” probabilities. In the “War Stars” example
advisors informed Mr. Fox about the probability of “War Stars” becoming a “success”. But even there,
how did his advisors arrive at the probabilities?
In answering this question it may be useful to make the following distinction between sources of
probabilities. Fran the point of view of decision analysis, relative frequency and subjective probabilities
will be most useful.
a.
Subjective Probability
Probability obtained in this manner are based on personal degrees of belief. A manager may claim
that the chance of losing money this year is only 1/100. Or a Vancouver resident claims that the
probability of the Canucks reaching the Stanley Cup finals is 9/10. In order to arrive at a subjective
“degree of belief” probability it may be useful to perform a “lottery”. For example, suppose you may
choose between the following two lotteries:

Toss a coin:
if it lands heads you will receive $1,000, but if it lands tails you will not
receive anything.

If you pass “Commerce 211” with a First Class you will receive $1,000, but if you do not,
you will not receive anything.
Let us agree that in the first lottery your chance of winning $1,000 is equal to 1/2. (See Question
1.1.g.) Suppose you were not indifferent between the two lotteries and chose the second. This means you
believe that the probability of your achieving a First Class in “Commerce 211” is greater than 1/2. By
performing a sequence of similar lotteries it is possible to assess your own belief of what is the probability
you receive a First Class. (See Question 1.1.h.)
b.
Objective Probability
i.
Deductive Logic. A technical term to indicate that the assignment of the probability is
determined logically from symmetry or geometric considerations associated with the
experiment. For example, without actually rolling a fair die many times, we could say that
the probability that a “five” appears is equal to 1/6.
ii.
Relative Frequency definition of probability. For example, suppose you have a box with
many red and white balls, but you do not know how many are red and how many are white.
Moreover, for some reason you are not able or allowed to count the balls one by one. To
“estimate” the probability of drawing a red or white ball, you might perform the following
experiment: Draw one ball at random, record its colour, and put it back into the box. Repeat
this, say 100 times. Suppose you have drawn 70 red and 30 white balls, then you could say
that the probability of drawing a red ball from the box is close to .7. More generally, the
probability defined by relative frequency is n/N, where ii is the number of times the event “a
red ball is drawn” occurs during N repeated experiments. In our example n=70, whereas
N=100, thus n/N 70/100 = .7. Of course, .7 is only an estimate of the “true” probability. The
statistical theory developed in Commerce 212 will enable us to measure the accuracy of this
estimate.
Objective and subjective probabilities are fundamentally different. If we asked a number of
persons to determine the probability objectively, each would arrive at the same answer, provided they
were given the same set of assumptions. But, if we asked them to determine the probability subjectively,
each individual would arrive at his or her own answer.
6.
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Finally, we often use a mathematical notation to express probabilities. For example, the probability
that a head appears in a cointossing experiment is equal to 1/2, can be denoted as follows:
H = Event that a head appears
P(H) = 1/2
Questions:
1.1.e.
A ball is randomly drawn from an urn containing four white and six red balls. What is the
probability that this ball is a red one? And what does “random” mean? What definition of
probability did you actually use?
1.1.f.
Suppose that in the past one out of 50 creditors of a firm defaulted. One could say that the
probability that a creditor will default in the future is .02. Is the “probability using historic
data” objective or subjective? Is it possible to have a “probability using historic data” that is
strictly objective?
1.1.g.
What is the probability that a head appears in a cointossing experiment? What is the main
assumption you made? What definition of probability did you use?
1.1.h.
Create a sequence of lotteries to discover what your neighbour in class believes his
probability of getting a first class for Commerce 211 is.
1.1.i.
What method(s) do you think Mr. Fox’s advisors used to arrive at the probability that “War
Stars” would be a success?
1.1.j.
Apply the new notation “P(...) = ...” to the “War Stars” example.
7.
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1.2.
Decision Analysis: Criteria of Choice
In the previous chapter a general introduction to Decision Analysis was given, a decision tree was
constructed for the “War Stars” example, some difficulties that may arise in more complicated and more
realistic situations were indicated, and finally some basic definitions were introduced.
However, no decision in Mr. Fox’s decision problem has yet been made!
In this chapter we discuss a few of many possible criteria of choice in differing decision making
settings.
A.
Decisions Under Certainty: This is the situation when the decision tree does not contain any event
forks. Decision problems under certainty are called deterministic. The case study in Chapter 1.4.,
the Vancouver Electronics Company (A), is the only deterministic example we will encounter in
this course. Linear programming is another example of deterministic decision making. A suitable
criteria might be maximization of profit or minimization of cost.
B.
Decisions Under Risk: “Expected Monetary Value” (EMV); This criterion makes use of known or
assessed probabilities. This course is mainly concerned with problems involving EMV. This
criterion is applied to the “War Stars” example in Chapter 1.3. An alternative is to use utility
theory (see 1.5).
