DECISION ANALYSIS: INTRODUCTION. Métodos Cuantitativos M. En C. Eduardo Bustos Farias 1

Transcription

1 DECISION ANALYSIS: INTRODUCTION Cuantitativos M. En C. Eduardo Bustos Farias 1

2 Agenda Decision analysis in general Structuring decision problems Decision making under uncertainty - without probability information Decision making under uncertainty - with probability information Value of information Summary Cuantitativos M. En C. Eduardo Bustos Farias 2

3 Making a Decision We all make decisions..everyday Some of them are trivial and some are really important Some decisions are simple and some are really complex What makes a Decision Hard? Uncertainty Tradeoffs.. Decision Analysis is a coherent procedure to decision making Cuantitativos M. En C. Eduardo Bustos Farias 3

4 Decision vs. Outcome Suppose you can choose between deal A and deal B Deal A: Flip a coin. When it lands: if the side facing up is Heads, you win $1000 otherwise you win nothing. Deal B: A die is rolled. If the side facing up is a One, you win $1000, otherwise you win nothing Which deal would you choose? Why? Cuantitativos M. En C. Eduardo Bustos Farias 4

5 Decision vs. Outcome Suppose the coin is flipped and the die is rolled. The results are a Tails and a One. Do you still think you made a good decision? If you were given another opportunity to choose between deal A and deal B before flipping the coin and rolling the die again, which would you choose? Cuantitativos M. En C. Eduardo Bustos Farias 5

6 Decision vs. Outcome Under your control Choice of alternatives Information Preferences Not under your control Uncertainty Cuantitativos M. En C. Eduardo Bustos Farias 6

7 Structuring Decision Problems Commitment to action Logic Decision basis Alternatives Information Preferences Payoffs States of Nature A decision problem is characterized by decision alternatives, information, preferences, states of nature, and resulting payoffs. Cuantitativos M. En C. Eduardo Bustos Farias 7

8 Structuring the Decision Problems Decision alternatives: different possible strategies the decision maker can employ. Information: decision maker s knowledge and experience. Preferences: decision maker s taste for risk States of nature: future events, not under control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive. Resulting payoff: for each decision alternative and state of nature, there is a resulting payoff. These are often represented in matrix form called a payoff table. Cuantitativos M. En C. Eduardo Bustos Farias 8

9 An Investment Example Tom has inherited $1000 from a distant relative. He decided to invest the $1000 for a year to get some hands-on investment experience before graduating from college. Tom s broker has selected three potential investments she believes would be appropriate for Tom: a mutual fund, a growth stock, and a certificate of deposit. Given the small amount of money that Tom has for investment, the broker suggests Tom to pick one out of the three investments. Which investment should Tom pick? Cuantitativos M. En C. Eduardo Bustos Farias 9

10 Decision Making under Uncertainty Decision analysis approach: Identify alternatives Define the states of nature mutually exclusive collectively exhaustive Determine the payoffs resulting from each decision and the state of nature States of Nature (Economic Condition) Decision Up Stable Down Alternatives Mutual Fund $250 $150 -$150 Stock $500 $100 -$600 C/D Account $60 $60 $60 Cuantitativos M. En C. Eduardo Bustos Farias 10

11 Decision Making under Uncertainty - without Probability Information If the decision maker does not know with certainty which state of nature will occur, then he is said to be doing decision making under uncertainty Three commonly used criteria for decision making under uncertainty when probability information regarding the likelihood of the states of nature is unavailable are: 1. The optimistic approach 2. The conservative approach 3. The minimax regret approach Cuantitativos M. En C. Eduardo Bustos Farias 11

12 Optimistic Approach An optimistic decision maker focuses on the best possible outcome(state of nature) corresponding to the decision. States of Nature (Economic Condition) Decision Up Stable Down Alternatives Mutual Fund $250 $150 -$150 Stock $500 $100 -$600 C/D Account $60 $60 $60 Best payoff for a decision Decision Alternatives Maximum Payoff Mutual Fund $250 Stock $500 C/D Account $60 maximum If Tom is an optimistic investor: Pick the stock investment Cuantitativos M. En C. Eduardo Bustos Farias 12

