ࡱ> s jbjb kk|&]V&V&V&V&&&&4n(n(n(n(h(,)n(>**(:*:*:*---I>K>K>K>K>K>K>,x@lB4w>&-,"---w>3V&V&:*:**333-V&R:*&:*I>&'V&V&V&V&-I>33s9R7=&"&I>)\>n(n(1+>Supplement to Chapter 12 Decision Criteria Section 12.2 briefly introduced three decision criteria the maximax criterion, the maximin criterion, and the maximum likelihood criterion before focusing on Bayes decision rule as the criterion to be used in the remainder of the chapter. We now will describe these first three decision criteria, as well as three additional criteria, in greater detail. This presentation is self-contained and so will include much of what was said in Section 12.2 in this expanded coverage. Recall that Max Flyer, the founder and sole owner of the Goferbroke Co., does not put much faith in the consulting geologists numbers in estimating that there is one chance in four of oil on the tract of land. Since these numbers led to the prior probabilities of the possible states of nature, Max would prefer to make his decision without relying on these prior probabilities if there is a good way of doing so. His daughter Jennifer, who has studied management science, agreed to begin by introducing him to some decision criteria which dont use these probabilities before going on to others which do use probabilities. Now let us eavesdrop on her description. The Maximax Criterion The maximax criterion is the decision criterion for the eternal optimist. It says to focus only on the best that can happen to us. Here is how it works. For each decision alternative, determine its maximum payoff from any state of nature. Determine the maximum of these maximum payoffs. Choose the alternative that can yield this maximum of the maximum payoffs. Figure 1 shows the application of this criterion to Goferbrokes problem when using the corresponding Excel template in one of this chapters Excel files. It begins with the payoff table (Table 12.3) without the prior probabilities. In step 1, we find the maximum in each row (700 and 90). Step 2 identifies 700 as the maximum of these two numbers. Since 700 is the payoff that can be yielded by drilling for oil, this is the decision alternative to be chosen in step 3.  Figure 1 The Excel template for the maximax criterion, applied to the first Goferbroke Co. problem. An Objection to This Criterion: Now suppose that the payoff table were the one shown in Figure 2. The maximax criterion again leads to choosing the alternative of drilling for oil. What a terrible decision! In the somewhat unlikely event that oil is there, drilling only does negligibly better than selling. In the far more likely event that the land is dry, drilling gives a disastrously large loss.  Figure 2 Application of the maximax criterion to a variation of the first Goferbroke Co. problem. Maxs Reaction: Max: At first, I thought this criterion might have possibilities because I view myself as something of an optimist. However, now I see that this criterion is really over the top in being totally optimistic. I need something that is more discriminating. Jennifer: Now that you see that you need to rein in your optimistic tendencies, let us look at a more conservative criterion. The Maximin Criterion The maximin criterion is the criterion for the total pessimist. In contrast to the maximax criterion, it says to focus only on the worst that can happen to us. Here are the steps. For each decision alternative, determine its minimum payoff from any state of nature. Determine the maximum of these minimum payoffs. Choose the alternative that can yield this maximum of the minimum payoffs. Applying this criterion to Goferbrokes problem gives Figure 3 (another Excel template). The basic difference from Figure 1 is that the numbers in columnH (-100 and 90) now are the minimum rather than the maximum in each row. Since 90 is the maximum of these two numbers, the alternative to be chosen is to sell the land.  Figure 3 The Excel template for the maximin criterion, applied to the first Goferbroke Co. problem. An Objection to This Criterion: The fact that this criterion is overly cautious is dramatically illustrated in Figure 4. The maximin criterion still says to sell the land. However, the Dry state of nature now means there is a little oil there so drilling gives virtually the same payoff as selling. Furthermore, the Oil state of nature means that there is a huge oil field there. With roughly one chance in four that the latter state of nature is the true one, the gamble of drilling for oil instead is the obvious best decision.  