Decision Theory Using Probabilities, MV, EMV, EVPI and Other Techniques

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1 1 Decision Theory Using Probabilities, MV, EMV, EVPI and Other Techniques Thompson Lumber is looking at marketing a new product storage sheds. Mr. Thompson has identified three decision options (alternatives) as he looks at the situation. 1. He can construct a large new plant in which he can manufacture the sheds. 2. He can construct a smaller plant in which he can manufacture the sheds. 3. He can construct no plant. Mr. Thompson believes that the reception of the new product will be either favorable (high demand) or unfavorable (low demand). These are known as the states of nature. These are two outcomes over which Mr. Thompson has no effective control. 1. State of Nature High Demand. 2. State of Nature Low Demand. Mr. Thompson must now determine the profit impact of each of the possible options (alternatives) as they are affected by the states of nature. This is a reasonably complex process which is accomplished with the aid of his accounting and marketing departments. Sales must be forecasted under all three alternatives and under both states of nature. Expenses must be projected under all three alternatives and under both states of nature. The result of the forecasts will be an anticipated net profit associated with each possible decision and each state of nature. As a result of this analysis, Mr. Thompson has determine that if he builds a larger plant and the product is favorable received, the net profit will be 200,000, but a smaller plant will result in a net profit of 100,000 because of not being able to deliver the product. These are conditional values because they are dependent on building the larger or smaller plant. The conditional values for an unfavorable market are determined to be a loss of 180,000 and a loss of 20,000 for the larger and smaller plant decisions. Of course, doing nothing results in a zero for either state of nature. A recap of the information may be expressed in a contingency table. (Table 1) Favorable - Unfavorable - Construct Large 200, ,000 Construct Small 100,000-20,000 Do Nothing Zero Zero

2 2 There are three possible environments in which decisions are made certainty, risk and uncertainty. 1) may be made under certainty. Two investment opportunities exist. Both are guaranteed and have the same risk. You can invest 1,000 at 6% at the credit union or at 10% in Treasury Bonds. The decision is clear with all things being equal. You invest in the Treasury Bonds. 2) may be made under risk. We know the probability of being dealt a club from a deck of cards is while the probability of rolling a 5 with a single, six-side die is 1/6 (0.1667). The decision maker will maximize his or her expected well being. There are usually two equivalent criteria: maximization of expected monetary value and minimization of expected losses. Here we would chose the higher probability if our intent was to maximize our chance of winning. If however, we attach a monetary outcome to the probabilities, we might have a different picture. Let s say it costs us 100 if we draw a club from the deck of cards or we get a 5 when rolling a single, six-sided die. Now we want to minimize our chance of loss, so we select the rolling of the die since that has the smaller chance of happening. If the 100 is a gain rather than a loss, the opposite would be true. Drawing a club would maximize the probability to 0.25 rather than 0.16 for rolling a 5 on a single, six-sided die. Risk is present in either case. In one, minimization of loss is the important consideration. In the other case, maximization of profits is the important consideration. 3) may be made under uncertainty. under uncertainty occur when the probability is not knowable. For example, the probability of a Democrat being President in 25 years is unknowable. Accessing the probability of the success of a new product is unknowable. We are uncertain yet some decision must be made. Let s look at the decisions facing Mr. Thompson under all three environments certainty, risk and uncertainty.

