CHAPTER Finding the right way – analysing decisions 11 Chapter objectives This chapter will help you to: ■ work out expected values using probabilities ■ appreciate attitudes to risk and apply decision rules ■ construct decision trees and use them to decide between alternative strategies ■ ask ‘what if’ question about conclusions from decision trees by employing sensitivity analysis ■ make use of Bayes’ rule to find posterior probabilities for decision trees ■ become acquainted with business uses of decision analysis In the previous chapter we looked at how probability can be used to assess risk. In this chapter we will consider how probability is used in the analysis of decisions. We will begin with expectation, the process of multiplying probabilities by the tangible results of the outcomes whose chances they measure to obtain expected values of the process or situation under investigation. We will move on to examine various quantitative approaches to taking decisions, including decision trees. 11.1 Expectation A probability assesses the chance of a certain outcome in general. Expectation is using a probability to produce a predicted or expected value of the outcome. 348 Quantitative methods for business Chapter 11 To produce an expected value we have to apply the probability to something specific. If the probability refers to a process that is repeated, we can predict how many times a certain outcome will occur if the process happens a specific number of times by multiplying the prob- ability by the number of times the process happens. Example 11.1 The probability that a customer visiting the Kenigar Bookshop makes a purchase is 0.35. If 500 customers visit the shop one day, how many should be expected to make a purchase? Expected number of customers making a purchase ϭ 0.35 * 500 ϭ 175 The result we obtained in Example 11.1 is a prediction, and like any prediction it will not always be true. We should not therefore interpret the result as meaning that out of every 500 customers that visit the store exactly 175 will make a purchase. What the result in Example 11.1 does mean is that in the long run we would expect that the average number of customers making a purchase in every 500 that visit the store will be 175. In many business situations outcomes are associated with specific financial results. In these cases the probabilities can be applied to the monetary consequences of the outcomes to produce a prediction of the average amount of money income or expenditure. These types of prediction are called expected monetary values (EMVs). Example 11.2 A rail operating company incurs extra costs if its long-distance trains are late. Passengers are given a voucher to put towards the cost of a future journey if the delay is between thirty minutes and two hours. If the train is more than two hours late the com- pany refunds the cost of the ticket for every passenger. The cost of issuing vouchers costs the company £500. The cost of refunding all the fares costs the company £6000. The probability that a train is between thirty minutes and two hours late is 10% and the probability a train is more than two hours late is 2%. What is the expected monetary value of the operating company’s extra costs per journey? To answer this we need to take the probability of each of the three possible outcomes (less than thirty minutes late, thirty minutes to two hours late, more than two hours late) and multiply them by their respective costs (£0, £500 and £6000). The expected monetary value is the sum of these results. EMV ϭ (0.88 * 0) ϩ (0.1 * 500) ϩ (0.02 * 6000) ϭ 0 ϩ 50 ϩ 120 ϭ 170 The company can therefore expect that extra costs will amount to £170 per journey. Chapter 11 Finding the right way – analysing decisions 349 At this point you may find it useful to try Review Questions 11.1 to 11.4 at the end of the chapter. 11.2 Decision rules From time to time companies are faced with decisions that are pivotal to their future. These involve developing new products, building new facilities, introducing new working practices and so on. In most cases the managers who take these decisions will not know whether they have made the right choices for many months or years to come. They have to take these decisions against a background of either uncertainty, where they cannot attach a probability to each of the outcomes, or risk, where they can put a probability to each of the outcomes. In this section we will look at decision rules, techniques available to managers taking decisions under conditions of both uncertainty and risk. All of these