2-1 2.1 Decision Analysis: Expected Value of Perfect Information (EVPI) The Expected Value of Perfect Information (EVPI) will be explained with the following example: Example: “Flip the Coin Game” Richard has to decide whether to play a game of “flip the coin” with Karin. The game is as follows: If the coin ends up heads Karin pays Richard one dollar and if the coin ends up tails Richard pays Karin eighty cents. What is the most Richard should pay a clairvoyant (look up this word if you do not know its meaning) to tell him whether the coin will turn up heads or tails prior to the next toss? In the figure below we portrayed Richard’s initial decision problem (the endpoints are Richard’s payoffs in cents). Heads (.5) 100 Play the game 10 Tails 10 -80 (.5) Do not play the game Figure 2.1. Decision Tree (without perfect information) It is not hard to calculate that without perfect information Richard should decide to play the game, assuming he is using EMV as his decision making criterion. The EMV for this decision is ten cents. Of course, if a clairvoyant would tell Richard the outcome of the next toss, Richard would play the game if told “heads”, but would not play the game if told “tails”. However, Richard has to pay the clairvoyant before the outcome will be disclosed. How much should Richard pay for this perfect information? IT IS IMPORTANT TO NOTE THAT THE CLAIRVOYANT CAN ONLY TELL WHAT THE OUTCOME OF THE NEXT TOSS WILL BE. HE IS NOT ABLE TO INFLUENCE THE TOSS. (Think about this before reading on!) IN OTHER WORDS, THE CLAIRVOYANT IS NOT ABLE TO CHANGE THE PROBABILITY OF ENDING UP HEADS OR TAILS IN THE GAME. If the clairvoyant will give the perfect information, what will be Richard’s expected monetary value? Clearly, the probability that the clairvoyant will answer “heads” is 1/2, as is the probability that he will answer “tails”. The following tree reflects Richard’s decision problem with perfect information: 2-2 Play 100 Heads 100 Do not play (.5) 0 50 Tails -80 Play (.5) 0 Do not play 0 Figure 2.2 Decision Tree (with perfect information) The above tree shows Richard’s EMV with perfect information is 50 cents. This is an increase of 40 cents from the EMV for his initial problem, of 10 cents. Thus, by receiving perfect information from the clairvoyant, Richard’s EMV improved by 40 cents. Forty cents is the most Richard should pay for perfect information. That is, he is indifferent between: - not having perfect information (and therefore deciding to play the game), and - having perfect information at a price of 40 cents and deciding to play if heads is indicated, and deciding not to play if tails is indicated. The above indifference can be illustrated in the following (combined) decision tree. However, note that the decision “buy perfect information” is not normally available and is included only for illustrating how to carry out this calculation Heads 100 Play 10 (.5) Tails 10 (.5) -80 No Perfect Info Do not play 0 10 Play 60 Heads 60 Do not play Perfect Info (.5) -40 10 Tails Play -120 (.5) -40 Do not play -40 Figure 2.3. Decision Tree with “Indifference Point” Notice that none of the branches are pruned in the first decision fork. Also notice that the four end points at the right bottom of the tree include the 40 cents cost of the perfect information. Comparing the Figures 2.1 and 2.2, notice that Figure 2.2 can be obtained from Figure 2.1 by placing the uncertain event (Heads or Tails) prior to the decision (Play or Not Play). In general, the steps to be considered for an EVPZ calculation are: 2-3 1) Construct the tree and determine the DIV for the decision problem, assuming that no perfect information is available. 2) Construct a separate (perfect information) tree by modifying the tree in step 1. i.e., move the chance event about which perfect information is going to be received closer to the origin of the tree. How close has to be determined for each situation separately, but following the rule of keeping the tree in correct time sequence will always be correct. If the perfect information is available at the beginning of the problem and effects all decisions then move it to the beginning of the tree. If it effects only decisions on part of the tree, move it to the beginning of that part. 3) Determine the EMV for the decision tree in step 2. 