Chapter 9

1. Chapter 9 Decision Making Under Uncertainty

2. Introduction • This chapter provides a formal framework for analyzing decision problems that involve uncertainty. • Our discussion includes the following: • criteria for choosing among alternative decisions • how probabilities are used in the decision-making process • how early decisions affect decisions made at a later stage • how a decision maker can quantify the value of information • how attitudes toward risk can affect the analysis

3. Introduction continued • Throughout, we employ a powerful graphical tool - a decision tree - to guide the analysis. • A decision tree enables a decision maker to view all important aspects of the problem at once: the decision alternatives, the uncertain outcomes and their probabilities, the economic consequences, and the chronological order of events. • We show how to implement decision trees in Excel by taking advantage of a very powerful and flexible add-in from Palisade called PrecisionTree.

4. Elements of decision analysis • Although decision making under uncertainty occurs in a wide variety of contexts, all problems have three common elements: • the set of decisions (or strategies) available to the decision maker, • the set of possible outcomes and the probabilities of these outcomes, and • a value model that prescribes monetary values for the various decision-outcome combinations. • Once these elements are known, the decision maker can find an optimal decision, depending on the optimality criterion chosen.

5. Payoff Tables • At the time the decision must be made, the decision maker does not know which outcome will occur. • However, once the decision is made, the outcome will eventually be revealed, and a corresponding payoff will be received. • This payoff might actually be a cost, in which case it is indicated as a negative value. • The listing of payoffs for all decision–outcome pairs is called the payoff table.

6. Good decisions vs. good outcomes • Before proceeding, there is one very important point we need to emphasize: the distinction between good decisions and good outcomes. • In any decision-making problem where there is uncertainty, the “best” decision can have less than optimal results - that is, you can be unlucky. • Regardless of which decision you choose, you might get an outcome that, in hindsight, makes you wish you had made a different decision. • The point is that decision makers must make rational decisions, based on the information they have when the decisions must be made, and then live with the consequences. • Second-guessing these decisions, just because of bad luck with the outcomes, is not appropriate.

7. Possible decision criteria • What do we mean when we call a decision the “best” decision? • One possibility is to choose the decision that minimizes the worst payoff. This criterion, called the maximincriterion, is appropriate for a very conservative (or pessimistic) decision maker. • Such criterion tends to avoid large losses, but it fails to even consider large rewards. • It is typically too conservative and is seldom used.

8. Possible decision criteria continued • At the other extreme, the decision maker might choose the decision that maximizes the best payoff. This criterion, called the maximax criterion, is appropriate for a risk taker (or optimist). • This criterion looks tempting because it focuses on large gains, but its very serious downside is that it ignores possible losses. • Because this type of decision making could eventually bankrupt a company, the maximax criterion is also seldom used.

9. Expected monetary value (EMV) • The expected monetary value, or EMV, for any decision is a weighted average of the possible payoffs for this decision, weighted by the probabilities of the outcomes. • Using the EMV criterion, you choose the decision with the largest EMV. This is sometimes called “playing the averages.” • EMV is a “sensible” criterion for making decisions under uncertainty.

10. Sensitivity analysis • Some of the quantities in a decision analysis, particularly the probabilities, are often intelligent guesses at best. • Therefore, it is important, especially in real-world business problems, to accompany any decision analysis with a sensitivity analysis. • Here we systematically vary inputs to the problem to see how (or if) the outputs - the EMVs and the best decision - change. • For our simple decision problem, this is easy to do in a spreadsheet.

11. Sensitivity analysis continued:Simple Decision Problem.xlsx • After entering the payoff table and probabilities, calculate the EMVs in column F as a sum of products, using the formula =SUMPRODUCT(C6:F6, C10:E10) in cell F6 and copying it down.