techniques assist managers by helping them analyse the decisions and the possible outcomes in a systematic way. The starting point is the pay-off table in which the results or pay-offs of the different possibilities or strategies that could be chosen are arranged according to the conditions or states of nature affecting the pay-off that might prevail. Example 11.3 Following the success of their CeeZee Seafood fast food restaurant in London, the pro- prietors, Soll and Perretts, are thinking of expanding the business. They could do this by investing in new sites or by franchising the operation to aspiring fast food entrepre- neurs who would pay a fee to Soll and Perretts. The estimated profits for each strategy depend on the future demand for healthy fast food, which could increase, remain stable, or decline. Another possibility for Soll and Perretts is to accept the offer of £20m that a major international fast food company has made for their business. The expected profits are shown in Table 11.1. Table 11.1 Expected profits (in £m) for Soll and Perretts State of future demand Strategy Increasing Steady Decreasing Invest 100 40 Ϫ30 Franchise 60 50 0 Sell 20 20 20 The pay-off table in Example 11.3 does not in itself indicate what strategy would be best. This is where decision rules can help. When you apply them remember that the decision you are analysing involves choosing between the available strategies not between the states of nature, which are by definition beyond the control of the decision-maker. 11.2.1 The maximax rule According to the maximax rule the best strategy is the one that offers the highest pay-off irrespective of other possibilities. We apply the max- imax rule by identifying the best pay-off for each strategy and choosing the strategy that has the best among the best, or maximum among the maximum, pay-offs. 350 Quantitative methods for business Chapter 11 Example 11.4 Which strategy should be selected in Example 11.3 according to the maximax decision rule? The best pay-off available from investing is £100m, from franchising, £50m and from selling, £20m, so according to the maximax rule they should invest. The attitude of the decision-maker has a bearing on the suitability of deci- sion rules. The maximax rule is appropriate for decision-makers who are risk-seekers; those who are prepared to accept the chance of losing money in gambling on the biggest possible pay-off. However, we should add that the attitude of the decision-maker may well be influenced by the financial state of the business. If it is cash-rich, the maximax approach would make more sense than if it were strapped for cash. In the former case it would have the resources to cushion the losses that may result in choosing the strategy with the highest pay-off. 11.2.2 The maximin rule If maximax is the rule for the optimists and the gamblers, maximin is for the pessimists and the risk-avoiders. The maximin rule is to pick the strategy that offers the best of the worst returns for each strategy, the maximum of the minimum pay-offs. Chapter 11 Finding the right way – analysing decisions 351 Example 11.5 Which strategy should be selected in Example 11.3 according to the maximin deci- sion rule? The worst pay-off available from investing is Ϫ£30m, from franchising, £0m and from selling, £20m, so according to the maximin rule they should sell. This approach would be appropriate for a business that does not have large cash resources and would therefore be especially vulnerable to taking a loss. It would therefore make sense to pass up the opportun- ity to gain a large pay-off if it carries with it a risk of a loss and settle for more modest prospects without the chance of losses. 11.2.3 The minimax regret rule This rule is a compromise between the optimistic maximax and the pessimistic maximin. It involves working out the opportunity loss or regret you would incur if you selected any but the best strategy for the conditions that come about. To apply it you have to identify the best strategy for each state of nature. You then allocate a regret of zero to each of these strategies, as you would have no regret if you had picked them and it turned out to be the best thing for that state of nature, and work out how much worse off you would be under that state of nature had you chosen another strategy. Finally look for the largest regret figure for each strategy and choose the strategy with the lowest of these figures, in doing so you are selecting the strategy with the minimum of the maximum regrets. Example 11.6 Which strategy should be selected