Assumptions, Page 1 of7
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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 odd-numbered 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?
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:
0.05 / 0.50 / 0.20 / 0.25S1 / S2 / S3 / S4 / B of B / B of W / E. L. / E. V. / M/M R
d1 / -200 / 450 / 100 / 75 / 450 / -200 / 106.25 / 253.80 / 550
d2 / 350 / 75 / 300 / -100 / 350 / -100 / 156.30 / 90.00 / 450
d3 / 100 / 250 / -100 / 350 / 350 / -100 / 150.00 / 197.50 / 400
d4 / 125 / 50 / 100 / 75 / 125 / 50 / 87.50 / 70.00 / 400
d5 / 100 / -50 / 50 / 25 / 100 / -50 / 31.25 / -3.75 / 500
Table 1: Payoff Table
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 d2 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 d2 & 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-of-nature 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 non-quantitative 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 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, d6, which has the following payoffs:
S1 / S2 / S3 / S4d6 / 500 / 500 / 500 / 500
Table 2: Alternative d6
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 non-quantitative 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.
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