3 3 1. Under Certainty: Look back at Table 1 as you consider the decisions facing Mr. Thompson. Clearly Mr. Thompson will make the decision to maximize his profits, given there is a 100% probability of one of the states of nature occurring. For example, if Mr. Thompson knows that his product will be well received (favorable state of nature), the decision to be made is clear build the large plant. The result of this decision is a 200,000 profit. If however, Mr. Thompson knows the product will not be favorable received (unfavorable state of nature), he should do nothing since this minimizes his loss. However, let us assume he has made the decision to build the new product. Now we only have two options (alternatives) available to us build the large plant or build the small plant. Let s examine the outcomes from Table 1 in light of his decision to build the product. Under the condition of unfavorable acceptance, it is quite clear the decision is to minimize the loss, so the smaller plant will be build, thus limiting the loss to -20,000 (See Table 1). 2. Under Risk: Here the decision maker will apply decisions using the estimate probabilities associate with each outcome. Here we will want to select the decision with the highest expected monetary value (EMV). There are two other possibilities we will examine the concept of perfect information and opportunity lost. Then we must look at our decision based on analyzing possible changes in our base assumptions. We do this using the concept of sensitivity analysis. Expected Monetary Value (EMV): The EMV is the weighted sum of possible payoffs for each alternative assuming the decision can be repeated many times. To assess the EMV, we must use probability. One of the three generally accepted methods of probability is that of subjective probability, which is often used in business settings. Mr. Thompson uses his best, educated estimate of the success of the two possible states of nature. In this instance, Mr. Thompson believes that each state of nature is as likely to occur as the other; therefore, the probability of a favorable market is 50% (0.50) and the probability of an unfavorable market is also 50% (0.50). Remember the probability of each of the events is measured between 0 and 1 and the sum of the probability of all events is equal to 1. This assessment of the probabilities meets both of these rules.

4 4 This can be show in tabular format as follows: (Table 2) EMV Computed Favorable - Unfavorable - Construct Large 200, ,000 10,000 Construct Small 100,000-20,000 40,000 Probability We arrive at the EMV (far right column in Table 2) by the following calculations. (200,000 times 0.50) + (-180,000 times 0.50) = 10,000 (100,000 times 0.50) + (-20,000 times 0.50) = 40,000 (Zero times 0.50) + (Zero times 0.50) = Zero. The decision is for Mr. Thompson to build a small plant and proceed with the new product. The other alternatives yield less expected return given the probabilities of for the states of nature. If, of course, the probabilities shift in the direction of the unfavorable state of nature to or 20 80, the decisions remains the same but the strength of the decision is lessened (28,000 for 60 40) and (4,000 for 80 20). You might try to make these calculations using the approach just outlined, but change the probabilities to 40% for favorable and 60% for unfavorable. Make the calculation a third time using 20% favorable and 80% unfavorable. Expected Value of Perfect Information (EVPI): Mr. Thompson is not especially confident in using his estimate of the probabilities (subjective approach). He knows if these probabilities change, the decision may change. Mr. Thompson knows a marketing research firm which has the technical expertise to assess the states of nature (market acceptability) more accurately. Of course, there would be a change for their services. The firm, Scientific Marketing, would change Mr. Thompson 65,000 for the study, but if the study is accurate Mr. Thompson can keep from making a very expensive mistake. The question is simply this, even if the information provided by Scientific Marketing is perfectly accurate, should Mr. Thompson pay 65,000 for the information? EVPI places an upper bound on what Mr. Thompson (any company) should pay for information. Two things are possible expected value of perfect information (EVPI) and expected value with perfect information.

5 5 The EVPI is the expected value with perfect information minus the maximum EMV. Let s try to un-confuse these concepts. First, we must calculate the expected value with perfect information. The best outcome for a favorable market (state of nature) is to build the large plant, which yields a net profit of 200,000 payoff (maximizes the profit). The best outcome (minimizes the loss) for the unfavorable state of nature is to do nothing thus yielding a zero payoff. Let s assume the states of nature are The calculations are as follows: You will multiply the best possible outcome times the probability of it occurring then add that result to the worst possible result times the probability of it occurring. The net result will be the Expected Value with Perfect Information. This value is not the EVPI. The value just calculated is the best less the worst assuming the probability of each outcome is given or estimated. The weight used is the probability of each outcome occurring. Okay, maybe we better look at the numbers because I think I just confused myself. The numbers work as follows: (200,000 times 0.50) + (Zero times 0.50) = 100,000 average return (payoff) assuming the decision could be repeated many times. The 100,000 is the expected value with perfect information. Next we will determine the EVPI, which is the expected value of perfect information. EVPI = expected value with perfect information less EMV (expected monetary value without perfect information). Side Note: Notice there is a difference between the terms EVPI and EMV. The EMV comes from Table 2 and includes all possible decisions (here three) and all possible states of nature (here two). The expected value with perfect information comes from assuming only two options will happen the best and worst as weighted by the probability of each outcome. Now let s calculate the EVPI, which is the expected value of perfect information. EVPI = 100,000-40,000 = 60,000. This means that Mr. Thompson should pay no more than 60,000 for the study by Scientific Marketing. Mr. Thompson can only improve his profit by 60,000. The price of 65,000 is too high for purchasing perfect information. Remember the assumption is for Mr. Thompson s subjective estimates of the states