C.
Decisions Under Uncertainty: In this setting, we assure we have no knowledge of and refuse to
make any assumptions about the probabilities of occurrence of the uncertain event. We distinguish
three possible criteria of choice for decisions under uncertainty and provide brief comments on
their interpretation and application to the “War Stars” example.
C.1.
Maximin (or Minimax) Criterion: Maximize minimum profit (or, in case the endpoints refer to
losses: minimize maximum loss). This very conservative criterion chooses the “least worst”
decision and is most useful when bad consequences must be avoided at all costs. If Mr. Fox
decides to produce the movie, the “worst” that can happen is a net loss of $4 million. If he decides
not to produce the movie the “worst” that can happen is a “net loss” of $0 (in this case the only
possibility). Clearly, the “least worse” is not to produce the movie.
C.2.
Plunger Criterion (Maximax or Minimin): Choose the decision with best of all consequences.
Usually used by very optimistic decision makers or gamblers, in situations where consequences do
not matter too much, or in desperate situations. If Mr. Fox wishes to apply this criterion he should
decide to produce “War Stars” (and hope the movie is going to be a success).
C.3.
Minimax Regret Criterion: This criterion is sometimes referred to as the “morning after” view.
The Regret is the difference between what you get from a decision and what you would have
gotten if you had known the outcome before making the decision. If the movie were a success, Mr.
Fox would choose to produce the movie. If the movie were known to be a failure, he would not
produce the movie. If Mr. Fox decides not to produce the movie his “maximum regret” is $6
million (namely, he lost the opportunity to produce a successful movie). If Mr. Fox decides to
produce the movie, his “maximum regret” is $4 million (namely, if the movie is a “failure” he
could have done better by not producing the movie and avoid the $4 million net loss). Clearly, the
minimum “maximum regret” criterion results in deciding to produce “War Stars”.
In practice we usually have a rough idea of the relevant probabilities and these criteria (C.1, C.2 and C.3)
are of little practical importance. Notice that you do not need to know the probabilities for any of these
criteria.
8.
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Questions:
1.2.a.
The decision tree below reflects the problem faced by Mr. S. Lake, Manager of the student
pub “The Pot”, just before a pending beer strike in the summer of 1978.
Strike (.1)
$4,000
No strike (.9)
$2,000
Strike (.1)
$6,000
Stockpile beer
Maintain present
Inventory policy
No strike (.9)
$6,000
In which way(s) is this diagram simplified and unrealistic?
1.2.b.
Determine the strategy Mr. Lake should adopt assuming he would like to
use:
 Maximin Criterion;
 Maximax (Plunger) Criterion;
 Minimax Regret Criterion.
Under what circumstances do you think the above criteria might be applied?
9.
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1.3
Decision Analysis: Expected Monetary Value (EMV)
In this chapter we discuss the strategy Mr. Fox should adopt assuming he wishes to use “Expected
Monetary Value” (EMV) as his criterion of choice. Should “War Stars” be produced or not?
The EMV approach prescribes that the decision maker select the alternative with the best expected
(average) payoff. The expected payoff (or the EMV) of an alternative is the sum of all possible payoffs of
that alternative, weighted by the probabilities of those payoffs occurring. For example, the EMV of the
event fork in the “War Stars” example (of which the tree is repeated below) can be calculated as follows:
Step 1:
Multiply each payoff (endpoint) by its corresponding probability.
Example: “Success”: (.5) ($6 million) = $3 million
“Failure”: (.5) ($4 million) = $2 million
Step 2:
Sum up the results of the multiplication of Step 1; the total is the EMV.
Example: ($3 million) + ($2 million) = $1 million.
Thus, the EMV of the event is $1 million.
At the initial decision point, the decision maker has a choice between “produce movie” and “do not
produce movie”. The first choice has an EMV of $1 million, whereas the second choice has an EMV of
$0 (which in this case is the same as the end point).
EMV
= $1 million
EMV = $1 million
“Success”
(.5)
Produce movie
“Failure”
(.5)
Do not produce movie
$6 million
$4 million
$0
Figure 1.2 Complete “War Stars” Decision Tree
The choice is clear: The optimal strategy for “War Stars” using EMV is to produce the movie!
Folding Back the Decision Tree (Backward Induction)
Notice that the calculation of EMV starts at the right most end of the tree and continues to the left
until the origin has been reached. The calculation of the EMV at a chance point is different from the EMV
calculation at a decision point.
At a chance point the EMV is calculated by a probability weighted average of all possible payoffs
of that alternative.
At a decision point the payoffs for each alternative are compared and the best one is selected as the EMV
(the expected payoff) for that decision point. All other alternatives are disregarded, or pruned. (See “//“ in
the tree of Figure 9.)