Figure 4 Application of the maximin criterion to a variation of the first Goferbroke Co. problem. Maxs Reaction: Max: I can see now that this criterion would never say to drill for oil unless we absolutely knew that there was some there. That is no way to run an oil prospecting company. Dont you have a criterion that strikes a happy medium between being totally optimistic and totally pessimistic? Jennifer: Yes, that is our next one. You now have seen that you are neither a total optimist nor a total pessimist. This next criterion asks you to rate yourself as to just where you fall in between. On a scale from 0 to 1, where 0 = totally pessimistic, 1 = totally optimistic, 0.5 = neutral (midway between), where would you place yourself? Max: I definitely am a little on the optimistic side. On this scale, I would put myself at about 0.6. Jennifer: OK, good. We call this number your pessimism-optimism index. So setting yours at 0.6, lets now look at how this criterion works. The Realism Criterion The realism criterion is basically a combination of the preceding two, where the pessimism-optimism index is used to combine them appropriately. The steps are given below. For each decision alternative, determine both its maximum payoff and minimum payoff from any state of nature. For each decision alternative, use the pessimism-optimism index to calculate its weighted payoff as Weighted payoff = index times maximum payoff + (1 - index) times minimum payoff. Choose the alternative with the largest weighted payoff. Figure 5 shows the application of this criterion to Goferbrokes problem on the corresponding Excel template. Note that columns H and I are just column H of Figures 1 and 3, respectively. Column J then uses the formula in step 2, with an index of 0.6, to calculate these weighted payoffs. Since the weighted payoff for drilling (380) is larger than for selling (90), the decision is to drill for oil.  Figure 5 The Excel template for the realism criterion, applied to the first Goferbroke Co. problem. This criterion provides a welcome middle ground between the maximax and minimax criteria. Furthermore, selecting a value for the pessimism-optimism index enables the decision maker to choose just how aggressive or cautious to be. The criterion even gives the decision maker the flexibility to be totally optimistic (index = 1) or totally pessimistic (index = 0) if desired, so the maximax and maximin criteria actually are special cases of this one. However, this criterion also has its flaws, as indicated below. An Objection to This Criterion: Figure 6 gives the payoff table for another problem (unrelated to Goferbrokes problem) that has been specially designed to show an extreme case where the realism criterion performs badly. Note that alternative 2 is much better than alternative 1 for just the first state of nature, whereas the reverse is true for the other four states of nature. Assuming that the first state of nature is not particularly more likely than any of the others, alternative 1 clearly is far better than alternative 2. Nevertheless, the realism criterion with an index of 0.6 chooses alternative 2.  Figure 6 Application of the realism criterion to another example. In fact, this criterion would choose alternative 2 with any value of the pessimism-optimism index. The reason is that this alternative has the larger value in both columns H and I of Figure 6. Maxs Reaction: Max: I rather like this criterion. It is more realistic than the first two criteria, and it even takes into account how optimistic or pessimistic I want to be. Furthermore, my problem only has two states of nature rather than the five in the example you just gave. Therefore, your objection to the criterion doesnt really apply to my problem, does it? Jennifer: Unfortunately, it does to some extent. The reason for having five states of nature in the example was to emphasize that the payoffs for one unlikely state of nature should not dictate the decision as strongly as this criterion allows. Max: I still dont see how this applies to my problem. Jennifer: Well, suppose you have a tract of land that is probably dry, but there is a small possibility of a lot of oil there. This possibility is your unlikely state of nature. However, suppose the possibility is so small that it clearly is not worthwhile to drill for oil. What do you think this criterion would tell you to do? Max: Oh oh. I suppose it would tell me to drill anywhere. Jennifer: Yes, it would! It just doesnt differentiate between very unlikely and somewhat likely states of nature. Now you have your consulting geologists report estimating that there is one chance in four of oil on your tract of land. Is this likely enough to make drilling worthwhile? This criterion just doesnt address this question. Max: Youre right. And this really is the key question, isnt it? I am beginning to think that I need to use the consulting geologists numbers somehow, unless you have a better criterion that doesnt need them. Jennifer: I do have one more that you might like better. The Minimax Regret Criterion The minimax regret criterion gets away from the focus on optimism versus pessimism. Instead, its focus is on choosing a decision that minimizes the regret that can be felt afterward if the decision does not turn out well. This is how regret is measured. After observing what the true state of nature turns out to be, the regret from having chosen a particular decision alternative is Regret = maximum payoff - actual payoff, where maximum payoff is the largest payoff that could have been obtained from any decision alternative for the observed state of nature. Table 1 shows the calculation of the regret for Goferbrokes problem. On the left is the payoff table, with the maximum payoff for each state of nature given just below this table. On the right, the above formula is used with these maximum payoffs to calculate the regret for each combination of a decision alternative and a state of nature. The table on the right is called the regret