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Assumptions
Q: What is an assumption?
A: You mean you don't know?
D: Despite having worked with assumptions in many of your classes, most of you can' t define what one is. I get guesses,
such as, "Something that we believe is true." which isn't quite correct. To illustrate what an assumption actually is, let
me tell you a little fairy tale:
Once upon a time, an evil sorcerer decided to open a theme park. He was going to copy the most popular shows and
rides from other theme parks, but add his own sadistic touches to each. When he visited the Disney theme parks, he
found that many people enjoyed their wide-screen movies (180º or 360º), so he decided to install one in his theme park.
What could he add, though, to torture people? Then he had an idea!
He went to an engineer and threatened to turn him into a toad unless the engineer developed a perfect hologram. Well,
everyone knows that a hologram is a 3-dimensional light image, but current state-of-the-art is far from perfect, the
images tend to flicker, and you can see through them. What the sorcerer wanted was something that would look (but not
be) solid. The engineer found the threat to be a sufficient motivation that he quickly fixed the problems with holograms
and delivered to the evil sorcerer a machine to project seemingly solid holograms.
The sorcerer proceeded to construct his theme park (any suggestions for an appropriate name?), and when it was time to
build the theatre, he put in the most comfortable chairs he could, except that every other chair in each row (all the oddnumbered ones) were those seemingly-solid holograms. Then he installed a video system so he could watch the fun.
Now let’s think about the visitors to the sorcerer’s theme park. They pay their $50 (2006 prices, in case I don’t keep this
thing updated) to get in, wait in line for an hour to get into the movie, and finally they are moving down the rows,
selecting a seat. When they pick a seat and go to sit down,
Q: What are they assuming about the chair?
A: That there really is a chair to sit in.
Q: What happens if the assumption is correct?
A: They sit down very comfortably.
Q: What happens if the assumption is incorrect?
A: They fall over on their ______ (fill in the blank with whatever word you use for that portion of your anatomy).
D: This is exactly what an assumption is. Not something that we believe to be true (our beliefs are irrelevant), but some
thing that MUST be true or else all our work results in our falling over on our _____.
All computer models have assumptions built into them. These assumptions may or may not match well with the problem
you are currently facing. Even if the assumptions do not match the problem situation, the computer model will still run
and will still give you an answer. Unfortunately, the answer will be completely worthless, because it is based on
assumptions that are not correct. This may be the scariest thing about computer models – they work even when they
shouldn’t. The computer has no brain, and therefore no means of determining whether or not a model should be used.
That is up to you. The computer will do exactly what you tell it to do. You must determine the appropriate model.
Q: So, what are the assumptions for payoff tables?
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A: Mutual Exclusiveness, Exhaustiveness and Determinism.
D: Don’t worry about the long names, as they are just more of that academic writing. For each assumption, you need to
know a definition that applies to this model – payoff tables. There are generic definitions, but they are not all that useful
because they tend to use technical words that confuse rather than tell you what to look out for. You also need to be able
to apply the assumption to a given situation, so that you can decide whether or not the assumption is acceptable.
Q: Does an assumption need to be completely correct to be acceptable?
A: No. Assumptions are rarely 100% correct. More commonly, we are asking ourselves if there is anything in the situation
that allows us to accept the assumption despite its inaccuracies.
Q: What happens if we reject an assumption?
A: If you reject an assumption, then you reject the model. As we mostly want to use the models (that’s why your company
bought them, after all) we spend an awful lot of time rationalizing, to make the situation fit the assumptions. This isn’t a
bad thing; it just means YOU have to be aware of the assumptions so you can allow for any potential inaccuracies
created by the lack-of-perfect-fit. Usually, there is some way to work around the assumptions, so you can still use the
computer models. To do that, though, you need to understand the assumptions, so:
Q: What is the assumption of Mutual Exclusiveness?
A: I usually get an answer something like, “If one thing, then not another,” which is pretty good. I also usually get a very
wrong answer, “Things are independent.” Independence and mutual exclusiveness are almost the exact opposite of each
other, as I will explain below.
A more specific definition would be “Only one decision alternative may be chosen, and only one state-of-nature will
occur.” Notice that I used different verbs (“chosen” vs. “occur”) for the different parts of the definition. This is
important because of an important distinction:
Q: What is the difference between decision alternatives and states-of-nature?
A: Control.
D: You control the selection of the decision alternatives, but you do not control which of the possible futures turns out to be
the one you face. This difference can help you when setting up a payoff table (or decision tree, our next model). If you
are looking at something and are not sure whether it is a decision alternative or state-of-nature, ask yourself whether or
not you control it. If you do control it, then it is a decision alternative. If not, it is a state-of-nature. Getting back to the
assumption, though,
Q: Who says we can pick only one alternative? Why can’t we pick two, or more, if we want to?
