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# Notes on decision analysis

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### Notes on decision analysis

2. 2. 1-2 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: fifty-fifty 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 fifty-fifty 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 pay-off 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 fifty-fifty 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. 3. 1-3 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 randomly-selected 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 randomly-selected 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. 4. 1-4 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 second-hand 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 above-mentioned 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.
6. 6. 1-6 Finally, we often use a mathematical notation to express probabilities. For example, the probability that a head appears in a coin-tossing 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 coin-tossing 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. 7. 1-7 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. 8. 1-8 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 Stock-pile 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. 9. 1-9 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.)
10. 10. 1-10 EMV: “Playing the long-run 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 Fifty-Fifty 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 long-run 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 break-even 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?
12. 12. 1-12 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 50-50 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 50-50 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 non-monetary 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?
13. 13. 1-13 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 short-term 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. 14. 1-14 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. 15. 1-15 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.
16. 16. 1-16 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
17. 17. 1-17 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.
18. 18. 1-18 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
19. 19. 1-19 \$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
20. 20. 1-20 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 T-account) 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 right-most 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. Break-even 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.
21. 21. 1-21 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?