6 6 of nature. If these underlying probabilities change then so too might the decision we make. Opportunity Loss: An alternate approach to maximizing EMV is to minimize expected opportunity loss (EOL), also referred to as regret. EOL is the cost of not picking the best alternative. It is important to notice that the minimum EOL will always result in the same decision as the maximum EMV, which means the following relationship holds: EVPI = minimum EOL. We need to calculate an opportunity loss table from our original profit Table 1. Table 1 is repeated here for ease of following the process. Table 1: Original Profit Table with EMV EMV Favorable - Unfavorable - Computed Construct Large 200, ,000 10,000 Construct Small 100,000-20,000 40,000 Probability We need to convert the expected profit in Table 1 to the Opportunity Lost if we make the wrong decision. Table 1 and 2 refer to value in terms of profit. Table 3 shifts our thinking to profit we lose if we make the wrong decision. The opportunity loss table is as follows. Table 3. EOL Favorable - Unfavorable - Computed Construct Large Zero 180,000 90,000 Construct Small 100,000 20,000 60,000 Do Nothing 200,000 Zero 100,000 Probability These values are calculated as follows:

7 7 If you know that the favorable state of nature will occur, you will clearly make the decision to construct the large plant (making 200,000). This means that you lose nothing (no profit) if you make this decision. You can set this value at zero and then calculate what you would lose if the state of nature is favorable and you make one of the other two decisions (small plant or do nothing). 200,000 less 100,000 = 100,000 which you would lose if the favorable state of nature occurs and you have chosen to construct the small plant. You follow the same process in filling out the balance of Table 3. For the favorable state of nature, if the decision has been made to do nothing, then you would lose 200,000 (200,000 less zero = 200,000). For the state of nature of unfavorable, the best decision would be to do nothing thus this value can be set at zero and the potential losses of profit can be stated with this as a base. In summary, under each of the states of nature you set the best decision as the base line and measure from that value to determine what you would lose if you make the wrong decision under each state of nature. In other words, it we know the product is going to be favorably received, we have no doubt about the decision we make (maximize the profit at 200,000). If we however hedge our bet and make the decision to construct the small plant, we will not be able to make the 200,000 and will lose the difference between the 200,000 and the 100,000 we can make when we construct the small plant. To determine the EOL (Expected Opportunity Lost), you would multiply the probabilities times each of the states of nature for each decision. For example, (Zero times 0.50) + (180,000 times 0.50) = 90,000 which is the far right column of Table 3 for the Decision to Build the Large. For the Decision to Build the Small the math yields the following: 100,000 times 0.50 = 50,000 20,000 times 0.50 = 10,000 Adding the two you have 50, ,000 = 60,000 which is shown in Table 3 in the last column as the EOL for the decision to build the small plant. The Decision to Do Nothing math yields the following: 200,000 times 0.50 = 100,000 Zero times 0.50 = Zero Adding the two you have 100,000 which is the EOL for the third decision to do nothing.

