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Decision Analysis
C O N T E N T S
1.1 A Decision Tree Model and its Analysis Bill Sampras’ Summer Job Decision
1.2 Summary of the General Method of Decision Analysis
1.3 Another Decision Tree Model and its Analysis Bio-Imaging Development Strategies
1.4 The Need for a Systematic Theory of Probability Development of a New Consumer Product
1.5 Further Issues and Concluding Remarks on Decision Analysis
1.6 Case Modules Kendall Crab and Lobster, Inc. Buying a House The Acquisition of DSOFT National Realty Investment Corporation
1.7 Exercises
PERHAPS THE MOST FUNDAMENTAL AND IMPORTANT TASK THAT A MANAGER FACES
is to make decisions in an uncertain environment. For example, a manufacturing manager must decide how much capital to invest in new plant capacity, when future demand for products is uncertain. A marketing manager must decide among a vari- ety of different marketing strategies for a new product, when consumer response to these different marketing strategies is uncertain. An investment manager must de- cide whether or not to invest in a new venture, or whether or not to merge with an- other firm in another country, in the face of an uncertain economic and political environment. In this chapter, we introduce a very important method for structuring and ana- lyzing managerial decision problems in the face of uncertainty, in a systematic and rational manner. The method goes by the name decision analysis. The analytical model that is used in decision analysis is called a decision tree.
C H A P T E R 1
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1.1 A DECISION TREE MODEL AND ITS ANALYSIS
Decision analysis is a logical and systematic way to address a wide variety of prob- lems involving decision-making in an uncertain environment. We introduce the method of decision analysis and the analytical model of constructing and solving a decision tree with the following prototypical decision problem.
BILL SAMPRAS’ SUMMER JOB DECISION
Bill Sampras is in the third week of his first semester at the Sloan School of Manage- ment at the Massachusetts Institute of Technology (MIT). In addition to spending time preparing for classes, Bill has begun to think seriously about summer employ- ment for the next summer, and in particular about a decision he must make in the next several weeks. On Bill’s flight to Boston at the end of August, he sat next to and struck up an in- teresting conversation with Vanessa Parker, the Vice President for the Equity Desk of a major investment banking firm. At the end of the flight, Vanessa told Bill directly that she would like to discuss the possibility of hiring Bill for next summer, and that he should contact her directly in mid-November, when her firm starts their planning for summer hiring. Bill felt that she was sufficiently impressed with his experience (he worked in the Finance Department of a Fortune 500 company for four years on short-term investing of excess cash from revenue operations) as well as with his over- all demeanor. When Bill left the company in August to begin studying for his MBA, his boss, John Mason, had taken him aside and also promised him a summer job for the fol- lowing summer. The summer salary would be $12,000 for twelve weeks back at the company. However, John also told him that the summer job offer would only be good until the end of October. Therefore, Bill must decide whether or not to accept John’s summer job offer before he knows any details about Vanessa’s potential job offer, as Vanessa had explained that her firm is unwilling to discuss summer job opportuni- ties in detail until mid-November. If Bill were to turn down John’s offer, Bill could ei- ther accept Vanessa’s potential job offer (if it indeed were to materialize), or he could search for a different summer job by participating in the corporate summer recruit- ing program that the Sloan School of Management offers in January and February.
Bill’s Decision Criterion
Let us suppose, for the sake of simplicity, that Bill feels that all summer job opportu- nities (working for John, working for Vanessa’s firm, or obtaining a summer job through corporate recruiting at school) would offer Bill similar learning, networking, and resumé-building experiences. Therefore, we assume that Bill’s only criterion on which to differentiate between summer jobs is the summer salary, and that Bill obvi- ously prefers a higher salary to a lower salary.
Constructing a Decision Tree for Bill Sampras’ Summer Job Decision Problem
A decision tree is a systematic way of organizing and representing the various deci- sions and uncertainties that a decision-maker faces. Here we construct such a deci- sion tree for Bill Sampras’ summer job decision. Notice that there are, in fact, two decisions that Bill needs to make regarding the summer job problem. First, he must decide whether or not to accept John’s summer
4 CHAPTER 1 Decision Analysis
A
Accept John
’s Offer
Reject John
’s Offer
Offer From Vanessa
No Offer From Vanessa
B
FIGURE 1. Representation of an event node.