12. Decision trees • The decision problem we have been analyzing is very basic. • You make a decision, you then observe an outcome, you receive a payoff, and that is the end of it. • Many decision problems are of this basic form, but many are more complex. • In these more complex problems, you make a decision, you observe an outcome, you make a second decision, you observe a second outcome, and so on. • A graphical tool called a decision tree has been developed to represent decision problems.

13. Decision trees continued • Decision trees can be used for any decision problems, but they are particularly useful for the more complex types. • They clearly show the sequence of events (decisions and outcomes), as well as probabilities and monetary values. • The decision tree for the simple problem appears on the next slide.

14. Decision trees continued

15. Decision trees conventions • Decision trees are composed of nodes (circles, squares, and triangles) and branches (lines). • The nodes represent points in time. A decision node (a square) represents a time when the decision maker makes a decision. A probability node (a circle) represents a time when the result of an uncertain outcome becomes known. An end node (a triangle) indicates that the problem is completed - all decisions have been made, all uncertainty has been resolved, and all payoffs and costs have been incurred.

16. Decision trees conventions continued • Time proceeds from left to right. This means that any branches leading into a node (from the left) have already occurred. Any branches leading out of a node (to the right) have not yet occurred. • Branches leading out of a decision node represent the possible decisions; the decision maker can choose the preferred branch. Branches leading out of probability nodes represent the possible outcomes of uncertain events; the decision maker has no control over which of these will occur.

17. Decision trees conventions continued • Probabilities are listed on probability branches. These probabilities are conditional on the events that have already been observed (those to the left). Also, the probabilities on branches leading out of any probability node must sum to 1. • Monetary values are shown to the right of the end nodes. (As we discuss shortly, some monetary values are also placed under the branches where they occur in time.)

18. Decision trees conventions continued • EMVs are calculated through a “folding-back” process, discussed next. They are shown above the various nodes. It is then customary to mark the optimal decision branch(es) in some way. We have marked ours with a small notch. • Decision trees allow you to use a folding-back procedure to find the EMVs and the optimal decision.

19. Folding-back procedure • Starting from the right of the decision tree and working back to the left: • At each probability node, calculate an EMV - a sum of products of monetary values and probabilities. • At each decision node, take a maximum of EMVs to identify the optimal decision.

20. Risk profiles • It is often useful to represent the probability distribution of the monetary values for any decision graphically. • Specifically, we show a “spike” chart, where the spikes are located at the possible monetary values, and the heights of the spikes correspond to the probabilities. • In decision-making contexts, this type of chart is called a risk profile. • By looking at the risk profile for a particular decision, you can see the risks and rewards involved. • By comparing risk profiles for different decisions, you can gain more insight into their relative strengths and weaknesses.

21. Risk profiles continued • Note that the EMV for any decision is a summary measure of the complete risk profile - it is the mean of the corresponding probability distribution. Therefore, when you use the EMV criterion for making decisions, you are not using all of the information in the risk profiles; you are comparing only their means.

22. Example 9.1:Background information • SciTools Incorporated specializes in scientific instruments and has been invited to submit a bid on a government contract. • The contract calls for a specific number of these instruments to be delivered during the coming year. • SciTools estimates that it will cost $5000 to prepare a bid and $95,000 to supply the instruments.

23. Example 9.1 continued:Background information • On the basis of past contracts, SciTools believes that the possible low bids from the competition (if there is competition) and the associated probabilities are: • In addition, they believe there is a 30% chance that there will be no competing bids.

24. Example 9.1 continued: Decision making elements • Although there is a wide variety of contexts in decision making, all decision making problems have three elements: • the set of decisions (or strategies) available to the decision maker • the set of possible outcomes and the probabilities of these outcome • a value model that prescribes results, usually monetary values, for the various combinations of decisions and outcomes. • Once these elements are known, the decision maker can find an “optimal” decision.