8 8 The results of the EVPI were 60,000 which means that this is the maximum Thompson could pay for the perfect information. Notice that the minimum EOL is also 60,000. Thus the relationship holds that EVPI = minimum EOL. Sensitive Analysis: Remember we have made all of our analyses based on a probability of each possible state of nature. What question that must be naturally asked is what happens if the probabilities change? Sensitive analysis investigates how our decision might change with different input data (different probabilities). We need to first set up some definitions. Definitions: P = Probability of a favorable market (state of nature). 1 P = Probability of an unfavorable market (state of nature). Formula Relationships Based on Probability (P): The formulas below utilize the same concept we used in Table 2 to determine the Expected Monetary Value (EMV). These formulas are stated in terms of P. Now you can chose a value for P and make the same calculations we did for Table 2 for EMV (last column on the right). Using these relationships if you substitute 0.50 for P you would get the same value shown in the last column in Table 2. From below the EMV for a large plant would be the following: EMV (large plant) = 380,000 (0.50) -180,000 = 10,000, which is the same value in Table 2 last column on the right. Go back to Table 2 and look at the value. The general formulas for each possible decision are stated below: EMV (large plant) = 200,000P - 180,000(1- P) = 200,000P - 180, ,000P EMV (large plant) = 380,000P - 180,000 EMV (small plant) = 100,000P - 20,000(1- P) = 100,000P - 20, ,000P EMV (small plant) = 120,000P - 20,000 EMV (do nothing) = Zero P + Zero (1-P) = Zero

9 9 Two probability points must be calculated. We could use trial and error, but it is much faster and simpler to use the formulas created above. The probability value of a favorable market (P) has a range of values. Below or above a point will have an impact on the decision that Mr. Thompson might make. You need to determine the cross-over point where Mr. Thompson s decisions might change. There are two points where the decision will move in one direction or the other. Calculation of those points is as follows: Point 1 This is the point where the probability of doing nothing crosses the probability of building the small plant. We will use the formula relationships created just above in determine EMV (doing nothing) = EMV (Small ) From the formulas just above, we have: 0 = 120,000 P - 20,000 P = 20, ,000 = Point 2 This is the point where the probability of building the small plant crosses the probability of building a large plant. EMV(Small ) = EMV (Large ) From the formulas just above, we have: 120,000P - 20,000 = 380,000P - 180, ,000 P = 160, , ,000 = To summarize: Range of P Value Do Nothing Less than Build Small to Build Large Above Interpretation: If the probability of a favorable market is less than , the best decision is to do nothing. If the probability of a favorable market is from through , the best decision is to build the small plant.

10 10 If the probability of a favorable market is above the best decision is to build the large plant. See example below using a 62% favorable market. Favorable - Unfavorable - EMV Computed Construct Large 200, ,000 55,600 Construct Small 100,000-20,000 54,400 Probability The decision at 62% favorable is to build the large plant (EMV =55,600). Graphically this looks like the following: EMV 200 Large Small Do Nothing (P) 0 to 1

11 11 3. under uncertainty: We have looked at Decision Under Risk and Under Certainty. We now need to look at Under Uncertainty. Some of these methods are repeats of information and process we have already examined, but they are listed here again for purposes of summary and emphasis. under uncertainty include five different criteria. 1) Maximax 2) Maximin 3) Equally Likely 4) Criterion of Realism 5) Minimax Before you let the terms confuse you let s take a look at each one separately. We will continue to use the Thompson case information. Maximax: This is an optimistic approach. In the case of Thompson, recall the following table. We have already looked at his approach but here we put that decision into a more formal table. Selection of Favorable - Unfavorable - Maximum Decision Construct Large 200, , ,000 Construct Small 100,000-20, ,000 In the right hand column you would simply select the maximum net profit associated with each possible decision alternative. In this case, under the favorable state of nature the decision to construct the large plant yields a net profit of 200,000 as compared to a -180,000 loss for the unfavorable state of nature and the decision to construct the small plant yields a net profit of 100,000 as opposed to a -20,000 loss for the unfavorable state of nature. Both of these values are listed in the far right column. The maximax decision then is to select the value which maximizes the net profit. In this instance, you would build the large plant, which yields a net profit of 200,000.