4) EVPI = (EMV in step 3) - (EMV in step 1). Note: This is not a cookbook set of rules for determining EVPIS Step 2 requires a lot of thought and understanding of the procedure. Example: Mount Para Productions has the opportunity to perform market research to obtain more information regarding the possibility of success of “War Stars”. What is the most Mr. Fox should spend on such market research? The best information this market research can provide is, of course, perfect information, i.e., the indication that the movie will surely be a “success” or a “failure”. Therefore, the marketing research should never cost more than the highest price to be paid for perfect information. Note, that the research cannot influence “War Stars” success. Clearly, we would like to know the EVPI for this problem: Step 1: See Figure 1.2; the EMV is $1 million. Step 2: Draw the tree under perfect information. $6m produce $6 million “success” $3m (.5) do not produce $0 $0 produce -$4 million “failure” (.5) do not produce $0 Step 3: EMV = $3 million Step 4: EVPI = $3 million - $1 million = $2 million Does this mean Mr. Fox should have the marketing research performed, even if it would cost him $1 .9 million? Not very likely! The $2 million is the absolute most he should pay for perfect information. Of course, the marketing research is not likely to be perfect. The EVPI is an upper limit on the price to be paid for perfect information, thus, it is also an upper limit on the price to be paid for any information. We will encounter some more realistic situations where EVPI plays a useful role, but hopefully the above examples give an idea of what 2-4 EVPI means and why it is a useful concept. Problems: 2.1 .a. You are one of two manufacturers with a no-calorie cola. You consider a $2 million advertising campaign or a $1 million campaign. Your competitor considers the same two alternatives. If each of you spends $1 million, net profits on the new cola will be $500,000 each. If one of you spends $2 million and the other $1 million, the one spending more nets $2 million and the other nets nothing. If each of you spend $2 million, you each sustain a net loss of $2 million, because of consumer reaction to excessive advertising. Chances of your competitor spending $1 million on advertising are estimated to be 0.5. 1) Determine the optimal strategy on the basis of the criterion of maximizing DIV. 2) What is the most you should pay an informant for perfect information on your competitor’s action? 2.1.b. The Dover Manufacturing Company has a policy of producing part #27 in lots of 1,000 units. The machine that manufactures this part has been inconsistent in its production of defective parts. For the last twenty production lots of part #27, this machine has produced the following percent defective parts: Percent Number of Lots Defective in the last 20 2 5 5 8 10 4 15 2 20 1 It costs Dover $5.00 to replace a defective part. A complicated adjustment can be made to the machine which will insure 2% defective parts. The adjustment costs $300 per run to make. How much should Dover be willing to spend to determine the percent defectives the machine will make on the next run? 2.1.c. It was 8:00 p.m. and Mr. Dobbs was wondering whether his planned trip to Vancouver, costing $100, would be a washout. He knew that the purchasing agent of Nichols and Sons was only in the city for one week in two. If the agent were in the city, Mr. Dobbs believed that the probability was .6 that he could get an order. If he got an order, his commission would be $200. 1) On the basis of this information, should Mr. Dobbs make the trip? 2) “I’d give my right arm to know whether I’ll get an order on this trip,” he grumbled. How much does Mr. Dobbs value his right arm? 3) Then he recalled a service known as Executive Checkoff, which for $8 could inform him whether Nichols’ purchasing agent was in Vancouver. Because of the hour, Mr. Dobbs knew that this was the only source of information available to him. Should Mr. Dobbs purchase the information from Executive Checkoff? 2.1.d. RCPM Officer Brown receives a tip on a narcotics sale to take place in one of the university student residences the following night. This tipster has been correct 40 percent of the time. If the officer does not raid the residence at the time of the supposed sale, he neither gains nor loses career progress points. If he leads a raid and the tip proves false, 2-5 he loses 50 career progress points; if the tip is correct, he earns 100 progress points. 1) If the officer wishes to maximize his expected career progress points, what should he do? 2) If a career progress point is worth $10 to Office Brown, what is the most he would pay for a perfect tip? 2.1.e. Westminster Food Producers Ltd. (WFP) produces a perishable food product at a cost of $10 per case. The product can be sold for $15 per case. For planning purposes the company is considering possible demands of 100, 200, or 300 cases. If the demand is less than production, the excess production is lost. If demand is more than production, the firm, in an attempt to maintain a good service image, will satisfy the excess demand with a special production run at a cost of $18 per case. The product, however, always sells at the $15 per case price. 1) Set up a decision tree for the problem facing WFP. 2) If P(100) = 0.2, P(200) = 0.2, and P(300) = 0.6, use the expected monetary value criterion to determine the solution. 