13 Optimistic Approach Maximax Approach for Profit Payoffs 1. Find the maximum payoff for each decision alternative. 2. Select the decision alternative that has the maximum maximum payoff. Minimin Approach for Cost Payoffs 1. Find the minimum payoff for each decision alternative. 2. Select the decision alternative that has the minimal minimum payoff. Cuantitativos M. En C. Eduardo Bustos Farias 13

14 Conservative Approach A conservative decision maker wishes to ensure a guaranteed minimum possible payoff, regardless of which state of nature occurs. States of Nature (Economic Condition) Decision Up Stable Down Alternatives Mutual Fund $250 $150 -$150 Stock $500 $100 -$600 C/D Account $60 $60 $60 Worst payoff for a decision Decision Alternatives Maximum Payoff Mutual Fund -$150 Stock -$600 C/D Account $60 maximum If Tom is a conservative investor: Pick the C/D Account investment Cuantitativos M. En C. Eduardo Bustos Farias 14

15 Conservative Approach Maximin Approach for Profit Payoffs 1. Find the minimum payoff for each decision alternative. 2. Select the decision alternative that has the maximum minimum payoff. Minimax Approach for Cost Payoffs 1. Find the maximum payoff for each decision alternative. 2. Select the decision alternative that has the minimum maximum payoff. Cuantitativos M. En C. Eduardo Bustos Farias 15

16 Minimax Regret Approach Another criterion that pessimistic or conservative decision makers frequently use is the minimax regret criterion - minimizing the biggest possible regret value or loss of opportunity. This approach requires the construction of a regret table or an opportunity loss table. Calculation of Regret Values for a State of Nature 1. Determine the best value (maximum payoff or minimum cost) for a state of nature. 2. Calculate the regret for each decision alternative as the absolute value of the difference between its payoff value and this best value. Cuantitativos M. En C. Eduardo Bustos Farias 16

17 Construct a Regret Table Payoff Table States of Nature (Economic Condition) Decision Up Stable Down Alternatives Mutual Fund $250 $150 -$150 Stock $500 $100 -$600 C/D Account $60 $60 $60 Best payoff for the state of nature Regret Table States of Nature (Economic Condition) Decision Up Stable Down Alternatives Mutual Fund $250 $0 $210 Stock $0 $50 $660 C/D Account $440 $90 $0 Cuantitativos M. En C. Eduardo Bustos Farias 17

18 Minimax Regret Approach Regret Table States of Nature (Economic Condition) Decision Up Stable Down Alternatives Mutual Fund $250 $0 $210 Stock $0 $50 $660 C/D Account $440 $90 $0 Maximum regret for a decision Decision Alternatives Maximum Regret Mutual Fund $250 Stock $660 C/D Account $440 minimum If Tom wants to minimize his largest possible loss: Pick the mutual fund investment Cuantitativos M. En C. Eduardo Bustos Farias 18

19 Minimax Regret Approach Minimax Regret Approach 1. Determine the best value (maximum payoff or minimum cost) for a state of nature. 2. Calculate the regret for each decision alternative as the absolute value of the difference between its payoff value and this best value. 3. Find the maximum regret for each decision alternative. 4. Select the decision alternative that has the minimum maximum regret. Cuantitativos M. En C. Eduardo Bustos Farias 19

20 What If Probability Information is Available? Federal Reserve predicts that the economy will go up with probability 40%, stay stable with probability 50%, and go down with probability 10%. How might Tom use this information? Cuantitativos M. En C. Eduardo Bustos Farias 20

21 We Encode Our Uncertainty Using Probability Uncertainty comes from our lack of knowledge Probability allows us to speak precisely about our ignorance Instead of the probability is say I assign the probability... to Your probability changes as your knowledge changes Always condition probabilities on your background state of information - all your knowledge and experience We condition A on B when we think about A given that B happened Cuantitativos M. En C. Eduardo Bustos Farias 21