25. Example 9.1 continued: Solution • There are three elements to SciTools’ problem. • The first element is that they have two basic strategies - submit a bid or do not submit a bid. • If they decide to submit a bid they must determine how much they should bid. • The bid must be greater than $100,000 for SciTools to make a profit. • The Bidding Data would probably persuade SciTools to bid either $115,000, $120,000, $125,000 or a number in between these.

26. Example 9.1 continued: Solution • The next element involves the uncertain outcomes and their probabilities. • We have assumed that SciTool knows exactly how much it will cost to prepare the bid and supply the instruments if they win the bid. In reality these are probably estimates of the actual cost. • Therefore, the only source of uncertainty is the behavior of the competitors - will they bid and, if so, how much? • The behavior of the competitors depends on how many competitors are likely to bid and how the competitors assess their costs of supplying the instruments.

27. Example 9.1 continued: Solution • From past experience SciTools is able to predict competitor behavior, thus arriving at the 30% estimate of the probability of no competing bids. • The last element of the problem is the value model that transforms decisions and outcomes into monetary values for SciTools. • The value model in this example is straightforward but in other examples it is often complex. • If SciTools decides right now not to bid, then its monetary values is $0 - no gain, no loss.

28. Example 9.1 continued: Solution • If they make a bid and are underbid by a competitor, then they lose $5000, the cost of preparing the bid. • If they bid B dollars and win the contract, then they make a profit of B - $100,000; that is, B dollars for winning the bid, less $5000 for preparing the bid, less $95,000 for supplying the instruments. • It is often convenient to list the monetary values in a payoff table.

29. Example 9.1 continued: SciTools’ Payoff Table • Often it is possible to simplify the payoff tables to better understand the essence of the problem. SciTools care only whether they win the contract or not. An alternative payoff table for SciTools is shown on the next slide. See the file SciTools Bidding Decision1.xlsx for these and other calculations.

30. Example 9.1 continued: SciTools’ Payoff Table

31. Example 9.1 continued: Risk profiles for SciTools • A risk profile simply lists all possible monetary values and their corresponding probabilities. • From the alternate payoff table we can obtain risk profiles for SciTools. • For example, if SciTools bids $120,000 there are two possibly monetary values, a profit of $20,000 or a loss of $5000, and their probabilities are 0.58 and 0.42, respectively. • Risk profiles can be illustrated on a bar chart (see next slide). There is a bar above each possible monetary value with height proportional to the probability of that value.

32. Example 9.1 continued: Risk profiles for SciTools

33. Example 9.1 continued: Calculating EMVs • Each EMV (other than the EMV of $0 for not bidding) is a sum of products of monetary outcomes and probabilities. • These EMVs indicate that if SciTools uses the EMV criterion for making its decision, it should bid $115,000. The EMV from this bid, $12,200, is the largest of the EMVs.

34. Example 9.1 continued: Calculating EMVs • It is very important to understand what an EMV implies and what it does not imply. If SciTools bids $115,000, its EMV is $12,200. • However, SciTools will definitely not earn a profit of $12,200. It will earn $15,000, or it will lose $5000. • The EMV of $12,200 represents a weighted average of these two possible values. • Nevertheless, we use this value as our decision criterion.

35. Example 9.1 continued: Developing the decision tree • This is a direct translation of the payoff table and EMV calculations. • The company first makes a bidding decision; then observes what the competition bids, if anything; and finally receives a payoff.

36. Example 9.1 continued: Folding Back the Decision Tree • The company should bid $115,000, with a resulting EMV of $12,200. Of course, this decision is not guaranteed to produce a good outcome for the company. • For example, the competition could bid less than $115,000, in which case SciTools would be out $5000. Alternately, the competition could bid more than $120,000, in which case SciTools would be kicking itself for not bidding $120,000 and getting an extra $5000 in profit. • Unfortunately, in problems with uncertainty, we can virtually never guarantee that the optimal decision will produce the best result. • All we can guarantee is that the EMV-maximizing decision is the most rational decision, given what we know when we must make the decision.