A: Actually, this is a problem with how the data is usually collected.
D: Consider the payoff table we were working (shown below as Table 1) and consider the following question:
d1
d2
d3
d4
d5
0.05
S1
-200
350
100
125
100
0.50
S2
450
75
250
50
-50
0.20
S3
100
300
-100
100
50
0.25
S4
B of B B of W
E. L.
E. V.
M/M R
75
106.25  253.80
 450  -200
 550
-100
-100  156.30
90.00
 350
 450
350
-100  150.00  197.50
 350
 400
75
125  50
87.50
70.00
 400
25
500
 100
 -50  31.25  -3.75
Table 1: Payoff Table
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Q: Where did the payoffs come from?
A: Someone sat down and worked out the profit or loss for each combination of decision alternative and state-of-nature. So,
Q: If we choose d2 and S1 occurs, how much money do we make?
A: $350 million.
Q: If we choose d3 and S1 occurs, how much money do we make?
A: $100 million.
Q: If we choose d2 and d3 and S1 occurs, how much money do we make?
A: I don’t know.
D: You might have thought the answer was $450 million ($350MM + $100MM), but that assumes independence, meaning
to invest in one alternative has no effect on the outcome of another alternative. That may or may not be true, but the data
does not indicate it. Each payoff was calculated without reference to the others, which means we have NO IDEA what
the interactions may be. It is possible the d 2 and d3 compete for some resource, so doing both means both payoffs would
decrease. It is possible that doing d2 makes it easier to do d3, so the payoff for d3 would increase. We just don’t know.
The only way to find out is to consider doing d 2 & d3 as a separate alternative, and work out the payoff for that
combination.
While that might sound attractive, keep in mind that your payoff table would get pretty big, pretty quickly. With just
five alternatives, if we wanted to list all possible combinations of two alternatives, then three, four and all five, that is
another 26 alternatives to add to your table. That might be a bit more than you want to analyze.
Q: Is mutual exclusiveness really a serious problem?
A: No, it is more of a warning.
D: Once you know about this assumption, you know not to pick two alternatives from one table. In the same way, when
you set up the states-of-nature, you know to set them up so that only one state-of-nature can occur. For example:
Q: If the weather in the near future was going to affect some decision you were facing, could you set up your states-ofnature in a payoff table as S1 = hot, S2 = cold, S3 = wet, and S4 = dry?
A: No, that is not a mutually exclusive set, since you would expect the weather to be wet & hot or wet & dry, etc.
D: When the states-of-nature are financial, say, sales or revenues, then it is pretty easy to be mutually exclusive. Just list
your states-of-nature as S1 = < $10MM, S2 = $10MM to 25MM and S3 = > $25MM, and you are done. With nonquantitative states-of-nature, you want to be a little more careful.
Q: What if we want to choose more than one alternative but don’t want to list out all the combinations, can we still use a
payoff table?
A: Yes.
D: This is what I alluded to earlier, when I said that you can usually get around the assumptions. Consider a situation where
your company offers in-house grants to explore new ideas. You are the one in charge of distributing the money, and you
have one million dollars to give away. You have 10 applications for money, each requiring different amounts of money,
between $100,000 and $300,000. Since this is clearly NOT a mutually exclusive situation (you must pick more than one
alternative to spend the million dollars), initially you might decide you can’t use a payoff table. You could, however, use
the payoff table to pick the single best alternative. After that, delete that alternative from the table and review your
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payoffs to see if any have changed as a result of your first choice. Once you have the corrected payoffs, you can use the
new table to pick the second best alternative, and keep repeating this until you run out of money to give away.
So you see, the assumption isn’t a problem, it is simply a warning, that if you want accurate results, you must know the
limitations (in this case, assumptions) of your model.
Q: What is the assumption of Exhaustiveness?
A: Exhaustiveness is a long-winded way of saying “complete,” so exhaustiveness means that you have a complete list of all
decision alternatives and all states-of-nature.
D: First, you need this assumption because if your lists are incomplete, then your analysis will almost certainly be wrong.
For instance, what if there were another alternative I overlooked on our payoff table, d 6, which has the following payoffs:
d6
S1
S2
S3
500
500
500
Table 2: Alternative d6
S4
500
Clearly, d6 is superior to any of the other alternatives, but if it not part of the table, we cannot analyze or choose it.
Second, exhaustiveness is rarely completely true. For example, suppose you were an investment counselor and you had
a list of five possible investment plans for one of your clients. Would a sixth alternative be to steal the money and run
off to Acapulco with it? Of course it is an alternative, but you wouldn’t put it in the table (especially not if you were
actually considering that; why leave a paper trail?). So, we automatically discard the weird or foolish alternatives, but
isn’t really the issue.