22 Expected Value Approach If a probability estimate for the occurrence of each state of Expected nature Value is available, Approach one may use the expected value approach. 1. Find the expected payoff for each decision alternative. - multiply the probability for each state of nature by the associated return and then sum these product. 2. Select the decision alternative that has the best expected payoff. States of Nature (Economic Condition) Decision Alternatives Up Stable Down Expected Value (EV) Mutual Fund $250 $150 -$150 =.4*250+.5*150+.1*-150 = 160 Stock $500 $100 -$600 =.4*500+.5*100+.1*-600 = 190 C/D Account $60 $60 $60 =.4*60+.5*60+.1*60 = 60 Probability Cuantitativos M. En C. Eduardo Bustos Farias 22

23 Conditional Probability Given events A and B, the probability that A happens given that we know B happens can be calculated by: Equivalently P {A B}: A given B {A B}: A joint B ( A B ) = P ( A P ( B ) B ) ( A B) = P( A B P( B) P ) Cuantitativos M. En C. Eduardo Bustos Farias 23

24 Prior and Posterior Probabilities - Bayesian Analysis Bayesian Analysis Prior Probability Prior Probability: Prior to obtaining additional information, the probability estimates for the state of nature Posterior Probability: Through Bayesian analysis, prior probabilities are revised, the outcome is posterior probability additional Information Posterior Probability Cuantitativos M. En C. Eduardo Bustos Farias 24

25 Bayes Theorem The famous Bayes Theorem: Given events B and A 1, A 1, A 1,, A n, where A 1, A 1, A 1,, A n are mutually exclusive and collectively exhaustive, posterior probabilities Pr(A i B) can be found by: ( B) P A i P B = ( A ) i P( B) P(A i ): prior probability P(A i B): posterior probability 1 = 1 P( B A ) i ( A ) + P( B A ) P( B A ) P B 1 P( B Ai ) P( Ai ) = P( B A) P( A) + P( B A ) P( A ) P( B A ) P( A Cuantitativos M. En C. Eduardo Bustos Farias n n n )

26 Decision Tree Approach A decision tree is a chronological representation of the decision problem: : decision node : state-of-nature node : branches leaving decision node represent different decision alternatives : branches leaving state-of-nature node represent the different states of nature Cuantitativos M. En C. Eduardo Bustos Farias 26

27 Construct a Decision Tree Root of tree: corresponds to the present time; tree is constructed outward into future The end of each limb of a tree: payoffs attained from the series of branches making up that limb. Cuantitativos M. En C. Eduardo Bustos Farias 27

28 Construct a New Decision Tree Cuantitativos M. En C. Eduardo Bustos Farias 28

29 What If More Information Can Be Obtained? Tom has learned that, for only $50, he can receive the results of a noted economist s forecast, which predicates up, stable, and down for the upcoming year. The economist offered the following verifiable statistics regarding the result of his model: When economy went up, his forecast predicted up 80% of the time, stable 10% of the time,and down 10% of the time. When economy was stable, his forecast predicted up 30% of the time, stable 60% of the time,and down 10% of the time. When When economy went down, his forecast predicted up 0% of the time, stable 10% of the time,and down 90% of the time. Cuantitativos M. En C. Eduardo Bustos Farias 29

30 Conditional Probability Should Tom purchase the forecast information from the economist? Using probability language, how is the performance of the economist s forecasts? When the economy went up, the economist predicted Pr( up up)=0.8 Pr( stable up)=0.1 Pr( down up)=0.1 When the economy stayed stable, the economist predicted Pr( up stable)=0.3 Pr( stable stable)=0.6 Pr( down stable)=0.1 When the economy went down, the economist predicted Pr( up down)=0 Pr( stable down)=0.1 Pr( down down)=0.9 Cuantitativos M. En C. Eduardo Bustos Farias 30

31 Bayesian Analysis - Tabular Approach up economical forecast for Tom: State of Nature Si Prior Probability P(Si) Conditional probability P("up" Si) Posterior Probability P(Si "up") Joint Probability P(Si and "up") up stable down P(get "up")= 0.47 Step1: Fill in Prior probabilities P(S i ) and conditional probabilities P( up S i ) Step 2: Calculate joint probabilities P( up S i )= P( up S i ) P(S i ) Step 3: Sum the Joint Probability Column to calculate the marginal probability P( up ) Step 4: Calculate the posterior probability using Bayes Theorem: P(S i up )= P( up S i )/ P( up ) Cuantitativos M. En C. Eduardo Bustos Farias 31