table. Note that the regret is 0 if you drill for oil and oil is found, because this is the best alternative for this state of nature. The same holds true for selling the land if the land is dry. However, if you sell the land and it contains oil, you have given up a payoff of another 610 by not drilling. Similarly, drilling when the land is dry is 190 worse than selling. Table 1 Calculation of the Regrets for the Goferbroke Co. Problem Payoff Table Regret Table AlternativeState of Nature Oil Dry AlternativeState of Nature Oil DryDrill for oil 700 -100 700 90Drill for oil -700 -(-100)Sell the land 90 90 0 190Maximum payoff: 700 90 700 90Sell the land -90 -90 610 0 After obtaining the regret table, the following steps are followed. For each decision alternative, determine its maximum regret from any state of nature by referring to the regret table. Determine the minimum of these maximum regrets. Choose the alternative that can yield this minimum of the maximum regrets. Figure 7 illustrates the application of these three steps to Goferbrokes problem on the Excel template for this criterion. The numbers in cells H17 and H18 are obtained in step 1. Step2 determines that the minimum of these numbers is 190, so step 3 chooses the corresponding alternative of drilling for oil. This alternative guarantees that the regret after learning the true state cannot exceed 190, whereas the regret can be as large as 610 with the other alternative.  Figure 7 The Excel template for the minimax regret criterion, applied to The first Goferbroke problem. An Objection to This Criterion: Now let us add a third decision alternative to the problem. Based on the consulting geologists report, an insurance company would be willing to sell the Goferbroke Co. an insurance policy to protect against the land being dry. If Goferbroke pays a massive premium and then drills for oil without finding any, the insurance company will pay an amount $700,000 larger than the premium. After deducting the cost of $100,000 for drilling, this would leave a profit of $600,000 if the land is dry. Unfortunately, the premium for this insurance policy is so exorbitant $6.7 millionthat Max could never buy the policy. Even if he finds oil for a gain of $700,000, the net loss of $6 million would put him out of business immediately. Nevertheless, let us go ahead and apply the minimax regret criterion when the bad alternative of buying the insurance policy is included in the problem. Continuing to use units of thousands of dollars, the payoff table for this problem is shown in the top half of Figure 8.  Figure 8 Application of the minimax regret criterion to the first Goferbroke Co. problem when an insurance option is included. The maximum payoff for each state of nature given in row 11 is used to calculate the regrets. This yields the regret table shown in rows 14-21. Applying the minimax regret criterion to this regret table presumably will lead either to choosing the new alternative, buy insurance, or the alternative that was chosen before, drill for oil (see Figure 7). Right? Wrong! For some reason, introducing a new alternative that is soundly rejected leads this criterion to switch its choice to the alternative that was rejected in Figure 7. This new option of buying insurance is a completely irrelevant alternative. Not only is its maximum regret approximately ten times that for the other alternatives, but Max would never consider such exorbitantly expensive insurance. However, a reasonable criterion certainly should not make its choice between the serious alternatives depend upon which (if any) irrelevant alternatives are included in the payoff table. Maxs Reaction: Max: That is pretty bizarre behavior for a criterion, all right. But I have two other reasons why I dont like this criterion very much. Jennifer: What are those? Max: First, I like to look ahead rather than worrying about past mistakes. Im really not the kind of person who loses a lot of sleep regretting a decision which turned out badly. It seems to me that this criterion really is designed for that kind of person. Or for someone who wants to avoid getting a poor evaluation from the boss because of making a disastrous decision. Jennifer: Youre right. So what is the second reason? Max: It seems to me that this criterion has the same problem as the realism criterion. If the possibility of a lot of oil on a tract of land is so small that it clearly is not worthwhile to drill for oil, this criterion probably would still say to drill. Or at least it would if we leave out irrelevant alternatives like buying that insurance policy. Jennifer: Yes, youre right again. Like the realism criterion, this criterion doesnt differentiate between very unlikely and somewhat likely states of nature. Max: OK. So where does this leave us? You really dont have a good criterion for me that doesnt need to use the consulting geologists numbers? Jennifer: Sorry. No. What constitutes a good criterion depends on ones philosophy of decision making. So one of these criteria might suit someone else fine. But you have concluded that none of these really work for you. Max: Yes. Jennifer: OK. With this conclusion, I think you are now ready to believe something I learned in my management science course. Many years