in Example 11.3 according to the minimax regret decision rule? If they knew that demand would increase in the future they should choose to invest, but if instead they had chosen to franchise they would be £40m worse off (£100m Ϫ £60m), and if they had chosen to sell they would be £80m (£100m Ϫ £20m) worse off. 352 Quantitative methods for business Chapter 11 11.2.4 The equal likelihood decision rule In decision-making under uncertainty there is insufficient information available to assign probabilities to the different states of nature. The equal likelihood approach involves assigning probabilities to the states of nature on the basis that, in the absence of any evidence to the con- trary, each state of nature is as likely to prevail as any other state of nature; for instance if there are two possible states of nature we give each of them a probability of 0.5. We then use these probabilities to work out the expected monetary value (EMV) of each strategy and select the strategy with the highest EMV. Example 11.7 Which strategy should be selected in Example 11.3 according to the equal likelihood decision rule? In this case there are three possible states of nature – increasing, steady and decreas- ing future demand – so we assign each one a probability of one-third. The investing strategy represents a one-third chance of a £100m pay-off, a one-third chance of a £60m These figures are the opportunity losses for the strategies under the increasing demand state of nature. The complete set of opportunity loss figures are given in Table 11.2. From Table 11.2 the maximum opportunity loss from investing is £80m, from franchising, £30m and from selling, £50m. The minimum of these is the £30m from franchising, so according to the minimax regret decision rule this is the strategy they should adopt. Table 11.2 Opportunity loss figures (in £m) for Example 11.3 State of future demand Strategy Increasing Steady Decreasing Invest 0 10 50 Franchise 40 0 20 Sell 80 30 0 At this point you may find it useful to try Review Questions 11.5 to 11.9 at the end of the chapter. 11. 3 Decision trees The decision rules we examined in the previous section help to deal with situations where there is uncertainty about the states of nature and no probabilities are available to represent the chances of their happening. If we do have probabilities for the different states of nature we can use these probabilities to determine expected monetary values (EMVs) for each strategy. This approach is at the heart of decision trees. As their name implies, decision trees depict the different sequences of outcomes and decisions in the style of a tree, extending from left to right. Each branch of the tree represents an outcome or a decision. The junctions, or points at which branches separate, are called nodes. If the branches that stem from a node represent outcomes, the node is called a chance node and depicted using a small circle. If the branches represent different decisions that could be made at that point, the node is a decision node and depicted using a small square. All the paths in a decision tree should lead to a specific monetary result that may be positive (an income or a profit) or negative (a cost or a loss). The probability that each outcome occurs is written alongside the branch that represents the outcome. We use the probabilities and the monetary results to work out the expected monetary value (EMV) of each possible decision. The final task is to select the decision, or series of decisions if there is more than one stage of decision-making, that yields the highest EMV. Chapter 11 Finding the right way – analysing decisions 353 pay-off and a one-third chance of a Ϫ£30m pay-off. To get the EMV of the strategy we multiply the pay-offs by the probabilities assigned to them: EMV(Invest) ϭ 1/3 * 100 ϩ 1/3 * 60 ϩ 1/3 * (Ϫ30) ϭ 33.333 ϩ 20 ϩ (Ϫ10) ϭ 43.333 Similarly, the EMVs for the other strategies are: EMV(Franchise) ϭ 1/3 * 40 ϩ 1/3 * 50 ϩ 1/3 * 0 ϭ 13.333 ϩ 16.667 ϩ 0 ϭ 30 EMV(Sell) ϭ 1/3 * 20 ϩ 1/3 * 20 ϩ 1/3 * 20 ϭ 20 According to the equal likehood approach they should choose to invest, since it has the highest EMV. 354 Quantitative methods for business Chapter 11 Example 11.8 The proprietors of the business in Example 11.3 estimate that the probability that demand increases in the future is 0.4, the probability that it remains stable is 0.5 and the probability that it decreases is 0.1. Using this information construct a decision tree to represent the situation and use it to advise Soll and Perrets. EMV for the Invest strategy ϭ 0.4 * 100 