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EMV: “Playing the longrun averages”
The expected Monetary Value (EMV) decision criterion is typically used when many similar
decisions have to be made under risk. Let us take the hypothetical situation where Mr. Fox will have to
decide often, say 100 times whether or not to produce a movie under similar circumstances. If in all cases
he decides to produce the movie, what would be his total net profit or loss? According to the FiftyFifty
chance of success in all cases, he would expect about one half of these 100 movies, say about 50, to
become a “success”, and the other half to become a “failure”. The total net profit would then be: 50 x ($6
million) + 50 x ($4 million) = $100 million, which means on the long run a net profit of $1 million per
movie. Notice that this amount is exactly the EMV at point 1 in the tree in Figure 1.2. The above
illustrates, that using EMV implies “playing the longrun averages”. The above calculation illustrates the
law of large numbers; an important result in probability.
Questions:
1.3.a.
Suppose the endpoint in Figure 1.2 for “success” is $3 million, what strategy should Mr. Fox
adopt, assuming he prefers to use EMV as his decision criterion?
1.3.b.
Use EMV to determine Mr. Fox’s decision if the probability of “success” was .3, i.e.,
P(“success”) = .3 and P(“failure”) = .7.
1.3.c.
Suppose the probability of “War Stars” becoming a success is p, where 0 ≤ p ≤ 1. For what
value of p would Mr. Fox be indifferent between producing and not producing the movie,
i.e., what are the breakeven probabilities? (Use EMV.)
1.3.d.
Referring to Problem 1.2.q, determine the strategy the manager of “The Pot” should adopt,
assuming he prefers to use EMV as his criterion of choice?
1.3.e.
Of course, Mount Para will not produce “War Stars” 100 times but EMV might still be an
appropriate decision making criterion. Why?
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1 .4. Case Study: Vancouver Electronics Company (A)
Vancouver Electronics Company (VEC) is a mediumsized manufacturer of electronic components
founded in 1967. It produces specialized electronic parts that are purchased for use in automated factories.
In August of 1975, Mr. Andrew Howard, founder of VEC, was contacted by Mr. William Stone,
president of Stone Manufacturing Company. Mr. Stone was preparing to build a new automated cement
mix factory in White Rock and wanted to know if VEC could supply him with 100 electronic ampometers
with housings at $1000 each for use in this factory.
Since his plant will be operating at fairly low capacity until January, Mr. Howard was inclined to
accept this new job. VEC has never produced ampometers before, but it has produced a closely related
product and no technical problems seemed to stand in the way. Mr. Howard called Mr. Peter Wong, his
Plant Engineer, into his office and told him of Mr. Stone’s offer.
“Considering the slack capacity in the plant, I think we should accept this offer,” Mr. Wong said.
“One worker could assemble one ampometer in about 10 hours, so at our current wage rates, that’s about
$50. Raw materials would be about $450 per ampometer. There are two possible ways we could go with
the housings for the ampometers. We could buy them at $300 each ... they are very similar to the ones we
bought for that small subcontract last year ... or we could buy a mold from Farentox Burnaby and make
them ourselves for about $50 each. The Farentox mold would cost us about $17,500, but then we’d have it
if we ever needed to make additional ampometer housings.”
Questions:
1.4.a.
According to the above text, is Mr. Howard faced with any uncertainties?
1.4.b.
What should Mr. Howard decide?
1 .4.c.
How would Mr. Howard’s decision change if Mr. Stone had asked for 50 rather than 100
units?
1.4.d.
At what number of units would Mr. Howard be indifferent as to which method be used?
1.4.e.
If Mr. Howard decides to purchase the mold and make the housings, how many units must he
sell to start making a profit? How many units must he sell in order to make a profit if he
desires to buy the housings instead?
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1.5
Decision Analysis: Utility
It is often assumed that expected profit or loss in dollars is the appropriate measure of the
consequences of taking an action, given a state of nature. However, there are many situations where this is
inappropriate. For example, suppose that an individual was offered the choice of accepting (1) a 5050
chance of winning $10,000 or nothing, or (2) receiving $4,000 with certainty. Many people would prefer
the $4,000 even though the expected payoff on the 5050 chance of winning $10,000 is $5,000. A
company may be unwilling to invest a large sum of money in a new product even if the expected profit is
substantial if there is a risk of losing their investment and thereby becoming bankrupt. People buy
insurance even though it is a poor investment with a negative EMV.
Do these examples invalidate the previous material? Fortunately, there is a way of transforming
monetary or even nonmonetary values into an appropriate scale that reflects the decision makers’
preferences. This scale is called the utility scale, and it can be used to measure the consequences of taking
an action, given an outcome.