ago, some eminent management scientists set down a set of rules that any reasonable criterion should satisfy. They then tried to develop a criterion that would satisfy all these rules. What they found instead is that this is impossible. There just doesnt exist a uniformly reasonable criterion that ignores whatever information you have about the relative likelihood of the various possible states of nature. Max: A pity. OK. You have convinced me that I need a criterion that uses the information from the consulting geologist. But I dont want to make a decision that relies on having exactly one chance in four of oil as being the gospel truth, because I know those numbers can be off quite a bit. Jennifer: You wont have to. We do have enough faith in the consulting geologists report to believe that there is a substantial chance of oil, but a much larger chance that the land is dry. That is the information that we need to use Max: I agree. So tell me how we can do this. The Maximum Likelihood Criterion The maximum likelihood criterion says to focus on the most likely state of nature as follows. 1. Identify the state of nature with the largest prior probability. 2. Choose the decision alternative that has the largest payoff for this state of nature. Your MS Courseware includes an Excel template for applying this criterion. Figure 9 shows how this would be done for Goferbrokes problem. Since Dry is the state of nature with the largest prior probability, we only consider the payoffs in column D (-100 and 90). The larger of these two payoffs is 90, so we choose the corresponding alternative, sell the land.  Figure 9 The application of the Excel template for the maximum likelihood criterion to the first Goferbroke Co. problem.The rationale for this criterion is a simple one. The final payoff will depend partially on which state of nature will turn out to be the true one. Although we dont know which state of nature will occur, we do know which one probably has the maximum likelihood. By basing our decision on the assumption that this state of nature will occur, we are giving ourselves a better chance of a favorable outcome than by assuming any other state of nature. This criterion also has received a number of criticisms outlined below. 1. This criterion chooses an alternative without considering its payoffs for states of nature other than the most likely one. What if any of these other payoffs would be disastrous? (Fortunately, this is not the case for Goferbrokes problem, where the payoff from selling the land is the same for both states of nature.) 2. For alternatives that are not chosen, this criterion also ignores their payoffs for states of nature other than the most likely one. What if any of these payoffs would be far better than could be obtained with the chosen alternative? Shouldnt we consider the fact that Goferbrokes payoff from drilling for oil and finding it is much, much more than from selling the land? 3. If the differences in the payoffs for the most likely state of nature are much less than for another somewhat likely state of nature, then it might make more sense to focus on this latter state of nature instead. For Goferbroke, the difference between the payoffs for drilling and selling when the land is dry is only 190, whereas it is 610 when the land contains oil, so perhaps it is more crucial to choose the best alternative for this latter state of nature instead. 4. If there are many states of nature and they are nearly equally likely, then the probability that the most likely state of nature will be the true one is fairly low. In this case, would it make sense to make the decision based on just one state of nature that has a fairly low probability? Maxs Reaction: Max: It sounds like this criterion wont fit some situations very well. But I kind of like it for my problem. It is simple for one thing. But more importantly, it doesnt require me to use the consulting geologists number that I know are pulled somewhat out of the air. I do have enough faith in his report to believe it is more likely that the land is dry than that there is oil there. So I am pretty comfortable in providing the information the criterion needs by specifying which is more likely. Jennifer: Sure, you can do that. But is it making the decision the way you want? Max: What do you mean? Jennifer: When you buy your tracts of land, is your main goal to find oil there? Or are you mostly interested in reselling the land? Max: Finding oil, of course. That is the whole point. Jennifer: Typically, for a tract of land you buy, will it be more likely that it is dry or that it contains oil? Max: Dry. Thats the nature of our business. You have to try a lot of sites to find that one big strike. Jennifer: So what do you think this criterion will tell you to do on those sites? Max: Oh, youre right! Now I see your point. Because it normally is more likely that the land is dry, it will keep telling me time after time to sell rather than drilling for oil. That is no way to run an oil prospecting business. Jennifer: Exactly! The key is the second and third criticisms I gave earlier. You really need to take into account how large the payoff might be if you do find oil. Max: I agree. This is not a good criterion for an oil prospector to use. I hope the next one is better. Jennifer: Youll be the judge of that. The Equally Likely Criterion It usually is difficult to place a lot of faith in the prior probabilities of the possible states of nature. Therefore, the equally likely criterion says to not even try to assign meaningful numbers to these probabilities. In the absence of further information, simply assume instead that the states of nature are equally likely and proceed as follows. 