12 12 Maximin: This is a pessimistic approach. Using the same table, the numbers would look this way. Again we have already looked at this approach, but this table presents the decision in a more organized manner. Selection of Favorable - Unfavorable - Minimum Decision Construct Large 200, , ,000 Construct Small 100,000-20,000-20,000 Next you select the alternative that maximizes the minimum outcome for every alternative. In this case, the decision should be to do nothing. This alternative makes the most of the projected losses associated with the unfavorable states of nature. Equally Likely: The equally likely approach computes the highest average outcome. This approach is also referred to as the Laplace approach. The table looks exactly like the very first table because the calculations yield the same results. This is the same as a outcome assumption for the states of nature. 200, ,000 = 20,000 2 = 10, ,000-20,000 = 80,000 2 = 40, = 0 2 = 0 Favorable - Unfavorable - Average Construct Large 200, ,000 10,000 Construct Small 100,000-20,000 40,000 Probability

13 13 The assumption is that the probabilities of each state of nature are equal and thus each state of nature is equally likely. In this case, the decision is to construct the small plant (40,000). Criterion of Realism: This criterion uses the weighted average approach. This is sometimes referred to as the Hurwicz criterion. This approach is a compromise between the pessimistic and optimistic approach. The calculations and the table follow. A coefficient of realism is selected (alpha - α). The coefficient is between 0 and 1. When the decision maker is optimistic about the future, α will be closer to 1. When the decision maker is pessimistic about the future, α will be closer to 0. Criterion of Realism = (α)(maximum for each decision) + (1-α)(minimum for each decision) Assume for a moment that the decision maker, Mr. Thompson, set the alpha at The calculations would then be as follows. (0.80)(200,000) + (.020)(-180,000) = 124,000. (0.80)(100,000) + (0.20)(- 20,000) = 76,000. Notice that the alpha is essential the same as probability. Criterion of Favorable - Unfavorable - Realism Construct Large 200, , ,000 Construct Small 100,000-20,000 76,000 Probability or Coefficient of Realism In this case, the decision should be to construct the large plant. This is a weighted average approach.

14 14 Minimax: The final approach under uncertainty is the minimax. This one is a bit more difficult to understand. We have been dealing with the net profit expected in the first four approaches under uncertainty. Now we shift to opportunity loss. To understand this, you must shift your thinking to profits we will lose if we make a decision other than the best decision for the state of nature. The starting table is as follows (which is a restatement of Table 1 Values). Favorable - Unfavorable - Construct Large 200, ,000 Construct Small 100,000-20,000 Do Nothing Zero Zero We next must realize that if we know the state of nature is going to be favorable, we would clearly make the decision to construct the large plant for a net profit of 200,000. In a similar manner, if we know the state of nature is going to be unfavorable, we would clearly make the decision to do nothing with a zero net profit impact. With perfect knowledge, we can now assert these two are the correct and best decision. However, we do not operate with perfect knowledge, so the question is how much money (net profit) would we lose if we make the wrong decision? The differences in the correct decision values and the other possible values are called opportunity lost. The revised table looks like the following. Opportunity Lost Table: Maximim for Favorable - Unfavorable - each Decision Construct Large Zero 180, ,000 Construct Small 100,000 20, ,000 Do Nothing 200,000 Zero 200,000 Let s review the value shown as 200,000 for doing nothing. How did I get that value?