32 State of Nature Si Bayesian Analysis - Tabular Prior Probability P(Si) Conditional probability P("stable" Si) Approach stable economical forecast for Tom: Posterior Probability P(Si "stable") Joint Probability P("stable"and Si) up stable down P(get "stable")= 0.35 down economical forecast for Tom: State of Nature Si Prior Probability P(Si) Conditional probability P("down" Si) Joint Probability P(Si and "down") Posterior Probability P(Si "down") up stable down P(get "down")= 0.18 Cuantitativos M. En C. Eduardo Bustos Farias 32

33 Bayesian Analysis - Tabular Posterior Probability Table Approach Posterior Probability Table Payoff Table Alternativ Mutual Stock C/D Nature\Forecast Up Stable Down es Fund Account Up Up $250 $500 $60 Stable Stable $150 $100 $60 Down Down ($150) ($600) $60 Sumproduct(Posterior Prob. * Payoff) Forecast Up Stable Down Mutual Fund Stock C/D Account Prob(get " ")= EVSI (expected Value of Sample Information)= Expected Payoff Under Sample Information Cuantitativos M. En C. Eduardo Bustos Farias 33

34 Expected Value of Sample Information Expected Value with Sample Information about the States of Nature EV wsi = $238.8 Expected Value without Sample Information about the States of Nature EV wosi = $190 = result from the expected value approach Expected Value of Sample Information (EVSI) EVSI = EV wsi EV wosi = $238.8 $190 = $48.8(additional expected payoff possible through knowledge of the sample information) Cuantitativos M. En C. Eduardo Bustos Farias 34

35 Expected Value of Perfect Information If Tom knew in advance how the economic situation would be in a year, what his optimal decision would be? If Tom knew in advance the economic situation would be His Optimal Decision would be With a payoff of Up Stock $500 Stable Mutual fund $150 Down CD $60 There is a 40% probability that this perfect information will indicate an up economic situation and Tom will invest in stock and earn $500. Similarly, with 50% probability, the perfect information will indicate a stable economy and Tom will invest in mutual fund and earn $150. With 10% probability, the perfect information will indicate a down economy and Tom will invest in C/D account and earn $60. Cuantitativos M. En C. Eduardo Bustos Farias 35

36 Expected Value of Perfect Information Expected Value with Perfect Information about the States of Nature EV wpi = $ $ $60 0.1=$281 Expected Value without Perfect Information about the States of Nature EV wopi = $190 = result from the expected value approach Expected Value of Perfect Information EVPI = EV wpi EV wopi = $281 $190 = $91 Cuantitativos M. En C. Eduardo Bustos Farias 36

37 Expected Value of Perfect Information Expected Value of Perfect Information 1. Determine the best payoff corresponding to each state of nature. 2. Compute the expected value of these best payoffs (EV wpi ). 3. Compute the expected value without the perfect information (EV wopi ) using the expected value approach. 4. EVPI = EV wpi EV wopi EVPI is the increase in the expected payoff that would result if one knew with certainty which state of nature would occur. The EVPI provides an upper bound on the expected value of any sample or survey information. Cuantitativos M. En C. Eduardo Bustos Farias 37

38 Efficiency of Sample Information Efficiency of sample information Efficienty of Sample Information = EVSI EVPI (Investment example: efficiency = $48.8/$91 = 53.6%) WHY < 1? EVPI provides an upper bound for the EVSI, efficiency of sample information is always a number between 0 and 1. WHY? Cuantitativos M. En C. Eduardo Bustos Farias 38

39 Summary Decision Analysis is a coherent procedure to decision making No probability information Optimistic approach Conservative approach Minimax regret approach Basis for decision making Alternatives information preferences Probability information Sample information Bayes Theorem to revise probabilities prior probability posterior probability Expected value approach Decision tree approach Efficiency of sample information Cuantitativos M. En C. Eduardo Bustos Farias 39