We will not study utility in depth in this course, but it is important to realize that the concept of
utility is useful in certain situations.
Questions:
1.5.a.
You may choose between the following two lotteries:
1)
P(Win $100) = 1/2
P(Lose $10)
2) P(Win $10,000) = 1/2
= 1/2
Win $100
(.5)
Lose $10
(.5)
P(Lose $9,910) = 1/2
$100
$10
Win $10,000
(.5)
Lose $9,910
(.5)
$10,000
$9,910
Assuming you are using EMV, show that you are indifferent between the two lotteries.
1.5.b.
This time not using EMV, but instead your intuitive feeling, which lottery would you choose?
1.5.c.
What would be your answer under 1.5.b. if you were a millionaire?
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1.6.
Decision Analysis: The Case of Petro Enterprises
An example of how to construct a more complicated decision diagram appears in the article “Better
Decisions with Preference Theory”, the Harvard Business Review, Nov/Dec 1967. In the article which we
will refer to as “The Case of Petro Enterprises”, it is important to realize the objective of the decision
maker. Of course this is important for every decision problem. The decision maker may want to maximize
profit, minimize net loss, minimize man hours lost, maximize units produced, minimize amount of
pollution, and so forth.
Although not relevant for Petro Enterprises, we take the opportunity now to indicate differences
between a maximization and a minimization problem. Consider the following decision tree:
EMV = 14
(.5)
Event B
(.3)
20
10
Event C
ACT I
Event A
(.2)
5
ACT II
Figure 1.3. Decision Tree for Maximization or Minimization Problem
The EMV at a chance point is calculated in exactly the same way for both a minimization and a
maximization problem. In Figure 10 the EMV at the chance event is: (.5)x(20) + (.3)x(10) + (.2)x(5) = 14.
The EMV at a decision point is calculated differently for “min” and “max” problems. If Figure 10
refers to a maximization problem, the EMV at the decision point is 15. (Choose Act II and disregard
(prune) decision I.) If Figure 1.3 refers to a minimization problem, the EMV is 14. (Choose Act I, and
prune Act II.) Why?
In the Case of Petro Enterprises, the objective is to maximize the asset position of the firm. Notice
that the ending asset positions include the beginning asset position of $130,000! This is particularly
important if we decide to use utility theory to value our end points instead of F24V.
CASE OF PETRO ENTERPRISES
Petro Enterprises is a fledgling organization founded to wildcat in the Texas oil fields. Petro has a
nontransferable shortterm option to drill on a certain plot of land. The option is the only business deal in
which the firm is involved now or that it expected to consider between now and December 31, 1967, the
time drilling would be completed if the option were exercised. Two recent dry holes elsewhere have
reduced Petro’s net liquid assets to $130,000 , and William Snyder, president and principal stockholder,
must decide whether Petro should exercise its option or allow it to expire. It will expire in two weeks if
drilling is not commenced by then. Snyder has three possible choices:
1.
2.
3.
Drill immediately
Pay to have a seismic test run in the next few days, and then, depending on the result of the
test, decide whether or not to drill.
Let the option expire.
In order to decide which of the three choices he will make, Snyder must resolve the following two
decisions.
14.
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Take the seismic test
Drill
Don’t take the seismic test
Don’t drill
He also faces two uncertainties that will affect his choices; these are:
Test Favorable
Oil
Test Unfavorable
No Oil
To create Mr. Snyder’s decision diagram requires connecting the forks together in proper sequence.
This is done in the diagram below.
Oil
Drill
Test favorable
No oil
Don’t drill
Oil
Take seismic test
Test unfavorable
Drill
No oil
Don’t drill
Don’t take seismic test
Oil
Drill
No oil
Don’t drill
Note the resultant sequence; he first decides whether or not to take the test. If he decides to take the
test, he then learns its outcome, decides on whether or not to drill, and finally learns (only if he drills)
whether oil is present. If he decides against the seismic test, he then makes the drilling decision, learning
whether oil is present only if he drills. The presence or absence of oil was determined eons before the time
of Mr. Snyder’s decision, but he will not know whether oil exists until after a decision to drill has been
made. Hence the event fork for the presence or absence of oil appears last.
Using a Decision Diagram to Analyze a Problem
Once the decision diagram is constructed, there are several remaining tasks necessary to analyze a
problem:
15.
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1)
Determine the appropriate probabilities that describe the relative likelihood of each branch on the
event forks. Since each event fork should have branches which are both “mutually exclusive” and
“collectively exhaustive”, the probabilities at each event fork should always sum to one.
2)
Determine a criterion which is an appropriate measure of the economic consequences of the
problem (for example, net cash flow) and evaluate this criterion at each end point of the diagram.