3) What is the EVPI and what does it mean? 2.1.f. CBB’s TV Productions is considering producing a pilot of a comedy series for a major TV network. While the network may reject the pilot and the series, it may also purchase the program for one or two years. While CBB may decide to produce the pilot, CBB also has an offer of $100,000 to transfer the rights for the series to a competitor. CBB’s profits are summarized in the following payoff table. Network’s Decision Reject 1 Year 2 Year Produce pilot -100 50 150 -- Profit in thousands of Sell to competitor 100 100 100 dollars If the probability estimates for the outcomes are P(Reject) = 0.2, P(1 Year) = 0.3, P(2 Year) = 0.5, what should the company do? What is the maximum CBB should be willing to pay for inside information on what the network will do? 2.1.g. Let d1 and d2 be two possible decisions and 21, 22, 23 be the outcome of an uncertain event that occurs if decision d1 is chosen. If either 21 or 22 occur then the decision maker must choose between decisions d3 and d4. The decision diagram, probabilities and payoffs for this problem are described in the tree below and the decision maker’s objective is to maximize EMV. 2-6 d3 100 21 .7 d4 70 d3 30 d1 22 .7 d4 50 23 -10 .7 d2 60 1) What policy should the decision maker choose and why? 2) Two information sources (not necessarily perfect) concerning the uncertain event are available. One costs 5 and the second costs 15. Which one (ones) should be considered for further analysis and why? 2.1.h. Guzzler Motors is trying to decide the level of production of its new automobile. It has the option of producing 300, 400, 500 or 600 automobiles, and on the basis of past experience assigns equal probabilities to demands of 300, 400, 500 and 600 in the year. Each automobile costs $7,000 to produce, and sells for $10,000. Because fashions change quickly, the automobiles still unsold at the end of the year will be sold at the greatly reduced price of $5,000 each. 1. Using EMV as the criterion, how many automobiles should Guzzler produce? 2. What is the most Guzzler should be willing to pay for perfect information about the level of demand? 2-7 2.2 Case Study: Hatton Realty Co. (Circa 1970) Mr. James Hatton was the proprietor of a real estate firm specializing in investment properties in the Vancouver area. Mr. Hatton’s business had done well during the real estate boom in the Vancouver area. Just recently a prospective client who held land for speculation, offered Mr. Hatton exclusive listing of three properties subject to an unusual set of restrictions. The location of these properties and the prices he wished to receive were as follows: Location Price Dunbar $20,000 Kitsilano $40,000 Point Grey $80,000 Mr. Hatton would receive a commission of 5% on any of the properties which he sold. The client was not yet willing to offer an exclusive listing on all three properties. He wished first to test Mr. Hatton’s ability to sell his type of land holding. Mr. Hatton could recall his exact words: “But, you must sell the Dunbar property first and sell it within a month. If you cannot do that I see no reason to transfer any further listings to your agency. If you do sell the property within the one month time limit, you may choose whether or not to list exclusively either the Kitsilano or the Point Grey holdings for a one month period. If you are able to sell this second property within one month you will be offered the exclusive listing of the third property for one month.” Ever since this conversation, Mr. Hatton had sought to analyze this proposition. He assessed the selling costs from his previous experience in advertising and selling similar properties in the past. These costs would be incurred even if no sale were made. These costs would not be incurred, of course, if Mr. Hatton refused a listing. Mr. Hatton also assessed his chances of sale within a month based upon his historical experience with similar parcels at the same general price level. He wrote this information in a simple table for further study: Probability Assessments Property Promotional Expense Sale No Sale Dunbar $800 .6 .4 Kitsilano $200 .7 .3 Point Grey $400 .5 .5 Mr. Hatton was sure that the probability of selling any of these three properties was not in any way dependent upon the sale or availability of the others. Problems: 2.2.a. Determine Mr. Hatton’s optimal decision, assuming he decides to use EMV as his decision making criterion. Be sure to include all possibilities. 2.2.b. What is the lowest probability of a Dunbar sale for which Mr. Hatton should accept the offer? 2.2.c. What is the EVPI on the Dunbar sale event? (That is, what is the most Mr. Hatton should pay for perfect information regarding the Dunbar sale?) 2.2.d. Find the EVPI of the Dunbar sale event if only the Dunbar property is available for sale. 2.2.e. (This is tricky.) What is the EVPI for the Kitsilano sale event? 