A
Accept John
’s Offer
Reject John
’s Offer
C
Accept Vanessa
’s Offer
Offer From Vanessa
No Offer From Vanessa
B
Reject Vanessa
’s Offer
FIGURE 1. Further representation of the decision tree.
node C represents the decision that Bill would face if he were to receive a summer job offer from Vanessa’s firm.
Assigning Probabilities
Another aspect of constructing a decision tree is the assignment or determination of the probability, i.e., the likelihood, that each of the various uncertain outcomes will transpire. Let us suppose that Bill has visited the career services center at Sloan and has gathered some summary data on summer salaries received by the previous class of
1.1 A Decision Tree Model and its Analysis 5
TABLE 1. Distribution of summer salaries.
Total Summer Pay Weekly Salary (based on 12 weeks) Percentage of Students Who Received This Salary $1,800 $21,600 05% $1,400 $16,800 25% $1,000 $12,000 40% 0,$500 0$6,000 25% 1,00$0 6,000$0 05%
A
Accept John
’s Offer
Reject John
’s Offer
C
Accept Vanessa
’s Offer
Offer From Vanessa
No Offer From Vanessa
B
$11,
D
$21, $16, $12, $6, $
E
$21, $16, $12, $6, $
Reject Vanessa
’s Offer
FIGURE 1. Further representation of the decision tree.
MBA students. Based on salaries paid to Sloan students who worked in the Sales and Trading Departments at Vanessa’s firm the previous summer, Bill has estimated that Vanessa’s firm would make offers of $14,000 for twelve weeks’ work to summer MBA students this coming summer. Let us also suppose that we have gathered some data on the salary range for all summer jobs that went to Sloan students last year, and that this data is conveniently summarized in Table 1.1. The table shows five different summer salaries (based on weekly salary) and the associated percentages of students who received this salary. (The school did not have salary information for 5% of the students. In order to be conservative, we assign these students a summer salary of $0.) Suppose further that our own intuition has suggested that Table 1.1 is a good ap- proximation of the likelihood that Bill would receive the indicated salaries if he were to participate in the school’s corporate summer recruiting. That is, we estimate that there is roughly a 5% likelihood that Bill would be able to procure a summer job with a salary of $21,600, and that there is roughly a 25% likelihood that Bill would be able
1.1 A Decision Tree Model and its Analysis 7
A
Accept John
’s Offer
Reject John
’s Offer
C
Accept Vanessa
’s Offer
Offer From Vanessa
No Offer From Vanessa
B
$12,000 $14,
D
$21, $16, $12, $6, $
E
$21, $16, $12, $6, $
Reject Vanessa
’s Offer
FIGURE 1. The completed decision tree.
Fundamental Aspects of Decision Trees
Let us pause and look again at the decision tree as shown in Figure 1.6. Notice that time in the decision tree flows from left to right, and the placement of the decision nodes and the event nodes is logically consistent with the way events will play out in reality. Any event or decision that must logically precede certain other events and decisions is appropriately placed in the tree to reflect this logical dependence. The tree has two decision nodes, namely node A and node C. Node A represents the decision Bill must make soon: whether to accept or reject John’s offer. Node C represents the decision Bill might have to make in late November: whether to accept or reject Vanessa’s offer. The branches emanating from each decision node represent all of the possible decisions under consideration at that point in time under the ap- propriate circumstances. There are three event nodes in the tree, namely nodes B, D, and E. Node B rep- resents the uncertain event of whether or not Bill will receive a job offer from Vanessa’s firm. Node D (and also Node E) represents the uncertain events governing the school’s corporate summer recruiting salaries. The branches emanating from each event node represent a set of mutually exclusive and collectively exhaustive outcomes from the event node. Furthermore, the sum of the probabilities of each out- come branch emanating from a given event node must sum to one. (This is because the set of possible outcomes is collectively exhaustive.) These important characteristics of a decision tree are summarized as follows:
Key Characteristics of a Decision Tree
- Time in a decision tree flows from left to right, and the placement of the de- cision nodes and the event nodes is logically consistent with the way events will play out in reality. Any event or decision that must logically precede cer- tain other events and decisions is appropriately placed in the tree to reflect this logical dependence.
- The branches emanating from each decision node represent all of the possi- ble decisions under consideration at that point in time under the appropri- ate circumstances.