40 DECISION ANALYSIS: DECISION TREES AND SENSITIVITY Cuantitativos M. En C. Eduardo Bustos Farias 40

41 Elements of a Decision Analysis Cuantitativos M. En C. Eduardo Bustos Farias 41

42 Background Information SciTools Incorporated specializes in scientific instruments and has been invited to make 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. Cuantitativos M. En C. Eduardo Bustos Farias 42

43 Background Information -- continued 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: Data for Bidding Example Low Bid Probability Less than $115, Between $115,000 and $120, Between $120,000 and $125, Greater than $125, In addition, they believe there is a 30% chance that there will be no competing bids. Cuantitativos M. En C. Eduardo Bustos Farias 43

44 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. Cuantitativos M. En C. Eduardo Bustos Farias 44

45 SciTools Problem 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. Cuantitativos M. En C. Eduardo Bustos Farias 45

46 SciTools Problem -- continued 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. Cuantitativos M. En C. Eduardo Bustos Farias 46

47 SciTools Problem -- continued 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. Cuantitativos M. En C. Eduardo Bustos Farias 47

48 SciTools Problem -- continued 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. Cuantitativos M. En C. Eduardo Bustos Farias 48

49 SciTools Payoff Tables Payoff Table for SciTools Bidding Example SciTools Bid No Bid No Bid Competitors Lowest Bid ($1000s) <115 >115, < >120, <125 >125 Probability (0.2) 0.7(0.4) 0.7(0.3) 0.7(0.1) Cuantitativos M. En C. Eduardo Bustos Farias Probability in parenthesis means the probability of competitors lowest bid if they bid. P(A i B)= P(A i B) / P(B) So, what are the P(A i B)s for all i? 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

50 Alternative Payoff Table for SciTools Bidding Example Monetary Value SciTools Bids ($2000) SciTools Wins SciTools Loses Probability That ScitTools Wins No Bid NA = ( ) = ( ) = *0.1 Decision tree construction. Cuantitativos M. En C. Eduardo Bustos Farias 50

51 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. There is a bar above each possible monetary value with height proportional to the probability of that value. Cuantitativos M. En C. Eduardo Bustos Farias 51

52 Expected Monetary Values We still don t know which choice SciTools should make. If SciTools knew what the competitors would do the decision would be easy. However, the decision must be made before this uncertainty is resolved. A common way used to make the choice is to calculate the expected monetary value (EMV) of each alternative and then choose the alternative with the largest EMV. EMV is a weighted average of the possible monetary values, weighted by their probabilities. Cuantitativos M. En C. Eduardo Bustos Farias 52

53 SciTools EMVs EMVs for SciTools Bidding Example Alternative No Bid Bid $115,000 Bid $120,000 Bid $125,000 EMV Calculation EMV 0(1) $0 15,000(0.86) + (-5000)(0.14) 20,000 (0.58) + (-5000)(0.42) 25,000(0.37) + (-5000)(0.63) What exactly does the EMV mean? $12,200 $9500 $6100 It means that if SciTools were to enter many gambles like this, where on each gamble the gains, losses and probabilities were the same, then on average it would win Cuantitativos M. En C. Eduardo Bustos Farias 53 $12,200 per gamble.

54 Decision Trees So far in this example we have gone through most of the steps in solving SciTools problem. All of this can be done efficiently using a graphical tool called a decision tree. To understand SciTools and other decision trees we need to know the following conventions that have been established for decision trees. Cuantitativos M. En C. Eduardo Bustos Farias 54

55 Decision Tree Conventions 1. Decision trees are composed of nodes (circles, squares and triangles) and branches (lines). 2. The nodes represent points in time. A decision node(a square) is a time when the decision maker makes a decision. A probability node (a circle) is a time when the result of an uncertain event becomes known. An end node (a triangle) indicates that the problem is completed - all decisions have been made, all uncertainty have been resolved and all payoffs have been incurred. Cuantitativos M. En C. Eduardo Bustos Farias 55

56 Decision Tree Conventions -- continued 3. Time proceeds from left to right. This means that 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. 4. 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. Cuantitativos M. En C. Eduardo Bustos Farias 56