3)
Use an expected value analysis to “fold back” the diagram, choosing that alternative course of
action which has the highest expected monetary value (EMV) at each decision point (act fork).
4)
From this expected value analysis, determine the best set of decisions or optimal strategy for the
decision problem.
Having described Snyder’s possible choices, we consider their potential economic consequences.
To conserve capital and maintain flexibility, Petro subcontracts all drilling and seismic tests; also, it
immediately sells the rights to any oil discovered, instead of developing the oil fields itself. It can have
the seismic test performed on short notice at overtime rates for a fixed fee of $30,000, and the well can be
drilled for a fixed fee of $100,000. A large oil company has promised that if Petro drills and discovers oil,
it will purchase all of Petro’s rights for a flat $400,000.
To complete the description, it is necessary to know the probabilities assigned to the various
contingencies. The company’s geologist has examined the geology in the region and states that there is a
.55 probability that if a well is sunk, oil will be discovered. Data on the reliability of the seismic test
indicate that if the test result is favorable, the probability of finding oil will increase to .85; but if the test
result is unfavorable, it will fall to .10. The geologist has computed that there is a .60 probability the
result will be favorable if a test is made. (There is a simple, but important, logical interrelationship
between these probabilities, but it is not discussed until Section 4.4 of notes)
This decision problem involving uncertainty can be structured in the form of the decision tree
shown in Figure 1.4. The tree shows the probabilities, based on the judgment of the company geologist,
for the various events; see the figures in parentheses on the event forks.
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Oil
Drill
No oil
$100,000
$100,000
$400,000
Oil
(.60)
(.10)
$400,000
Test unfavorable
No oil
Drill
(.40)
$30,000
$0
(.15)
Don’t drill
Test favorable
Take seismic test
$400,000
(.85)
$400,000
$0
(.90)
$100,000
Don’t drill
$100,000
Oil
Don’t take seismic test
$430,000
(.55)
$400,000
Drill
No oil
$ 30,000
(.45)
$100,000
Don’t drill
$130,000
Figure 1.4.
The Decision Diagram with Probabilities, Cash Flows and Ending Cash Positions
$340,000
Drill
$100,000
Test favorable
$30,000
Test unfavorable
(.40)
No oil
(.15)
Don’t drill
(.60)
Take seismic test
Oil
(.85)
$400,000
$ 40,000
Oil
(.10)
$400,000
No oil
Drill
$100,000
(.90)
Don’t drill
Don’t take seismic test
$250,000
Drill
$100,000
Don’t drill
$400,000
$0
$100,000
$400,000
$0
$100,000
Oil
(.55)
$400,000
No oil
(.45)
$430,000
$ 30,000
$130,000
Figure 1.5
The Complete Decision Diagram with Expected Cash Positions of Final Decision
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Expected Value Analysis
How would Snyder’s problem be analyzed assuming that he is interested in ‘playing the averages’
and maximizing the mathematical expectation of his asset position (which is equivalent in this case to
maximizing the mathematical expectation of profit. Why?) These steps would be followed:
1)
Determine the asset positions Petro Enterprises would have if it arrived at each of the nine
end positions on the decision tree in Figure 1.4.
2)
Determine Petro’s best strategy by working backward through the tree; that is, at each fork
which represents a chance event (called an ‘event fork’) compute the expected value, and at
each fork which represents a choice of action (an ‘act fork’) choose that act which has the
highest expected value.
Computing Asset Positions
Having diagrammed the decision problem, we can put the cash flow associated with each act and
event on the diagram as is shown in Figure 1.4. For example, taking the seismic test costs $30,000, so an
outflow of this amount is indicated by writing ‘$30,000’ beside ‘Take seismic test’. Similarly, the
presence of oil results in an inflow of $400,000, so this figure appears by ‘Oil’.
The nine end positions of the tree represent the terminals of nine possible sequences of acts and
events. Corresponding to each is an asset position for Petro Enterprises. These asset positions can be
computed by summing the various cash flows from the origin of the diagram to each end position and
adding the total to the form’s current asset position of $130,000. The results of these calculations are
shown at the nine end positions show an asset position of $400,000. This is the sum of the receipts for the
oil and the current asset position, minus the costs of taking the seismic test and drilling.
The economic quantity which the decision maker uses to describe the result of a particular path on
his decision tree is called his criterion. In this case Snyder has chosen a criterion of net liquid assets, since
his liquid asset position determines his ability to consider future deals. Other businessmen in other
situations might well select earnings, net cash flow, or some other criterion. Obviously, the use of
different criteria can lead to different decisions in some situations.
Expectations and Choice
The terminal forks in Figure 1.5 event forks representing uncertainty about the results of drilling.