- The branches emanating from each event node represent a set of mutually exclusive and collectively exhaustive outcomes of the event node.
- The sum of the probabilities of each outcome branch emanating from a given event node must sum to one.
- Each and every “final” branch of the decision tree has a numerical value as- sociated with it. This numerical value usually represents some measure of monetary value, such as salary, revenue, cost, etc.
Notice that in the case of Bill’s summer job decision, all of the numerical values associated with the final branches in the decision tree are dollar figures of salaries, which are inherently objective measures to work with. However, Bill might also wish to consider subjective measures in making his decision. We have conveniently as- sumed for simplicity that the intangible benefits of his summer job options, such as opportunities to learn, networking, resumé-building, etc., would be the same at ei- ther his former employer, Vanessa’s firm, or in any job offer he might receive through the school’s corporate summer recruiting. In reality, these subjective measures would not be the same for all of Bill’s possible options. Of course, another important sub- jective factor, which Bill might also consider, is the value of the time he would have to spend in corporate summer recruiting. Although we will analyze the decision tree ignoring all of these subjective measures, the value of Bill’s time should at least be considered when reviewing the conclusions afterward.
Solution of Bill’s Problem by Folding Back the Decision Tree
If Bill’s choice were simply between accepting a job offer of $12,000 or accepting a dif- ferent job offer of $14,000, then his decision would be easy: he would take the higher salary offer. However, in the presence of uncertainty, it is not necessarily obvious how Bill might proceed. Suppose, for example, that Bill were to reject John’s offer, and that in mid- November he were to receive an offer of $14,000 from Vanessa’s firm. He would then be at node C of the decision tree. How would he go about deciding between obtain- ing a summer salary of $14,000 with certainty, and the distribution of possible salaries he might obtain (with varying degrees of uncertainty) from participating in the school’s corporate summer recruiting? The criterion that most decision-makers feel is most appropriate to use in this setting is to convert the distribution of possible salaries to a single numerical value using the expected monetary value (EMV) of the possible outcomes:
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A
Accept
John
’s Offer
Reject John
’s Offer
C
Accept Vanessa
’s Offer
Offer From Vanessa
No Offer From Vanessa
B
$12,000 $14,
$14,
$13,
$13,
$11,
$11,
D
$21, $16, $12, $6, $
E
$21, $16, $12, $6, $
Reject
Vanessa
’s Offer
FIGURE 1. Solution of the decision tree.
weighted average of the EMVs of each of the outcomes, weighted by the probabilities corresponding to each of the outcomes. In this case, this means multiplying the proba- bility of an offer (0.60) by the $14,000 value of decision node C, then multiplying the prob- ability of not receiving an offer from Vanessa’s firm (0.40) times the EMV of node D, which is $11,580, and then adding the two quantities. The calculations are:
This number is then placed above the node, as shown in Figure 1.7. The last step in solving the decision tree is to evaluate the remaining node, which is the first node of the tree. This is a decision node, and its evaluation is accomplished by comparing the better of the two EMV values of the branches that emanate from it. The upper branch, which corresponds to accepting John’s offer, has an EMV of $12,000. The lower branch, which corresponds to rejecting John’s offer and proceeding onward, has an EMV of $13,032. As this latter value is the highest, we cross off the branch cor- responding to accepting John’s offer, and place the EMV value of $13,032 above the ini- tial node. The completed solution of the decision tree is shown in Figure 1.7. Let us now look again at the solved decision tree and examine the “optimal de- cision strategy” under uncertainty. According to the solved tree, Bill should not ac- cept John’s job offer, i.e., he should reject John’s job offer. This is shown at the first decision node. Then, if Bill receives a job offer from Vanessa’s firm, he should accept this offer. This is shown at the second decision node. Of course, if he does not receive a job offer from Vanessa’s firm, he would then participate in the school’s corporate summer recruiting program. The EMV of John’s optimal decision strategy is $13,032. Summarizing, Bill’s optimal decision strategy can be stated as follows:
EMV 5 0.60 3 $14,000 1 0.40 3 $11,580 5 $13,032.
Bill’s Optimal Decision Strategy:
- Bill should reject John’s offer in October.
- If Vanessa’s firm offers him a job, he should accept it. If Vanessa’s firm does not offer him a summer job, he should participate in the school’s corporate summer recruiting.
- The EMV of this strategy is $13,032.