The assumption of exhaustiveness is warning to you that you can never be certain you have included everything you
ought to have included. If you are an honest person and doing your best for your company, this should concern you.
You’ll do the best you can, and ask others for ideas, and look at other companies that have faced similar situations, but
you must still consider the possibility that there is another alternative that might be better than anything you have
considered. This is even worse with the states-of-nature.
As with mutual exclusiveness, If your states-of-nature are quantitative/financial, then exhaustiveness is pretty easy to
satisfy, as S1 = < $10MM, S2 = $10MM to 25MM and S3 = > $25MM is exhaustive, covering everything from negative
to positive infinity. You might, though, be worried about whether you have used the correct groupings. With nonquantitative states-of-nature, such as anticipating reactions by your competition, you will be worried that you may have
overlooked something they might do, which could really mess up all your plans.
Ultimately, though, you need this assumption because it reminds you that at some point you simply have to make your
decision. When the time to make the decision arrives, stop worrying. Whether or not you have included everything is no
longer relevant. Analyze the data you have, and make a choice from the alternatives in front of you, then sit back and
see what happens. Don’t let yourself be paralyzed by uncertainty. When the time comes to make a decision, act with
confidence. After all, there is a military saying that one sign of a good leader is that s/he can make decisions with
confidence. If those decision happen to be right, then so much the better.
You will never have all the alternatives or states-of-nature listed, and I really don’t have any suggestions for how you
can create as complete a list as possible. All I can say is to do the best you can, and then assume exhaustiveness.
Q: What is the assumption of Determinism?
A: Determinism is the assumption that all of your data is 100% accurate.
D: Hopefully, you realize that this assumption is complete garbage. If decision making deals only with the future, then all
your data must deal with the future as well (some of it may be historical, but you are using that data to project into the
future). Data that deals with the future are called forecasts. The one thing we know about a forecast is that it is wrong.
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Q: Why would we use wrong data to make a decision?
A: Right or wrong, it is the best data available.
D: Please, please, please, please note that determinism does NOT mean you have the best data available. If you don’t have
the best data available, then you are an idiot. “Best data available,” however, is an interesting phrase, with “available”
being the most important word. Availability hinges on two points: time and money. You have only a certain amount of
time within which to collect your data, and you have a limited amount of money with which to purchase data. Thus your
“best data available” would not be the same as someone else’s, even though you were working on the same problem.
Getting back to my earlier comment, if you know you have to make a decision, you know there is data out there that you
have time to collect and can afford, but choose not to, well, what would you call yourself? So, determinism is the
assumption that the data you have, which is the best available, is 100% accurate.
Q: Why is the “best data available” always wrong?
A: We use point estimates rather than ranges.
D: This gets back to the idea I brought earlier, about degrees of wrongness. Any payoff number we use is certainly wrong
because if our number is off by even a penny, then in a technical sense, the forecast is wrong. Of course, if you forecast
a payoff of $450 million and the true payoff turned out to be $450 million and one penny, then I don’t think your boss
would be too concerned with your lack of forecast accuracy. Mathematically, however, you were wrong, and the odds
are exactly zero of selecting precisely the correct, single number as a forecast. We could get around this by using data
ranges (such as $380MM to $495MM) rather than the single number of $450MM, but then we have another problem:
how do we analyze twenty ranges? How do you average them or select the best or worst? The mathematics we normally
use can’t do this. This is why we need the assumption of determinism.
As a side-note, there is a branch of mathematics that deals with the algebra of imprecise estimates. It is called “fuzzy
logic” and was very popular in the 1980’s, particularly in relation to expert systems. Unfortunately, because dealing
with fuzzy (uncertain) data, tends to give fuzzy (that is, numerous) recommendations. This pretty much leaves the
decision maker right back where s/he started: making a choice from a set of alternatives. So, fuzzy logic never really
caught on in the business world.
Determinism works a bit like exhaustiveness; it tells you to quit worrying about possibilities and get to work analyzing
the data you have. This lets you make a recommendation as if your data were perfect, even though we are concerned
about the accuracy of our data. Unlike exhaustiveness, though, we can do something about our concerns. The way we
deal with determinism is to perform a Sensitivity Analysis.
Q: What’s a Sensitivity Analysis?
A: A means of comparing expected variation with allowable variation.
D: Gee, that was helpful, wasn’t it?
The most important thing about a sensitivity analysis is when it is done. You do not even try to do a sensitivity analysis
until after you have completed your analysis and know which alternative you are going to recommend. Now, go back to
the idea of degree of wrongness and ask
Q: Would you feel better about your recommendation if you knew your data, though wrong, was only off by a few pennies?