At each terminal fork we compute the expected value of the firm’s asset position, which is simply the
weighted average of the numbers at the end positions emanating from the fork. Taking the topmost
terminal fork again for illustration, the expected value is $340,000 (i.e., .85 x $400,000 + .15 x $0).
An analysis based on mathematical expectation assumes that Snyder would accept a $340,000 sure
asset position in exchange for a .85 chance of assets of $400,000 plus a .15 chance of $0 in assets and vice
versa. In other words, the asset position and the chance event are equivalent. Using utilities instead of
asset positions might be more realistic, but for the moment we will go along with it because it allows us to
replace the event fork by its mathematical expectation. As a matter of fact, since each terminal event fork
is assumed to be equivalent to its mathematical expectation, we can discard the terminal set of forks and
replace them by their mathematical expectations. We are left with the reduced diagram shown in Figure
1.6.
Now the terminal forks are act forks where Snyder’s choice is between drilling and not drilling. If
he is maximizing EMV, his choice is easy. He simply chooses the act with the highest EMV. Following a
favorable seismic test result, for example, the choice is between drilling, with an EMV of $340,000, and
not drilling, with an EMV of $100,000. Obviously, Snyder should decide to drill. Hence, if he were to
arrive at the position of the diagram following a favorable seismic test result, we know he would choose
to drill and thus would look forward to an asset position whose expected value is $340,000. It follows that
the fork is equivalent to an expected value of $340,000, so we put $340,000 at the base of the act fork.
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Once the results of similar choices have been placed at the base of each of the terminal act forks in
Exhibit 1.6, we can replace each act fork by its equivalent mathematical expectation, as shown in Figure 1
.4
Now we are faced with the reduction of the event fork representing the result of the test. The
procedure is the same as with any event fork; we take the mathematical expectation of the numbers at the
end positions  in this case $244,000 (i.e., .60 x $340,000 + .40 x $100,000).
After replacing the event fork by the expected value of its end positions, we are left with the single
act fork in Exhibit 1.7. The resultant act choice is easy; since $250,000 is greater than $244,000, Snyder
should not have his firm take the seismic test. Instead, he should drill immediately.
It is not_necessary to redraw the tree after reduction, as was done for illustrative purposes in
Figures 1 .6 and 1.7. We can simply write the appropriate mathematical expectation at the base of each
event or act fork and then prune the branch or branches not chosen.
The Optimal Strategy
In the example shown above there were several alternative strategies or sets of decisions that Mr.
Snyder could have chosen before analyzing the problem. Three of the most reasonable were:
1)
2)
3)
Take the seismic test; drill if the test if favorable, don’t drill if the test is unfavorable.
Don’t take the seismic test, but drill immediately.
Don’t take the seismic test and don’t drill.
The analysis has shown us that the expected asset position (expected monetary value) is highest for
strategy #2. Therefore, we shall refer to this as the optimal strategy, or more colloquially, the best set of
possible decisions. The analysis of this problem has assumed that Mr. Snyder was willing to “play the
averages”, that is, he was willing to use the expected value of the economic consequences as a basis for
evaluating uncertain events. In many situations this is a reasonable assumption. However, in other
situations, particularly those involving large possible losses, a decision maker may be very “risk averse”,
that is, he may be unwilling to evaluate uncertain events using their expected value because of what he
perceives as significant risks.
$340,000
Test favorable
Drill
Don’t drill
$340,000
$100,000
(.60)
Test unfavorable
Take seismic test
(.40)
$30,000
Don’t take seismic test
$100,000
$ 40,000
Drill
Don’t drill
Drill
$250,000
Don’t drill
Figure 1.6
First Reduction of the Decision Diagram
$100,000
$250,000
$130,000
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$340,000
Test favorable
$244,000
(.60)
Test unfavorable
Take seismic test
Take seismic test
(.40)
$244,000
$100,000
Don’t take seismic test
$250,000
Don’t take seismic test
$250,000
Figure 1.7
Further Reductions of the Decision Diagram
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1 .7
Decision Analysis:  Constructing a Decision Tree
The following is a set of “cookbook rules” for constructing a decision tree for a case study.
1)
Read the case carefully.
2)
Read the case once more. Identify and indicate (maybe underline) the decisions to be made and the
events that may occur.
3)
List choices and events. Collect the indicated choices (decisions to be made) and events in a “Taccount”:
Choices
Events
4)
Chronological sequence of_choices and events. Order the choices and events (in the Taccount)
according to a time sequence, simply by numbering the choices and events.1
5)
Draw the tree. Once the chronological sequence of decisions and events is determined the tree
follows immediately.
6)
Inspect the tree. Simply “climb” through the tree and check whether the tree is complete and the
sequences make sense.
7)
Calculate the Partial Cash Flows. Many branches have direct consequences in terms of cash flow.