Note that the output from constructing and solving the decision tree is a very concrete plan of action, which states what decisions should be made under each pos- sible uncertain outcome that might prevail. The procedure for solving a decision tree can be formally stated as follows:
Procedure for Solving a Decision Tree
- Start with the final branches of the decision tree, and evaluate each event node and each decision node, as follows:
- • For an event node, compute the EMV of the node by computing the weighted average of the EMV of each branch weighted by its probability. Write this EMV number above the event node.
- • For a decision node, compute the EMV of the node by choosing that branch emanating from the node with the best EMV value. Write this EMV number above the decision node, and cross off those branches emanating from the node with inferior EMV values by drawing a double line through them.
- The decision tree is solved when all nodes have been evaluated.
- The EMV of the optimal decision strategy is the EMV computed for the start- ing branch of the tree.
As we mentioned already, the process of solving the decision tree in this manner is called folding back the decision tree. It is also sometimes referred to as backwards induction.
Sensitivity Analysis of the Optimal Decision
If this were an actual business decision, it would be naive to adopt the optimal deci- sion strategy derived above, without a critical evaluation of the impact of the key data assumptions that were made in the development of the decision tree model. For example, consider the following data-related issues that we might want to address:
- Issue 1: The probability that Vanessa’s firm would offer Bill a summer job. We have subjectively assumed that the probability that Vanessa’s firm would offer Bill a summer job to be 0.60. It would be wise to test how changes in this proba- bility might affect the optimal decision strategy.
- Issue 2: The cost of Bill’s time and effort in participating in the school’s cor- porate summer recruiting. We have implicitly assumed that the cost of Bill’s time and effort in participating in the school’s corporate summer recruiting would be zero. It would be wise to test how high the implicit cost of participating
1.1 A Decision Tree Model and its Analysis 11
1.1 A Decision Tree Model and its Analysis 13
In the spreadsheet representation of Figure 1.8, the data for the decision tree is given in the upper part of the spreadsheet, and the “solution” of the spreadsheet is computed in the lower part in the “EMV of Nodes” table. The computation of the EMV of each node is performed automatically as a function of the data. For example, we know that node E of the spreadsheet has its EMV computed as follows:
The EMV of node D is computed in an identical manner. As presented earlier, the EMV of node C is the maximum of the EMV of node E and the value of an offer from Vanessa’s firm, and is computed as
Similarly, the EMV of nodes B and A are given by
and
All of these formulas can be conveniently represented in a spreadsheet, and such a spreadsheet is shown in Figure 1.8. Note that the EMV numbers for all of the nodes in the spreadsheet correspond exactly to those computed “by hand” in the solution of the decision tree shown in Figure 1.7. We now show how the spreadsheet representation of the decision tree can be used to study how the optimal decision strategy changes relative to the three key data issues discussed above at the start of this subsection. To begin, consider the first issue, which concerns the sensitivity of the optimal decision strategy to the value of the probability that Vanessa’s firm will offer Bill a summer job. Denote this probability by p, i.e.,
probability that Vanessa’s firm will offer Bill a summer job.
If we test a variety of values of p in the spreadsheet representation of the decision tree, we will find that the optimal decision strategy (which is to reject John’s job offer, and to accept a job offer from Vanessa’s firm if it is offered) remains the same for all val- ues of p greater than or equal to. Figure 1.9 shows the output of the spread- sheet when , for example, and notice that the EMV of node B is $12,016, which is just barely above the threshold value of $12,000. For values of p at or below the EMV of node B becomes less than $12,000, which results in a new opti- mal decision strategy of accepting John’s job offer. We can conclude the following:
- As long as the probability of Vanessa’s firm offering Bill a job is 0.18 or larger, then the optimal decision strategy will still be to reject John’s offer and to accept a summer job with Vanessa’s firm if they offer it to him.
This is reassuring, as it is reasonable for Bill to be very confident that the probability of Vanessa’s firm offering him a summer job is surely greater than 0.18. We next use the spreadsheet representation of the decision tree to study the sec- ond data assumption issue, which concerns the sensitivity of the optimal decision strategy to the implicit cost to Bill (in terms of his time) of participating in the school’s corporate summer recruiting program. Denote this cost by c, i.e.,
c 5 implicit cost to Bill of participating in c 5 the school’s corporate summer recruiting program.
p 5 0.17,
p 5 0.
p 5 0.
p 5
EMV of node A 5 MAX 5 EMV of node B, $12,000 6.