57 Decision Tree Conventions -- continued 5. 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 particular probability node must sum to Individual monetary values are shown on the branches where they occur, and cumulative monetary values are shown to the right of the end nodes. (Two values are often found to the right of each node: the top one is the probability of getting to that end node, and the bottom one is the associated monetary value). Cuantitativos M. En C. Eduardo Bustos Farias 57

58 SciTools Decision tree Cuantitativos M. En C. Eduardo Bustos Farias 58

59 Decision Tree Conventions -- continued This decision tree illustrates these conventions for a single-stage decision problem, the simplest type of decision problem. In a single-stage decision problem all decisions are made first, and then all uncertainty is resolved. Later we will see multistage decision problems, where decisions and outcomes alternate. Once a decision tree has been drawn and labeled with the probabilities and monetary values, it can be solved easily. Cuantitativos M. En C. Eduardo Bustos Farias 59

60 Folding Back Procedure The solution for the decision tree is on the next slide. The solution procedure used to develop this result is called folding back on the tree. Starting at the right on the tree and working back to the left, the procedure consist of two types of calculations. At each probability node we calculate EMV and write it below the name of the node. At each decision node we find the maximum of the EMVs and write it below the node name. After folding back is completed we have calculated EMVs for all nodes. Cuantitativos M. En C. Eduardo Bustos Farias 60

61 Result of Folding Back to Obtain Optimal Decision Cuantitativos M. En C. Eduardo Bustos Farias 61

62 The PrecisionTree Add-In Decision trees present a challenge for Excel. The PrescisionTree Add-in makes the process relatively straightforward. This add-in enables us to build and label a decision tree, but it performs the folding-back procedure automatically and then allows us to perform the folding back procedure automatically and then allows us to perform sensitivity analysis on key input parameters. Cuantitativos M. En C. Eduardo Bustos Farias 62

63 The PrecisionTree Add-In -- continued The first things you must do to use PrecisionTree is to add it in. Install the Palisade s Decision Tools suite with the Setup program on the CD-ROM. Then to run PrecisionTree, there are three 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. If Excel is currently running, the procedure in the previous bullet will launch PrecisionTree on top of Excel. If Excel is already running and the Desktop Tools toolbar (shown on the next slide) is showing, you can start PrecisionTree by clicking on its icon. Cuantitativos M. En C. Eduardo Bustos Farias 63

64 SCITOOLS.XLS This file contains the results of the PrecisionTree procedure, but you should work through the steps on your own, starting with a blank spreadsheet. You ll now when PrecisionTree is ready for use when you see its toolbar. Cuantitativos M. En C. Eduardo Bustos Farias 64

65 The PrecisionTree Add-In -- Using PrecisionTree: continued Inputs. Enter the inputs shown in columns A and B of this table. New tree. Click on the new tree button (the far left button) on the PrecisionTree toolbar, and then click on any cell below the input section to start a new tree. Click on the name box of this new tree to open a dialog box. Type in a descriptive name for the tree, as shown on the next slide. Cuantitativos M. En C. Eduardo Bustos Farias 65

66 The PrecisionTree Add-In -- continued Decision nodes and branches. From here on, keep the finished tree shown earlier in mind. This is the finished product toward which we re building. To obtain decision nodes and branches, click on the (only) triangle end node to open the dialog box shown on the next slide. Cuantitativos M. En C. Eduardo Bustos Farias 66

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68 The PrecisionTree Add-In -- continued We re calling this decision Bid? and specifying that there are two possible decisions. The tree expands as shown here. The boxes that say branch show the default labels for these branches. Click on either of the to open another dialog box where you can provide a more descriptive name for the branch. Do this to label the two branches No and Yes. Also, you can enter the immediate payoff/cost for either branch right below it. Since there is a $5000 cost of bidding, enter the formula =BidCost right below the Yes branch in cell B19. The tree should now appear as shown on the next slide. Cuantitativos M. En C. Eduardo Bustos Farias 68