Usually the partial cash flows are entered along the branches. (In large trees these can get in the
way so the final tree for analysis shall not include them.)
8)
Insert the Probabilities at the event forks.
9)
Evaluate the End Points. The sum of the partial cash flows out each branch (plus possibly a starting
cash flow) equals the end point value. (Of course units other than monetary ones are possible as
well.)
10)
Fold Back and Prune the Tree. Starting at the rightmost end positions the tree will be folded back
by calculating the EMV at each decision point or chance event. (Keep the difference between
decision and event points in mind.)
11)
State the Optimal Strategy. As a result of Step 10, the optimal strategy can now be stated. For
example: Choose “Act II”, if “Event A” occurs, choose “Act X”, if “Event B” occurs, choose “Act
Y”, etc.
12) Further Analysis: 13)
1
Sensitivity to incorrect probability assessments and on cash flow evaluation.
Breakeven probabilities
EVPI (See Chapter 2.3)
Evaluation of intangibles
Write a Coherent Report describing the decisions to be made and providing a rationale for your
analysis.
There are two important expectations to the need to represent forks in sequence that are worth mentioning, because they
allow the analyst flexibility. They are:
1)
A series of events may be shown in any order as long as there are no intervening acts.
2)
A series of acts may be shown in any order as long as there are no intervening events.
The above exceptions are mentioned because they can occasionally be exploited to simplify an analysis. For example,
sometimes it is easier or better to obtain a manager’s probability judgment if events are represented in one order as opposed to
another. In the meantime, remember that one can never go wrong by sticking to the time sequence.
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1.8
Case Study:_ Vancouver Electronics Company (B)
Late in August 1975, Mr. Andrew Howard, founder of Vancouver Electronics Company (VEC)
was trying to decide whether or not to accept a contract from Stone Manufacturing Company for 100
ampometers (see Vancouver Electronics Company (A)). Mr. Peter Wong, VEC’s Plant Engineer, is
discussing the problem with Mr. Howard.
“I think we should have Farentox prepare the mold, Andrew, and make the 100 housings
ourselves,” began Mr. Wong. “All of my projected costs show that we will definitely make more money
than if we buy the housings.”
“The problem with that, Peter,” replied Mr. Howard, “is that there is a chance that the Farentox
mold won’t give me a housing that’s acceptable to Stone. If this happens, then we’re back where we
started, purchasing the housings, and we’ve sunk $17,500 in a useless mold.”
“I don’t think there’s much chance of that,” said Mr. Wong. “We’ve had good luck with this type
of casting in the past, and it should work this time. If we go ahead with it, I can give you ten sample
housings by early next week, and Stone can check them then. True, if Stone doesn’t like them we’ll have
to buy the housings, but as I say, I’m almost sure this won’t happen.”
“There is still another option open to us,” Mr. Howard responded. “In the event that the housings
are unacceptable to Stone, we can forfeit the contract and pay Stone a penalty of $1,000.”
Problems:
1.8.a.
What is the uncertainty Mr. Howard is faced with?
1.8.b.
Draw a decision tree that accurately portrays Mr. Howard’s decision problem. (You may like
to use the “cookbook rules” in Chapter 1.7.)
1.8.c.
If Mr. Howard decides to use expected monetary value as his decision making criterion, what
policy should he adopt?
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1.9
Case Study: Vancouver Electronics Company (C)
A few minutes before Mr. Howard was going to call Mr. Stone with his decision regarding the
production of electronic ampometers he received a phone call. It was Mr. Stone.
“Mr. Howard, I have a problem, and it concerns the number of ampometers I’m going to need. We
spoke earlier of 100, but now, due to the outside possibility of my automated factory being smaller than
originally planned, I may need only 50. I won’t be able to tell you the quantity for two weeks, but I will
have to know by tomorrow whether or not you will accept the contract. If you accept, I can test ten of
your trial housings. I’ll be able to let you know if they’re okay within a week of delivery.” If you accept
the contract I would like to receive the finished ampometers no later than January, 1976.”
Mr. Howard had little bargaining power with Mr. Stone since he could take his ampometer contract
to Burnaby Components, a new struggling electronics company that was eager for any business it could
get. Therefore, Mr. Howard knew he had to accept Mr. Stone’s conditions. Even though he was not happy
with this recent turn of events, Mr. Howard still felt there was a tidy profit to be made on this contract.
Before Mr. Andrew Howard made his decision, he felt that there was additional information he
should consider. He was concerned that he didn’t have a better feel for the likelihood of the various
outcomes. He called Peter Wong, and asked him to come to his office. Mr. Wong arrived about five
minutes later.
“Peter,” Mr. Howard said, “if we buy the mold, the first thing that we’re going to find out is
whether or not the housings are acceptable. Can you give me a better idea of how likely it is that we’ll be
successful?”