ϩ 0.5 * 40 ϩ 0.1 * (Ϫ30) ϭ £57m EMV for the Franchise strategy ϭ 0.4 * 60 ϩ 0.5 * 50 ϩ 0.1 * 0 ϭ £49m EMV for the Sell strategy ϭ £20m The proprietors should choose to invest. Strategy Invest Franchise Sell Increases Decreases Stable Pay-off (£m ) 100 40 Ϫ30 60 50 0 20 Future demand 0.4 Increases 0.4 0.5 Stable 0.5 0.1 Decreases 0.1 Figure 11.1 Decision tree for Example 11.8 The probabilities of the states of nature in Example 11.8 were pro- vided by the decision-makers themselves, but what if they could commis- sion an infallible forecast of future demand? How much would this be worth to them? This is the value of perfect information, and we can put a figure on it by working out the difference between the EMV of the best strategy and the expected value with perfect information. This latter amount is the sum of the best pay-off under each state of nature multiplied by the probability of that state of nature. Chapter 11 Finding the right way – analysing decisions 355 Example 11.10 Sam ‘the Chemise’ has a market stall in a small town where she sells budget clothing. Unexpectedly the local football team have reached the semi-finals of a major tourna- ment. A few hours before the semi-final is to be played a supplier offers her a consign- ment of the team’s shirts at a good price but says she can have either 500 or 1000 and has to agree to the deal right away. If the team reach the final, the chance of which a TV commentator puts at 0.6, and Sam has ordered 1000 shirts she will be able to sell all of them at a profit of £10 each. If the team do not reach the final and she has ordered 1000 she will not sell any this sea- son but could store them and sell them at a profit of £5 each next season, unless the team change their strip in which case she will only make a profit of £2 per shirt. The probability of the team changing their strip for next season is 0.75. Rather than store the shirts she could sell them to a discount chain at a profit of £2.50 per shirt. If Sam orders 500 shirts and the teams reach the final she will be able to sell all the shirts at a profit of £10 each. If they do not make the final and she has ordered 500 she will not Example 11.9 Work out the expected value of perfect information for the proprietors of the fast food business in Example 11.3. We will assume that the proprietors’ probability assessments of future demand are accurate; the chance of increasing demand is 0.4 and so on. If they knew for certain that future demand would increase they would choose to invest, if they knew demand was definitely going to remain stable they would franchise, and if they knew demand would decrease they would sell. The expected value with perfect information is: 0.4 * 100 ϩ 0.5 * 50 ϩ 0.1 * 20 ϭ £67m From Example 11.8 the best EMV was £57m, for investing. The difference between this and the expected value with perfect information, £10m, is the value to the proprietors of perfect information. The decision tree we used in Example 11.8 is a fairly basic one, representing just one point at which a decision has to be made and the ensuing three possible states of nature. Decision trees really come into their own when there are a number of stages of outcomes and decisions; when there is a multi-stage decision process. A decision tree like the one in Figure 11.2 only represents the situ- ation; the real point is to come to some recommendation. This is a little more complex when, as in Figure 11.2, there is more than one point at which a decision has to be made. Since the consequences for the first decision, on the left-hand side of the diagram, are influenced by the later decision we have to work back through the diagram using what is called backward induction or the roll back method to make a recommen- dation about the later decision before we can analyse the earlier one. We assess each strategy by determining its EMV and select the one with the highest EMV, just as we did in Example 11.8. 356 Quantitative methods for business Chapter 11 have the option of selling to the discount chain as the quantity would be too small for them. She could only sell them next season at a profit of £5 each if the team strip is not changed and at a profit of £2 each if it is. Sam could of course decline the offer of the shirts. Draw a decision tree to represent the situation Sam faces. Profit (£ ) Final 0.6 Store Changed strip 0.75 10,000 2500 2500 0 2000 5000 5000 1000 No changed strip 0.25 Changed strip 0.75 No changed strip 0.25 Sell Final 0.6 No final 0.4 No final 0.4 Order 1000 Order 500 No order Figure 11.2 Decision tree for Example 11.10 [...]