1. For each decision alternative, calculate the average of its payoffs over all the states of nature. (With equally likely states of nature, this average is the expected payoff in the statistical sense.) 2. Choose the alternative with the largest average payoff. Using the corresponding Excel template in your MS Courseware, Figure 10 shows the application of this criterion to Goferbrokes problem. The average payoff for each alternative is given in column H. Since drilling for oil has an average payoff of 300, versus only 90 for selling the land, the choice is to drill.  Figure 10 The application of the Excel template for the equally likely criterion to the first Goferbroke Co. problem. This criterion is sometimes called the Laplace Principle, because it was first enunciated (to our knowledge) about 200 years ago by the famous French mathematician, the Marquis Pierre-Simon de Laplace. (Sometimes called the Isaac Newton of France, Laplace was a key founder of the modern theory of probability.) Some modern decision makers agree with Laplace that decision making should be based on the reality that it is impossible to accurately predict the future. Events occur randomly. A typical random event is the occurrence of a state of nature. It is unrealistic to try to assign prior probabilities to states of nature, since this would go beyond our ability to predict the future. Nature gives us no advance information that it will do anything but randomize over its states. Therefore, to stay within the bounds of rationality, we should simply assume that the states of nature are equally likely. Critics of this line of reasoning make three main points. 1. Treating the states of nature as equally likely amounts to assigning each one the following prior probability: Prior probability = . Assigning this value to each prior probability is just as arbitrary as assigning any other values to these probabilities. 2. In some situations, there is good evidence that certain states of nature are more likely than others. Using this information should improve the decision. 3. There often are alternative ways of itemizing the possible states of nature. For example, the state of having oil could be broken down into several states involving different amounts of oil. Changing the number of states changes the prior probability of each one, which might then change the resulting decision. It is undesirable to have the decision depend on the arbitrary way in which the possible states of nature are itemized. Maxs Reaction: Max: I like the fact that this criterion doesnt force me to rely on the consulting geologists numbers. Jennifer: You mean his estimate that there is 1 chance in 4 of having oil on this tract of land? Max: Yes. I just dont trust his numbers. Jennifer: Would you have any more faith in his numbers if he had said 1 chance in 2 of oil? Max: No, not particularly. Whatever numbers he comes up with, the real chance of oil could be quite a bit lower or quite a bit higher. Jennifer: But if 1 chance in 2 of oil is the right ballpark, would you want to drill? Max: Certainly. Those are great odds in this business. Why? Jennifer: Because this criterion is always giving you odds of 1 chance in 2 of oil. Max: Really? I didnt catch that. You mean it would tell me to drill regardless of how promising or unpromising the land looked? Jennifer: Yes, pretty automatically. I guess I didnt make clear that this equally likely business says to assume that the odds of having oil are the same as for being dry. Max: Whoa. Now I get it. That wont do at all! I may not trust the consulting geologists numbers completely, but this criterions numbers seem completely worthless in my business. I need real odds based on solid evidence, not numbers pulled completely out of the air. Jennifer: If 1 chance in 2 of oil is not the right ballpark, then you certainly are correct. So you prefer the consulting geologists numbers? Max: Definitely. But I dont want my decision to depend on his numbers being exactly correct. Jennifer: OK. Lets look at a criterion that uses his numbers. Then well talk about how to analyze the situation if his numbers are off some. Max: Good. Now you can return to Section 12.2 for a full description of one more decision criterion Bayes decision rule. You will see that Max reacts favorably to this decision criterion and so decides to adopt this one. REVIEW QUESTIONS Why might it be desirable to use a decision criterion that doesnt rely on the prior probabilities of the respective states of nature? What is the maximax criterion? What is the maximin criterion? What is the pessimism-optimism index? How is it used with the realism criterion? Why are the maximax and minimax criteria special cases of the realism criterion? How is regret measured in the minimax regret criterion? What is being