15 15 Remember this is money we will lose if we make the wrong decision. If the state of nature of favorable occurs, we could make 200,000 if we build the large plant. If however, we have chosen to do nothing, we will not make that 200,000 because we made the wrong decision. This is profit we are foregoing or not going to make because we made the wrong decision. The other values are determined in a similar manner. All values associated with each of the three possible decisions are written in the far right column. Our final decision is to select the minimum of the maximum opportunity loss thus the minimax decision is decision to build the small plant and thus minimize the opportunity loss. We have just given you several approaches to decision making using some important Decision Theory tools. Now let s summarize all of the decision and their outcomes. Summary of Under Certainty Decision s Favorable State Of Nature Unfavorable State Of Nature Large 200, ,000 Small 100,000-20,000 Nothing Zero Zero Known = Favorable. Decision is to build large plant. Known = Unfavorable. Decision is to do nothing. Under Risk EMV Approach Decision s Favorable State Of Nature Unfavorable State Of Nature EMV Computed Large 200, ,000 10,000 Small 100,000-20,000 40,000 Nothing Zero Zero Zero Probability With a probability, select the EMV that generates the highest expected profit. The decision to build the small plant is the correct one under risk since the EMV is best at 40,000.

16 16 EVPI Approach How much will Thompson be willing to pay for perfect information? Scientific Marketing charge is 65,000, but the maximum Thompson should pay is 60,000. Opportunity Loss Approach With a probability, we calculate a weighted average of EOL. EOL Computed Favorable - Unfavorable - Construct Large Zero 180,000 90,000 Construct Small 100,000 20,000 60,000 Do Nothing 200,000 Zero 100,000 Probability Here the decision is to select the minimum of the maximum weighted average expected opportunity loss (EOL). In this case, the decision to construct the small plant with a minimum expected loss of 60,000 and a probability of Sensitivity Analysis Using sensitivity analysis, we can determine the following. Interpretation: Range of P Value Do Nothing Less than Build Small to Build Large Above If the probability of a favorable market is less than.1667, the best decision is to do nothing. If the probability of a favorable market is between.1667 and.6154, the best decision is to build the small plant. If the probability of a favorable market is above.6154 the best decision is to build the large plant.

17 17 Under Uncertainty Maximax (Optimistic) Selection of Favorable - Unfavorable - Maximum Decision Construct Large 200, , ,000 Construct Small 100,000-20, ,000 This is the same concept as making decisions under certainty. The maximum profit or maximax is the decision to construct the large plant with a 200,000 net profit. This is the most optimistic approach. Maximin (Pessimistic) Selection of Favorable - Unfavorable - Minimum Decision Construct Large 200, , ,000 Construct Small 100,000-20,000-20,000 This is the same as making a decision under certainty with knowledge that the unfavorable state of nature would occur. This is the most pessimistic approach and selects the option of doing nothing for a zero net profit. Equally Likely (Equal Probabilities for Each State of Nature) Favorable - Unfavorable - Average Construct Large 200, ,000 10,000 Construct Small 100,000-20,000 40,000 Probability

18 18 This possibility assumes that both states of nature will occur equally likely; therefore, the probability is the same for either state of nature which is The decision is to construct the small plant. Criterion of Realism (Realistic) Criterion of Favorable - Unfavorable - Realism Construct Large 200, , ,000 Construct Small 100,000-20,000 76,000 Probability or Coefficient of Realism This approach works off of an estimate by the decision maker as to an alpha value which reflects a positive or negative estimate of the state of nature. In this case the weighted average yields the decision to construct the large plant. Minimax (Opportunity Loss) This approach shifts thinking to opportunity loss. How much would we lose if we make the wrong decision? Maximim for Favorable - Unfavorable - each Decision Construct Large Zero 180, ,000 Construct Small 100,000 20, ,000 Do Nothing 200,000 Zero 200,000 The decision which maximizes the profit we will lose if we make the incorrect decision is to construct the small plant. We have limited our states of nature to two and our decision alternatives to three. However, many decisions are much more complex and include several more states of nature and several more decision possibilities. The solution would work the same way, but the tables would expand considerably. Developing an MS Excel spread sheet is the best approach to use.

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