EMV of node B 5 (0.60) 3 (EMV of node C) 1 ( 1 2 0.60) 3 (EMV of node D)
EMV of node C 5 MAX 5 EMV of node E, $14,000 6.
EMV of node E 5
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Spreadsheet Representation of Bill Sampras' Decision Problem
Value of John's offer $12, Value of Vanessa's offer $14, Probability of offer from Vanessa's firm 0. Cost of participating in Recruiting $
Distribution of Salaries from Recruiting Weekly Salary Total Summer Pay Percentage of Students (based on 12 weeks) who Received this Salary $1,800 $21,600 5% $1,400 $16,800 25% $1,000 $12,000 40% $500 $6,000 25% $0 $0 5%
EMV of Nodes Nodes EMV A $12, B $12, C $14, D $11, E $11,
Data
FIGURE 1. Output of the spreadsheet of Bill Sampras’ summer job problem when the probability that Vanessa’s firm will make Bill an offer is 0.18.
If we test a variety of values of c in the spreadsheet representation of the decision tree, we will notice that the current optimal decision strategy (which is to reject John’s job offer, and to accept a job offer from Vanessa’s firm if it is offered) remains the same for all values of c less than. Figure 1.10 shows the output of the spreadsheet when. For values of c above , the EMV of node B becomes less than $12,000, which results in a new optimal decision strategy of ac- cepting John’s job offer. We can conclude the following:
- As long as the implicit cost to Bill of participating in summer recruiting is less than $2,578, then the optimal decision strategy will still be to reject John’s offer and to accept a summer job with Vanessa’s firm if they offer it to him.
This is also reassuring, as it is reasonable to estimate that the implicit cost to Bill of participating in the school’s corporate summer recruiting program is much less than $2,578. We next use the spreadsheet representation of the decision tree to study the third data issue, which concerns the sensitivity of the optimal decision strategy to the distrib- ution of possible summer job salaries from participating in corporate recruiting. Recall that Table 1.1 contains the data for the salaries Bill might possibly realize by participat- ing in corporate summer recruiting. Let us explore the consequences of changing all of the possible salary offers of Table 1.1 by an amount S. That is, we will explore modifying Bill’s possible summer salaries by an amount S. If we test a variety of values of S in the spreadsheet representation of the model, we will notice that the current optimal decision strategy remains optimal for all values of S less than. Figure 1.11 shows the output of the spreadsheet when. For values of S above , the EMV of node E will become greater than or equal to $14,000, and consequently Bill’s optimal decision strategy will change: he would reject an offer from Vanessa’s firm if it material- ized, and instead would participate in the school’s corporate summer recruiting pro- gram. We can conclude:
S 5 $2,419 S 5 $2,
S 5 $2,
c 5 $2,578 c 5 $2,
c 5 $2,
16 CHAPTER 1 Decision Analysis
We can summarize our findings as follows:
- For all three of the data issues that we have explored (the probability p of Vanessa’s firm offering Bill a summer job, the implicit cost c of participating in corporate summer recruiting, and an increase S in all possible salary values from corporate summer recruiting), we have found that the optimal decision strategy does not change unless these quantities take on unreasonable values. Therefore, it is safe to proceed with confidence in recommending to Bill Sampras that he adopt the optimal decision strategy found in the solution to the decision tree model. Namely, he should reject John’s job offer, and he should accept a job of- fer from Vanessa’s firm if such an offer is made. In some applications of decision analysis, the decision-maker might discover that the optimal decision strategy is very sensitive to a key data value. If this hap- pens, it is then obviously important to spend some effort to determine the most rea- sonable value of that data. For instance, in the decision tree we have constructed, suppose that in fact the optimal decision was very sensitive to the probability p that Vanessa’s firm would offer Bill a summer job. We might then want to gather data on how many offers Vanessa’s firm made to Sloan students in previous years, and in particular we might want to look at how students with Bill’s general profile fared when they applied for jobs with Vanessa’s firm. This information could then be used to develop a more exact estimate of the probability p that Bill would receive a job of- fer from Vanessa’s firm. Note that in this sensitivity analysis exercise, we have only changed one data value at a time. In some problem instances, the decision-maker might want to test how the model behaves under simultaneous changes in more than one data value. This is a bit more difficult to analyze, of course.