minimized with the minimax regret criterion? For each of the first four decision criteria, what type of person might find it appealing? Is it possible to develop a decision criterion that doesnt use the prior probabilities and still is reasonable for every situation? On which state of nature does the maximum likelihood criterion focus? What are some criticisms of the maximum likelihood criterion? What assumption about the states of nature is made by the equally likely criterion? What are some criticisms of the equally likely criterion? Glossary Equally Likely criterion: A criterion for decision making that assigns equal probabilities to all the states of nature. Maximax Criterion: A very optimistic criterion for decision making without using probabilities. Maximin criterion: A very pessimistic criterion for decision making without using probabilities. Maximum likelihood criterion: A criterion for decision making with probabilities that focuses on the most likely state of nature. Minimax regret criterion: A criterion for decision making without using probabilities that instead minimizes the regret that can be felt afterward if the decision does not turn out well. Pessimism-optimism index: An index that measures where the decision maker falls on a scale from totally pessimistic to totally optimistic. Realism criterion: A criterion for decision making without using probabilities that instead uses the decision makers pessimism-optimism index. Problems Although optional, an Excel template is available in your MS Courseware that can aid in doing each part of the following problems. This is indicated by the symbol E in front of each problem number. 1. You are given the following payoff table (in units of thousands of dollars) for a decision analysis problem without probabilities: State of NatureAlternativeS1S2A117A263A344(a) Which alternative should be chosen under the maximax criterion? (b) Which alternative should be chosen under the maximin criterion? (c) Which alternative should be chosen under the realism criterion when the pessimism-optimism index is 0.5? When this index is 0.25? When the index is 0.75? (d) Which alternative should be chosen under the minimax regret criterion? 2. Follow the instructions of Problem1 with the following payoff table: State of NatureAlternativeS1S2S3A1153020A2122428A352535A41820253. Silicon Dynamics has developed a new computer chip that will enable it to begin producing and marketing a personal computer if it so desires. Alternatively, it can sell the rights to the computer chip for $15 million. If the company chooses to build computers, the profitability of the venture depends upon the company's ability to market the computer during the first year. It has sufficient access to retail outlets that it can guarantee sales of 10,000 computers. On the other hand, if this computer catches on, the company can sell 100,000 machines. For analysis purposes, these two levels of sales are taken to be the two possible outcomes of marketing the computer, but it is unclear what their prior probabilities are. The cost of setting up the assembly line is $6 million. The difference between the selling price and the variable cost of each computer is $600. After developing the payoff table, determine which decision alternative should be chosen under each of the following criteria. (a) Maximax criterion. (b) Maximin criterion. (c) Realism criterion with a pessimism - optimism index of 0.5. (d) Minimax regret criterion. 4. Refer to Problem 12.3. After developing the payoff table, determine how many cases of strawberries Jean should purchase under the following criteria. (a) Maximax criterion. (b) Maximin criterion. (c) Realism criterion with a pessimism - optimism index of 0.5. (d) Minimax regret criterion. (e) Maximum likelihood criterion. (f) Equally likely criterion. 5. You are preparing to apply the minimax regret criterion to a decision analysis problem. You have identified the two states of nature and three decision alternatives (A1, A2, A3) that you definitely want to consider. A fourth decision alternative (A4) also has occurred to you, but you consider this alternative to be a poor one that would never be chosen. However, for completeness, you include this fourth decision alternative in the problem formulation and obtain the following payoff table. State of NatureAlternativeS1S2A11217A21315A31512A4020 (a) Apply the minimax regret criterion to this payoff table to determine which decision alternative should be chosen. (b) Remove the irrelevant decision alternative A4 from the payoff table to determine which decision alternative should be chosen. (c) What objection to this criterion is revealed by your answers to parts (a) and (b)? 6. Refer to Problem 12.4. Warren Buffy does not have great confidence in the accuracy of his prior probabilities. Therefore, he decides to try some decision criteria that do not use these probabilities, as well as two that do. Which investment should he make under each of the following criteria? (a) Maximax criterion. (b) Maximin criterion. (c) Realism criterion with a pessimism - optimism index of 0.5. (d) Minimax regret criterion. (e) Maximum likelihood criterion. (f) Equally likely criterion. 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