2-8 2.3 Case Study: Weston Manufacturing Company Directors’ Meeting At their quarterly meeting held on November 5, 1982, the directors of the Weston Manufacturing Company were informed that the company would probably finish the fiscal year ending December 31 with an operating loss of nearly $50,000 on sales of approximately $1.8 million. This would be the second year in succession of unprofitable operation, the only two such years in the company’s fifty-seven year history. The loss of $49,971.48 in 1964 followed a profit of more than $150,000 in 1963. The financial statement for the period ending October 31, 1982 showed a loss of $48,915.56. This compared with the loss of $73,059.68 for the same period in 1981. Scott Howell, the Chairman of the Board expressed disappointment in October’s profit of $15,261.58 on net shipments of $217,245.97. Unfortunately, one big order did not come out as expected due to final design changes. Shipments for November were projected to be $85,000 resulting in a loss of approximately $14,000. If shipments in December reached $240,000 as estimated, a profit of $13,000 was expected. After dispensing the financial projections for the remainder of the year, Scott Howell began to provide the directors with the background of negotiations with the Sheridan Electric Products Corporation, a national manufacturer of heavy-duty industrial electrical appliances, with headquarters in Dayton, Ohio, less than one hundred miles from Weston. “In 1975, Sheridan asked us to determine whether a flat bed car with 330 ton capacity and a bed height of 26 inches could be built. The car was needed to move one of a series of new transformers from the construction area to the testing shop, a distance of 2 1/2 miles on the company’s track. The low bed height was required because of vertical clearance constraints in the area of the construction shop. John Sanders did some figuring and wrote them that such a car could be built for about $20,000.” “It seemed to us that Sheridan was on the verge of placing the order but then decided instead to rent a car from the Baltimore and Ohio Railroad each time one of the large transformers needed to be moved. No reason was given for this decision, but I do know that we were the only people Sheridan had contacted with a view to having a car built for them. Over the next five years, Don Archer occasionally stopped by the Sheridan plant and found their interest in the purchase of the flat bed car to vary from time to time.” “Last year Sheridan indicated interest in resuming serious talks. Bert Stokes drew up some plans according to the gauge, capacity, height, and other specifications received from Fred Shillkof, Sheridan’s Chief Engineer. Shillkof approved the plans and, as usual, we took this as an assurance that the track was a normal, level, industrial installation, permitting the proposed simple nonoscillating design for the car. In spite of a general increase in costs in the interim, John was able to submit the original bid of $20,000. The production costs were actually $15,000. The order was placed and the car was shipped March 23, on schedule.’ “Unfortunately, Sheridan’s track foundation was not adequate, and the car derailed on a banked portion of the track. Sheridan would not accept the car and returned it, at a cost of $550 to us. Bert then undertook an engineering re-study and concluded that the Cost of rebuilding the car with oscillating trucks would be about $16,000. A further review of costs developed no useful shortcuts. On July 18, a revised total price of $36,000 was offered to Sheridan. If we decided to rebuild the car, the modifications could be completed in less than a week. 2-9 “As you know, I was more or less out of action for most of the fall due to a prolonged serious illness in the immediate family. Shortly after I returned, early in October, having received no reply to our July proposal, I sent a wire requesting that Bert and I meet with Sheridan’s Chief Engineer, the Purchasing Agent and the General Manager.” “I have asked Bert to report to the Board on that meeting. He will be along in a few minutes. In the meantime, has anybody any questions?” “Yes, Scott, there is one point I’d like to clear up,” said O’Brien, “did anyone from Weston see the Sheridan track?” “Bert and I inspected the track when we visited in October,” replied Howell. “It was totally unsuited for a nonoscillating car. It turned out that Sheridan’s people thought the car would flex, but any engineer could see that would be impossible for a car with 330-ton capacity.” There was a knock at the door and Albert Stokes entered. No introductions were necessary, so Howell asked Stokes to proceed immediately with the report of their joint visit to Dayton. “On October 22, Scott Howell and I met at Dayton, Ohio, with Sheridan’s Purchasing Agent, Mr. Robert Casey, and Mr. James Woodruff, their General Manager. Mr. Woodruff informed us that according to a report for his Traffic Department dated October 8, there had been four instances since March when the Weston car could have been used instead of paying the B & 0 Railroad $300 rental charges. However, they foresee that a car of 330 tons capacity could be used about 12 times a year-- equivalent to $3,600 rental charges. “Woodruff expressed the feeling that on the basis of a car life of 20 years, $36,000 was a greater investment than the company would consider. They felt that a suitable car should cost about $25,000. This figure was based on savings that would accrue