1.2 SUMMARY OF THE GENERAL METHOD OF DECISION ANALYSIS
The example of Bill Sampras’ summer job decision problem illustrates the format of the general method of decision analysis to systematically analyze a decision prob- lem. The format of this general method is as follows:
Principal Steps of Decision Analysis
- Structure the decision problem. List all of the decisions that have to be made. List all of the uncertain events in the problem and all of their possi- ble outcomes.
- Construct the basic decision tree by placing the decision nodes and the event nodes in their chronological and logically consistent order.
- Determine the probability of each of the possible outcomes of each of the un- certain events. Write these probabilities on the decision tree.
- Determine the numerical values of each of the final branches of the decision tree. Write these numerical values on the decision tree.
- Solve the decision tree using the folding-back procedure:
1.3 Another Decision Tree Model and its Analysis 17
(a) Start with the final branches of the decision tree, and evaluate each event node and each decision node, as follows: (a) • For an event node, compute the EMV of the node by computing the weighted average of the EMV of each branch weighted by its probabil- ity. Write this EMV number above the event node. (a) • For a decision node, compute the EMV of the node by choosing that branch emanating from the node with the best EMV value. Write this EMV number above the decision node and cross off those branches em- anating from the node with inferior EMV values by drawing a double line through them. (b) The decision tree is solved when all nodes have been evaluated. (c) The EMV of the optimal decision strategy is the EMV computed for the starting branch of the tree.
- Perform sensitivity analysis on all key data values. For each data value for which the decision-maker lacks confidence, test how the optimal decision strat- egy will change relative to a change in the data value, one data value at a time.
As mentioned earlier, the solution of the decision tree and the sensitivity analy- sis procedure typically involve a number of mechanical arithmetic calculations. Un- less the decision tree is small, it might be wise to construct a spreadsheet version of the decision tree in order to perform these calculations automatically and quickly. (And of course, a spreadsheet version of the model will also eliminate the likelihood of making arithmetical errors!)
1.3 ANOTHER DECISION TREE MODEL AND ITS ANALYSIS
In this section, we continue to illustrate the methodology of decision analysis by con- sidering a strategic development decision problem encountered by a new company called Bio-Imaging, Incorporated.
BIO-IMAGING DEVELOPMENT STRATEGIES
In 2004, the company Bio-Imaging, Incorporated was formed by James Bates, Scott Tillman, and Michael Ford, in order to develop, produce, and market a new and po- tentially extremely beneficial tool in medical diagnosis. Scott Tillman and James Bates were each recent graduates from Massachusetts Institute of Technology (MIT), and Michael Ford was a professor of neurology at Massachusetts General Hospital (MGH). As part of his graduate studies at MIT, Scott had developed a new technique and a software package to process MRI (magnetic resonance imaging) scans of brains of patients using a personal computer. The software, using state of the art computer graphics, would construct a three-dimensional picture of a patient’s brain and could be used to find the exact location of a brain lesion or a brain tumor, estimate its vol- ume and shape, and even locate the centers in the brain that would be affected by the tumor. Scott’s work was an extension of earlier two-dimensional image processing
1.3 Another Decision Tree Model and its Analysis 19
TABLE 1. Estimated revenues of Bio-Imaging, if the three-dimensional prototype were operational and if Bio-Imaging were awarded the SBIR grant.
Scenario Probability Total Revenues High Profit 20% $3,000, Medium Profit 40% 0,$500, Low Profit 40% 0,500,00$
TABLE 1. Estimated revenues of Bio-Imaging, if the three-dimensional prototype were operational, under financing from Nugrowth Development.
Scenario Probability Total Revenues High Profit 20% $10,000, Medium Profit 40% 0$3,000, Low Profit 40% 10,000,00$
dered whether the cost of the Nugrowth offer (80% of future profits) might be too high relative to the benefits ($1,000,000 in much-needed capital). Clearly James needed to think hard about the decisions Bio-Imaging was facing.