69 The PrecisionTree Add-In -- 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, following SciTools 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. The tree to this point should appear as shown on the next slide. Note that there are no monetary values below these decision branches because no immediate payoffs or costs are associated with the bid amount decision. Cuantitativos M. En C. Eduardo Bustos Farias 69

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71 The PrecisionTree Add-In -- continued Probability nodes and branches. We now need a probability node and branches from the rightmost end nodes to capture the competition bids. Click on the top one of these end nodes to bring up the same dialog box as shown below. Now, however, click on the red circle box to indicate that this is a probability node. Label it Any competing bid?, specify two branches, and click on OK. Then label the two branches No and Yes. Next, repeat this procedure to form another probability node following the Yes branch, call it Win bid?, and label its branches as shown on the next slide. Cuantitativos M. En C. Eduardo Bustos Farias 71

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73 The PrecisionTree Add-In -- continued Copying probability nodes and branches. You could build the next node and branches or take advantage of PrecisionTree s copy and paste function. Labeling probability branches. You should now have the decision tree as shown on the next slide. The structure is the same as the finished model but the monetary values are not correct. First enter the probability of no competing bid in cell D18 with the formula =PrNoBid and enter its complement in cell D24 with the formula =1-D18. Next, enter the probability that SciTools wins the bid in cell E22 with the formula =SUM(B10:B12) and enter its complement in cell E26 with the formula =1-E22. Cuantitativos M. En C. Eduardo Bustos Farias 73

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75 The PrecisionTree Add-In -- continued For the monetary values, enter the formula = ProdCost in the two cells, D19 and E23, where SciTools wins the contract. Enter the other formulas on probability branches. Using the previous step and the final decision tree as a guide, enter formulas for the probabilities and monetary values on the other probability branches, that is, those following the decision to bid $120,000 or $125,000. We re finished! The completed tree shows the best strategy and its associated EMV. Cuantitativos M. En C. Eduardo Bustos Farias 75

76 Risk Profile of Optimal Strategy Once the decision tree is completed, PrecisionTree has several tools we can use to gain more information about the decision analysis. First we see a risk profit and other information about the optimal decision. To do so, click on the fourth button from the left on the PrecisionTree toolbar. Then fill in the resulting dialog box shown on the next slide. The Policy Suggestion option allows us to see only that part of the tree that corresponds to the best decision, also shown on the next slide. Cuantitativos M. En C. Eduardo Bustos Farias 76

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78 Risk Profile of Optimal Strategy -- continued The Risk Profile option allows us to see a graphical risk of the optimal decision. As the risk profile shown on the next slide shows, there are only two possible monetary outcomes if SciTools bids $115,000. This graphical information is even more useful when there are a larger number of possible monetary outcomes. Cuantitativos M. En C. Eduardo Bustos Farias 78

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80 Sensitivity Analysis We have already stressed the importance of the follow-up sensitivity analysis for any decision problem, and PrecisionTree makes this relatively easy to perform. First we enter any values not the input cells and watch how the tree changes. But we can get more systematic information by clicking on PrecisionTree s sensitivity button. This brings up the dialog box shown on the next slide. It requires EMV cell to analyze at the top and one or more input cells in the middle. Cuantitativos M. En C. Eduardo Bustos Farias 80

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82 Sensitivity Analysis -- continued The cell to analyze is usually the EMV cell at the far left of the decision tree but it can be any EMV cell. Next, for any input cell such as the production cost cell, we enter a minimum value, a maximum value, a base value, and a step size. When we click Run Analysis, PrecisionTree varies each of the specified inputs and presents the results in several ways in a new Excel file with Sensitivity, Tornado, and Spider Graph sheets. Cuantitativos M. En C. Eduardo Bustos Farias 82

83 Sensitivity Analysis -- continued The Sensitivity sheet includes several charts, a typical one of which appears here. This shows how the EMV varies with the production cost for both of the original decisions. This type of graph is useful for seeing whether the optimal decision changes over the range of input variable. Cuantitativos M. En C. Eduardo Bustos Farias 83

84 Sensitivity Analysis -- continued The Tornado sheet shows how sensitive the EMV of the optimal decision is to each of the selected inputs over the ranges selected. Here we see that production cost has the largest effect on EMV, and bid cost has the smallest effect. Cuantitativos M. En C. Eduardo Bustos Farias 84