... 500 10,000 Final 0.6 Store 2500 Changed strip 0.75 No changed strip 0.25 2000 5000 5000 No order Figure 11. 3 Amended decision tree for Example 11. 10 Changed strip 0.75 1000 No changed strip 0.25 No final 0.4 2500 0 358 Quantitative methods for business Chapter 11 We can indicate as shown in Figure 11. 3 that the option of selling the stock if she were to order 1000 and the team does not reach the final,... 0.5 Building is unsound 0.5 Figure 11. 5 Probability tree for Example 11. 14 Chapter 11 Finding the right way – analysing decisions 361 Profit (£m) Sound 0.889 15 Unsound 0 .111 1 Build B Sell Sound 0.45 4 Sound 0.182 Build Unsound 0.55 15 C Unsound 0.818 Survey 1 4 Sell A No survey Build Sound 0.5 D Unsound 0.5 Sell 15 1 4 Figure 11. 6 Amended decision tree for Example 11. 14 depends on the probability that... 0.1, and the probability 366 Quantitative methods for business Chapter 11 that s/he makes a claim as a result of vehicle theft is 0.05 The typical payment for a major accident claim is £4500, for a minor accident claim is £800, and for theft £4000 The probability that a policyholder makes more than one claim in a year is zero What is the expected value of claims per policy? 11. 4 A film production company... Figure 11. 5 we can work out: P(Surveyor’s conclusion is sound) ϭ 0.40 ϩ 0.05 ϭ 0.45 P(Surveyor’s conclusion is unsound) ϭ 0.10 ϩ 0.45 ϭ 0.55 Using Bayes’ rule: P (A|B ) ϭ P (B and A ) P(B ) 362 Quantitative methods for business Chapter 11 We can work out: P(Building is sound|Surveyor’s conclusion is sound) ϭ 0.40/0.45 ϭ 0.889 P(Building is unsound|Surveyor’s conclusion is sound) ϭ 0.05/0.45 ϭ 0 .111 P(Building... given that the surveyor’s report predicts the building is sound This probability 360 Quantitative methods for business Chapter 11 Profit (£m) 15 Sound Build 1 Unsound B Sound 4 Sell Sound 15 Build Unsound C 1 Unsound Survey Sell 4 A No survey Build D 15 Sound 0.5 1 Unsound 0.5 4 Sell Figure 11. 4 Decision tree for Example 11. 14 Joint probability Surveyor’s conclusion is sound 0.8 0.40 Surveyor’s conclusion... key export commodity There are rumours that the government will reduce the subsidy for the next crop The 368 Quantitative methods for business Chapter 11 Cloppock farmers have to decide whether to increase or decrease the number of hectares it farms, or to keep it the same The pay-offs (in Soom, the national currency) for these strategies under the same subsidy regime and under reduced subsidies are:... (a) Advise the company using a decision tree (b) How sensitive is the choice of large- or small-scale production to a change in the probability of the drug being approved for NHS use? 372 Quantitative methods for business Chapter 11 11.18 Seela Energy wants to build a nuclear power station on a remote coastline in Northern Europe They need new capacity, as without it they will incur a loss of €100m because... of the project in which case the profit will fall to £3m Financial experts put the probability of devaluation at 0.5 370 Quantitative methods for business Chapter 11 (a) Construct a decision tree to portray the situation the company faces (b) Calculate the Expected Monetary Value for each project and use them to suggest which project the company should take (c) The President of Sloochai gives a key.. .Chapter 11 Finding the right way – analysing decisions 357 Example 11. 11 Find the EMV for each decision that Sam, the market trader in Example 11. 10, could take if she had ordered 1000 shirts and the team did not make it to the final EMV(Store) ϭ 0.75 * 2000 ϩ 0.25 * 5000... to go for a full launch on the basis of the performance of the product in the test market If they went directly to a full launch then, assuming it proved a success, they would earn profits on the product earlier, but they would risk a large financial outlay on full-scale production and promoting the product on a national basis According to Beattie the main advantage 364 Quantitative methods for business . final 0.4 Order 500 No order Figure 11. 3 Amended decision tree for Example 11. 10 358 Quantitative methods for business Chapter 11 We can indicate as shown in Figure 11. 3 that the option of selling. to invest, since it has the highest EMV. 354 Quantitative methods for business Chapter 11 Example 11. 8 The proprietors of the business in Example 11. 3 estimate that the probability that demand. pay-off for each strategy and choosing the strategy that has the best among the best, or maximum among the maximum, pay-offs. 350 Quantitative methods for business Chapter 11 Example 11. 4 Which