Data Estimates of Revenues and Probabilities
Given the intense competition in the market for medical imaging technology, James knew that there was substantial uncertainty surrounding the potential revenues of Bio-Imaging over the next three years. James tried to estimate these revenues under a variety of possible scenarios. Table 1.2 shows James’ data estimates of revenues un- der three scenarios (“high profit,” “medium profit,” and “low profit”) in the event that the three-dimensional prototype were to become operational and if they were to receive the SBIR grant. Under the “high profit” scenario the program would pre- sumably be very successful in the marketplace, yielding total revenues of $3,000,000. In the “medium profit” scenario, James estimated the revenues to be $500,000, while in the “low profit” scenario, he estimated that there would be no revenues. James as- signed his estimated probabilities of these three scenarios to be 20%, 40%, and 40% for the “high profit,” “medium profit,” and “low profit” scenarios, respectively. Table 1.3 shows James’ data estimates of revenues of Bio-Imaging in the event that the three-dimensional prototype were to become operational and if they were to accept the financing offer from Nugrowth Development. Given the higher resources that would be available to them ($1,000,000 of capital), James estimated that under the “high profit” scenario the program would yield total revenues of $10,000,000. In the “medium profit” scenario, James estimated the revenues to be $3,000,000; while in the “low profit” scenario, he estimated that there would be no revenues. As before, James assigned his estimated probabilities of the three scenarios to be 20%, 40%, and 40% for the “high profit,” “medium profit,” and “low profit” scenarios, respectively. Table 1.4 shows James’ data estimates of revenues of Bio-Imaging in the event that the three-dimensional prototype were not successful and if they were to receive
20 CHAPTER 1 Decision Analysis
TABLE 1. Estimated profit of Bio-Imaging, if the three-dimensional prototype were unsuccessful and if Bio-Imaging were awarded the SBIR grant.
Scenario Probability Total Revenues High Profit 25% $1,500, Low Profit 75% 1,500,00$
the SBIR grant for the two-dimensional software program. In this case James consid- ered only two scenarios: “high profit” and “low profit.” Note that the revenue esti- mates are quite low. Under the “high profit” scenario the program would yield total revenues of $1,500,000. In the “low profit” scenario, James estimated that there would be no revenues. James assigned his estimated probabilities of the scenarios to be 25% and 75% for the “high profit” and the “low profit” scenarios, respectively. James also gave serious thought and analysis to various other uncertainties facing Bio-Imaging. After consulting with Scott, he assigned a 60% likelihood that they would be successful in producing an operational version of the three-dimensional software pro- gram. Moreover, after consulting with Michael Ford, James also estimated that the like- lihood of winning the SBIR grant after successful completion of the three-dimensional software program to be 70%. However, they estimated that the likelihood of winning the SBIR grant with only the two-dimensional software program would be only 20%.
Construction of the Decision Tree
Let us now construct a decision tree for analyzing the decisions faced by Bio-Imag- ing. The first decision that must be made is whether Bio-Imaging should accept the offer from Medtech or instead continue with the research and development of the three-dimensional software program. This decision is represented by a decision node with two branches emanating from it, as shown in Figure 1.12. (The label “A” is placed on the decision node so that we can conveniently refer to it later.) If Bio-Imaging were to accept the offer from Medtech, there would be nothing left to decide. If instead they were to continue with the research and development of the three-dimensional software program, they would find out after six months whether or not they would succeed in making it operational. This event is represented by an event node labeled “B” in Figure 1.13. If Bio-Imaging were successful in developing the program, they would then face the decision whether to apply for an SBIR grant or to instead accept the offer from Nugrowth Development. This decision is represented as node C in Figure 1.14. If they were to apply for an SBIR grant, they would then either win the grant or not. This event is represented as node E in Figure 1.14. If they were to win the grant, they would then complete the development of the three-dimensional software product and market the product accordingly. The revenues that they would then receive would be in accordance with James’ estimates given in Table 1.2. The event node G in Figure 1.14 represents James’ estimate of the uncertainty regarding these revenues. If, however, Bio-Imaging were to lose the grant, the offer from Nugrowth would then not be available either, due to the inherent delays in the grant decision process at NIH. In this case, Bio-Imaging would not have the resources to continue, and would have to abandon the project altogether. This possibility is represented in the lower branch emanating from node E in Figure 1.14. On the other hand, if Bio-Imaging were to accept the offer from Nugrowth De- velopment, they would then face the revenue possibilities in accordance with James’ estimates given in Table 1.3. The event node H in Figure 1.14 represents James’ esti- mate of the uncertainty regarding these revenues.