85 Sensitivity Analysis -- continued Finally, the Spider Chart shows how much the optimal EMV varies in magnitude for various percentage changes in the input variables. We again see that the production cost has a relatively largest effect, whereas the other two inputs have relatively small effects. Cuantitativos M. En C. Eduardo Bustos Farias 85

86 Sensitivity Analysis -- continued Each time we click on the sensitivity button, we can run a different sensitivity analysis. An interesting option is to run a two-way analysis. Then we see how the selected EMV varies as each pair of inputs vary simultaneously. A typical result is shown on the next slide. For each of the possible values of production cost and probability of no competitor bid, this chart indicates which bid amount is optimal. Cuantitativos M. En C. Eduardo Bustos Farias 86

87 Sensitivity Analysis -- continued As we see, the optimal bid amount remains $115,000 unless the production cost and the probability of no competing bid are both large. Then it becomes optimal to bid $125,000. Cuantitativos M. En C. Eduardo Bustos Farias 87

88 Bayes Rule Cuantitativos M. En C. Eduardo Bustos Farias 88

89 Background Information If an athlete is tested for certain type of drug use, the test will come out either positive or negative. However, these tests are never perfect. Some athletes who are drug-free test positive (false positives) and some who are drug users test negative (false negatives). We will assume that 5% of all athletes use drugs 8% of all tests on drug-free athletes yield false positives 3% of all tests on drug users yield false negatives. The question then is what we can conclude from a positive or negative test result. Cuantitativos M. En C. Eduardo Bustos Farias 89

90 Solution Let D and ND denote that a randomly chosen athlete is or is not a drug user, and let T+ and T- indicate a positive or negative test result. We know the following probabilities First, since 5% of all athletes are drug users, we know that P(D) = 0.05 and P(ND) = These are called prior probabilities because they represent the chance that an athlete is or is not a drug user prior to the results of a drug test. Cuantitativos M. En C. Eduardo Bustos Farias 90

91 Solution -- continued Second, from the information on drug test accuracy, we know the conditional probabilities P(T+ ND) = 0.08 and P(T- D)= But a drug-free athlete either tests positive or negative, and the same is true for a drug user. Therefore, P(T- ND) = 0.92 and P(T+ D) = These four conditional probabilities of test results given drug user status are often called the likelihoods of the test results. Given these priors and likelihoods we want posterior probabilities such as P(D T+) or P(ND T-). Cuantitativos M. En C. Eduardo Bustos Farias 91

92 Solution -- continued These are called posterior probabilities because they are assessed after the drug test results. This is where Baye s rule enters. Bayes Rule says that a typical posterior probability is a ratio. The numerator is a likelihood times a prior, and the denominator is the sum of likelihoods times priors. Cuantitativos M. En C. Eduardo Bustos Farias 92

93 DRUGBAYES.XLS This file shows how easy it is to implement Bayes rule in a spreadsheet. The given priors and likelihoods are listed in the ranges B5:C5 and B9:C10. Cuantitativos M. En C. Eduardo Bustos Farias 93

94 Calculations We calculate the products of likelihoods and priors in the range B15:C16. The formula in cell B15 is =B$5*B9 and it is copied to the rest of B15:C16 range. Their row sums are calculated in the range D15:D16. These represent the unconditional probabilities of the two possible results. They are also the denominator of Bayes rule. Finally we calculate the posterior probabilities in the range B21:C22. The formula in B21 is =B15/$D15 and it is copied to the rest of the range B21:C22. Cuantitativos M. En C. Eduardo Bustos Farias 94

95 Resulting Probabilities A negative test result leaves little doubt that the athlete is drug-free; this probability is A positive test result leaves a lot of doubt of whether the athlete is drug-free. The probability that the athlete uses drugs is Since only 5% of athletes use drugs it takes a lot of evidence to convince us otherwise. This plus the fact that the test produces false positives means the athletes that test positive still have a decent chance of being drug-free. Cuantitativos M. En C. Eduardo Bustos Farias 95