A: Of course you would, unless for some bizarre reason being off by a few pennies would be disastrous.
D: Unfortunately, there is no way to know that. So what we do instead is try to build confidence by showing that your data
is sufficiently accurate. Since we can’t tell you how far wrong the data is, try instead to find out how far wrong the data
can be before we have a problem.
Q: What do you mean by “have a problem?”
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A: In this setting, “having a problem” means we would have chosen a different alternative.
D: We can find out how far wrong our data can be by doing something very simple: change the data and see if we change
our recommendation. This is why you must first have a recommendation made before beginning a sensitivity analysis.
As a result of your analysis, you should have a feel for which numbers in the data were most significant to you in
reaching your recommendation. Go to those numbers. If the piece of data was for the alternative you recommended,
then make it a little worse (decrease a profit or increase a cost). If it was for an alternative you didn’t recommend, then
make it better (increase a profit or decrease a cost). After you have made the change, look at the whole table and decide
whether or not you would still make the same choice (this doesn’t take long). If you didn’t change your mind, stick with
that same number and make a further change. Keep repeating this until you finally change your recommendation. You
have just found out how far wrong that piece of data can be before you have a problem. To put that in more formal
words, you have found the allowable variation for that piece of data. Now you must repeat this process for every one of
the most important pieces of data that you found during the analysis. If that sounds boring, it is.
Q: How does knowing the allowable variation for each important piece of data help?
A: It doesn’t, not by itself. Remember that we are supposed to compare expected variation to allowable variation.
Q: What is expected variation?
A: Literally what is says. It is determining the degree of confidence (or lack thereof) that can be placed in the data you
collected. It might help to think of “confidence” as the inverse of “expected variation.” The higher your confidence is in
the data, the lower the expected variation of that data is, and vice-versa. When we know the allowable variation, we’ve
done only half the job, though it is the harder half. The reason the next step is easier is because we do not actually assign
a value to expected variation. Now, the key word in “expected variation” is “expected,” which brings up the question
Q: Whose expectations?
A: Yours.
D: That may sound odd, but you are the person in the company with the most knowledge about this decision situation. You
have collected the data, it is your recommendation to make, who would be better to decide how much variation to expect
(that is, how much confidence do you have in the data)? All you have to do for this step (expected variation) is to look at
the allowable variation range for each number and ask your self whether or not you think the forecasted value will
change more than that. If you say, “No,” then that piece of data is acceptably accurate. If you say, “Yes,” then that piece
of data is not acceptably accurate, but you do not necessarily have a problem.
Repeat this evaluation for each of the numbers where you set an allowable variation range, then think about what you
have learned. If you are comfortable with the amount of your data that was acceptably accurate, then you have passed
the sensitivity analysis and the assumption of determinism is also acceptable. If you are not comfortable with the amount
of acceptably accurate data, then you have failed the sensitivity analysis, determinism may have to be rejected and you
have another decision to make.
Note that all of this is based on your judgment. That may sound awkward, but remember our premise: you are trying to
do your job well, and it is your data. That makes you the best person in the company to make these sorts of judgments.
Scary, isn’t it?
Q: What do we do if we fail the sensitivity analysis?
A: One of three things:



ask for more time (and money) to collect better data,
change your recommendation, or
stay with your first recommendation.
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D: The first choice is a gamble, because it might annoy your boss. Before you ask, make sure you actually can collect
better data. Otherwise you will end up looking pretty bad when you come back and base your recommendation on the
data you started with. If this isn’t an option, then consider the alternative that you recommended. If you were taking a
bit of gamble, and there is another alternative that is somewhat more conservative, then you might consider changing
your recommendation strictly because you lack confidence in your data. If that is not the case, then I would say to stick
with your first recommendation. After all, it was the alternative you preferred, based on the best data available and the
best analysis you could make of that data. I would, however, warn my superiors that I was worried about the accuracy of
the data and the actual outcome could be significantly different from the forecasted value.
Q: Where does this leave us with the assumption of determinism?
A: You are finished with it.
D: Determinism, as an assumption, is needed to allow you to perform the necessary calculations and then to warn you to
perform a sensitivity analysis. The other assumptions for your model are acceptable, then a problem with determinism
is not sufficient reason to reject the model. After all, if you switched to some other model, you would have the exact
same problems with the data itself. If you are in the fortunate position of being able to collect more data, then do so and
repeat the whole process. If not, then the decision must still be made, you have the best data available and you are still
the person most qualified to make the decision. This is where you earn your salary.
Well, all this is simply background to the process of actually making the decision. As I warned you in the first lecture,
we spend a lot of time understanding the model before we use it. The next lecture will teach you to analyze a payoff
table.
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