37. Example 9.1 continued: Sensitivity analysis • The next step in the SciTools decision analysis is to perform a sensitivity analysis, for example by using the sensitivity analysis tools of the PrecisionTree add-in. (See the file SciTools Bidding Decision 1.xlsx and analysis on next slide.) • We first calculate the EMVs in column G, exactly as described previously. Then we find the maximum of these in cell B21, and we use the following nested IF formula in cell B22 to find the decision from column B that achieves this maximum: =INDEX(B16:B19,MATCH(B21,G16:G19,0)) • This long formula simply checks which EMV in column G matches the maximum EMV in cell B21 and returns the corresponding decision from column B.

38. Example 9.1 continued: Sensitivity analysis

39. Example 9.1 continued: Sensitivity analysis • After we have the formulas in cells B21 and B22 set up, the data table is easy. We allow the probability of no competing bid to vary from 0.2 to 0.7. • The data table shows how the optimal EMV increases over this range. Also, its third column shows that the $115,000 bid is optimal for small values of the input, but that $125,000 becomes optimal for larger values. • The main point here is that if we set up a spreadsheet model that links all of the EMV calculations to the inputs, then using data tables to perform sensitivity analyses on selected inputs is easy.

40. The PrecisionTree add-in • Decision trees present a challenge for Excel. We must somehow take advantage of Excel’s calculating capabilities (to calculate EMVs, for example) and its graphical capabilities (to depict the decision tree). • Fortunately, there is a powerful add-in, PrecisionTree, developed by Palisade Corporation, that makes the process relatively straightforward.

41. The PrecisionTree add-in continued • The first thing you must do to use PrecisionTree is to “add it in.” We assume you have already installed the Palisade DecisionTools suite. Then to run PrecisionTree, you have two options: • If Excel is not currently running, you can launch Excel and PrecisionTree by clicking on the Windows Start button and selecting the PrecisionTree item from the Palisade Decision Tools group in the list of Programs. • If Excel is currently running, the first procedure will launch PrecisionTree on top of Excel.

42. The Decision tree model • PrecisionTree is quite easy to use - at least its most basic items are. • We will lead you through the steps for the SciTools example. Figure on the next slide shows the results of this procedure, just so that you can see what you are working toward. (See the file SciTools Bidding Decision 2.xlsx.)

43. The Decision tree model continued

44. Building the decision tree • Inputs. Enter the inputs shown in columns A and B of Figure below. • New tree. Click on the Decision Tree button on the PrecisionTree ribbon, and then select cell A14 below the input section to start a new tree. You will immediately see a dialog box where, among other things, you can name the tree. Enter a descriptive name for the tree, such as SciTools Bidding, and click on OK. You should now see the beginnings of a tree, as shown in Figure on the next slide.

45. Building the decision tree continued • Decision nodes and branches. From here on, keep the tree in figure above in mind. This is the finished product you eventually want. To obtain decision nodes and branches, select the (only) triangle end node to open the dialog as shown on the next slide.

46. Building the decision tree continued

47. Building the decision tree continued

48. Building the decision tree continued • More decision branches. The top branch is completed; if SciTools does not bid, there is nothing left to do. So click on the bottom end node (the triangle), following SciTools’s decision to bid, and proceed as in the previous step to add and label the decision node and three decision branches for the amount to bid.

49. Building the decision tree continued • Probability nodes and branches. Using the same procedure (and using decision tree figure as a guide), create probability nodes extending from the “bid $115,000” decision. You should have the skeleton in Figure below.

50. Building the decision tree continued • Copying probability nodes and branches. You could now repeat the same procedure from the previous step to build probability nodes and branches following the other bid amount decisions, but because they are structurally equivalent, you can save a lot of work by using PrecisionTree’s copy and paste feature. • Enter probabilities on probability branches. You should now have the decision tree, but the probabilities and monetary values are not yet correct.