“I don’t think we have too much to worry about, Andrew,” Mr. Wong replied. “As I mentioned
before, we’ve had good luck with this type of casting  I’d say that there’s an 80% chance of the mold
producing good housings.”
After Mr. Wong left, Mr. Howard called Stone Manufacturing, and asked Mr. Stone, “Mr. Stone,
you said earlier that there’s a chance that you won’t be needing 100 ampometers, but only 50. Can you
give me some idea of how much of chance there is of that?”
“Well, I don’t think there’s too much of a chance  the problem is that two of the board members
think the new White Rock plant should be a small one, and their principal plant in Portland, Oregon
should be expanded. The rest of the board members are pretty much convinced that it’s in their best
interests to expand Canadian operations and build a large facility in White Rock. My best guess at this
point in time is that there’s about an 85% chance of their building the factory that would require 100
ampometers. Of course, this will be resolved when the board meets in two weeks.”
After thanking Mr. Stone for his information, Mr. Howard addressed himself to the problems of assessing
the value of the mold and deciding whether or not to manufacture the housings. After much consideration
he decided to be very conservative and assume that no more housings would be needed. He felt that in the
fairly touchy liquid asset position of Vancouver Electronics, he should not make any assumptions
concerning the future demand for a specialty product like electronic ampometers. His best estimate of the
salvage value of the $17,500 mold was $500.
Finally, Farentox, the mold manufacturer informed Howard: “Sure Mr. Howard, we could supply
you with the mold within two days. But I would just like to remind you that starting next Monday we will
be closed for three weeks, our usual holiday period for all personnel, and we will be tied up on the Morris
Contract for six months.”
Somewhat disappointed by the last message, Mr. Howard started to analyze his problem and the
effect of the recent turn of events.
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Problems:
1.9.a.
Diagram Mr. Howard’s decision problem, including all information you think necessary for
him to make a decision. Recall that he must decide whether or not to buy the mold within
two days.
1 .9.b.
If Mr. Howard decides that he will use expected monetary value as his decision making
criterion, which policy should he adopt?
1.9.c.
Why was Howard disappointed by the information he received from Farentox?
1.9.d.
Suppose the date required for delivery of the ampometers was instead January, 1977 what
should Howard do now? i.e. Find his best decision rule using the EMV criterion.
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1.10
Case Study: Central Valley Vineyards
Mr. Robert Burns, owner of Central Valley Vineyards, located in the Okanagan Valley, pondered a
difficult decision late in August 1975. Burns raises grapes, and his problem concerned what to do with his
“leftover” grapes.
The grape that Burns grows is a multipleuse variety that can be utilized for canning, fresh table
consumption, wine production, or for sundrying into raisins. The acreage of canning and table grapes is
invariably contracted for at the beginning of the season, and then the remainder of the crop may be shifted
late in the season to either wine grapes or to raisins. This is known as “going wet” or “going dry”,
respectively.
This decision is usually made in August, near the end of the season, and once it has been made, it is
irrevocable. The weather conditions after the decision is made are critical, and they are difficult to predict.
Raisins in British Columbia are sundried completely in the open, and rain during this time can inflict
heavy losses on a farmer who is going dry. If the farmer is going wet, the grapes remain on the vines for
several weeks longer, and rain does not do as much damage.
Robert Burns has 100 acres of “uncommitted” grapes and wishes to consider three alternatives: 1)
allocate all of his acreage to raisins; 2) allocate all of the acreage to wine grapes; 3) allocate
approximately half of the acreage to each use. As for the weather, he feels that he can simplify his
problem by assuming that the rainfall situation will either be none at all, light or heavy. Burns has
available the past 20 years of SeptemberOctober weather records, as listed in the table on the next page.
Mr. Burns summarized the rainfall conditions for the above Sept/Oct periods in the categories
“no”, “light”, “heavy” according to whether the average rainfall was less than 1 cm, between 1.0 and 5.0
cm and greater than 5.0 cm. He then constructed the table below to show his expected dollar profit per
acre under the various acreage alternatives and weather conditions.
Rainfall
Conditions
“No”
“Light”
“Heavy”
Acreage Allocation
Raisins
Wine
Both
50
40
60
40
30
50
10
20
20
Problems
1.10.a.
Estimate the probability of “no”, “light” and “heavy” rainfall. Which method of probability
assessment did you use?
1.1O.b.
Which alternative course of action should Mr. Burns take if he decides to use EMV as his
decision making criterion?
1.10.c.
Is the optimal decision rule sensitive to how you grouped the data?
1.10.d.
In what way is this problem unrealistic. How would you change it?
1.10.e.
Do you believe using the average rainfall during Sept./Oct. is an adequate measure, of
precipitation for making a decision? Why or why not?
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