to them if they did not have to pay the rental charges. “We left the meeting with the understanding that we would review the design to see if costs could be reduced below the quotation of July 18. Scott had also suggested that they consider whether this car would not serve additional uses for Sheridan in moving and storing the new large transformers. We said we would keep in touch, although Woodruff and Casey indicated that there was ‘no great rush’.” “Bert, why wasn’t Fred Shillkof at this meeting?” asked Hall. “I don’t know, Max,” Stokes replied. “Neither Woodruff nor Casey gave any reason for his absence. He hasn’t been fired, and he wasn’t off sick. I know because we walked past his office on our way from the meeting, the door was open and he was working at his desk.” “One more question, Bert?” said O’Brien. “If we rebuilt the car with oscillating trucks according to the revised design, what are the chances that it will again derail?” “Very small, even though that track of theirs is not so hot. I’d say not more than one chance in a hundred -- not worth considering.” There being no further questions forthcoming, Stokes collected his papers together and left the room. “Well, gentlemen,” said Howell, “where do we go from here?” The ensuing silence was broken by Sanders: “We quoted them a figure of $36,000 based on the present estimate of the costs of modification and the production cost of the original car. Now though, with an indication that a suitable car at $25,000 might be acceptable to them, we could reconsider. But I would like to remind you that it would have been impossible to have built an oscillating car originally for $25,000. Maybe we should split the difference and make a bid of $30,000. However, I would say that we would have less than an even chance of getting the order at $30,000, say around two in five; whereas at 2-10 $25,000 the odds would be about nine to one in our favor. By the same token, at $36,000 we’d be lucky to have one chance in ten.” “What about trying to sell the car to someone else, if Sheridan turns us down?” asked Hall. Don Archer shook his head: “Not very good--as is we might have a one in twenty chance of selling the car. If we can find a customer, we might get between $10,000 and $18,000, and my best estimate is about $15,000. The market is pretty small, so I don’t think our selling costs would run over $200. We should be able to survey the market in less than two weeks. Of course, we could try to sell it elsewhere before we approach Sheridan again.” If we rebuild the car with an oscillating truck, there are more firms who might be interested, and the chances are about one in five of finding a customer. We should get between $17,000 and $25,000 with an average of about $20,000, but, in this market, the selling costs would be around $500, with all prospective customers contacted in less than four weeks. “Those are very reasonable prices, but that would be about the most we could expect in either case.” All agreed that the possibilities of bargaining further with Sheridan were nil. Sanders added: “As scrap, the car might be worth about $3,000 to us. If we rebuild the car with oscillating trucks and it still doesn’t work, its scrap value might go up to around $4,000. “I think that just about says it, John,” says Howell. “I certainly don’t want to absorb any loss in view of our recent poor profit picture but would do so in preference to a legal battle. Our lawyers have assured me that we could force Sheridan to pay a substantial cancellation charge on the grounds that the track was substandard. Another and perhaps stronger reason for demanding a cancellation charge would be the claim that Shillkof could have warned us when he saw and approved the plans. Legally, we are on sure ground, but this is our first contract with Sheridan, and possible future business from such a large company could substantially help us to halt, even reverse, our present sales decline. Besides, getting your name involved with a wrangle in court never does you any good in this business, no matter how right you are in the eyes of the law. For the same reason we have to give Sheridan the right of first refusal on a modified car. Only if they turn down our bid, can we consider selling it elsewhere. Anyway, the next move seems to be ours, and should be made soon. The sale of this car could significantly alter the profit projections discussed earlier. Given Bert and Don’s estimates, I see no reason why we cannot resolve this transaction before the end of the fiscal year. I would like you to give some intensive thought to this matter, and for us to reach a decision before the end of the week. Now, John, let’s have that general report of yours.” Problems: 2.3.a. Draw a decision tree for the problem facing Weston. 2.3.b. Determine what decision they should make to maximize EMV. 2.3.c. Find the EVPI associated with the external sale of the unmodified car.
Pages to are hidden for
"The Expected Value of Perfect Information (EVPI) will be"Please download to view full document