The Business Master (4th Edition)

home *** CD-ROM | disk | FTP | other *** search

/ The Business Master (4th Edition) / The Business Master - 4th Edition.iso / files / busition / treeplan / treeplan.doc < prev next >

Wrap

Text File | 1993-08-12 | 256.8 KB | 6,034 lines

TreePlan for DOS Decision Tree Software Version 2.1 +---------- | / +----------()------------ | \ / | [] +---------- \ | +---------- Program and User's Manual by Michael R. Middleton University of San Francisco Copyright 1993 Michael R. Middleton All Rights Reserved Published by Decision Support Services 2105 Buchanan Street, #1 San Francisco, CA 94115 Voice/Fax (415) 673-6217 Compuserve 71330,3445 TreePlan User's Manual Page i Warranty ------------------------------------------------------------------ Users of TreePlan must accept this disclaimer of warranty: TreePlan copyrighted software and documentation are provided on an 'as is' basis. The author and Decision Support Services disclaim all warranties, expressed or implied, including, without limitation, the warranties of merchantability and of fitness for any purpose. The user assumes all risks as to the quality and performance of this software and documentation. The author and Decision Support Services assume no liability for indirect, consequential, or incidental damages which may result from the use or misuse of TreePlan. Some states do not allow the exclusion of the limit of liability for consequential or incidental damages, so the above limitation may not apply to you. This agreement shall be governed by the laws of the State of California and shall inure to the benefit of the author and Decision Support Services and any successors, administrators, heirs, and assigns. Any action or proceeding brought by either party against the other arising out of or related to this agreement shall be brought only in a state or federal court of competent jurisdiction located in the city and county of San Francisco, California. The parties hereby consent to in personam jurisdiction of said courts. Shareware ------------------------------------------------------------------ Shareware is a method of distributing software that gives you a chance to try software before buying it. If you try a shareware program and continue using it, you are expected to register. Individual programs differ on the details about the evaluation period, registration fee, and support. Copyright laws apply to both shareware and commercial software, and the copyright holder may choose to retain all rights. The only meaningful difference between shareware and commercial software is the method of distribution. Shareware authors specifically grant the rights to copy and distribute the software and documentation, either to all users and by all methods, or with some restrictions. You should try to find software that suits your needs and budget, whether it's commercial or shareware. Both types have good programs and bad, but shareware makes finding the right program easier, because you can try before you buy. Also, because shareware distribution costs are much lower, shareware prices are often lower, too. Finally, shareware has the ultimate money-back guarantee: if you don't use it, you don't pay for it. TreePlan User's Manual Page ii Registration ------------------------------------------------------------------ TreePlan is a shareware product and may be provided at no charge to the user for evaluation. If you find this program useful and continue to use TreePlan after a 90-day evaluation period, you must make a registration payment to Decision Support Services. The registration fee will license one copy for use on any one computer at any one time. To register TreePlan by cash, check, or money order, please print the ORDER.DOC text file and mail directly to Decision Support Services. The registration fee is $29.00 (California residents, add sales tax) plus $4.00 shipping to U.S. and Canada, $6.00 elsewhere. Make money order or check (in U.S. funds drawn on a U.S. bank) payable to Decision Support Services, and mail to Decision Support Services, 2105 Buchanan Street, #1, San Francisco, CA 94115. For credit card orders, please see the VISA&MC.DOC text file. Upon registering TreePlan, you will be sent by postal mail a formatted, laser-printed, bound copy of the User's Manual and disks containing the most recent version of the software and documentation. Registered users will be notified of the next major version and are entitled to 90 days of technical support via postal mail, phone, fax, or Compuserve 71330,3445. A site license agreement is required to use TreePlan on a computer network or to permit copying of the documentation for users within an organization. Technical support is provided only to one representative of the organization. The User's Manual may not be copied unless a site license agreement has been signed. Please see the SITELICE.DOC text file for additional information and an order form. Distribution ------------------------------------------------------------------ Please share copies of the TreePlan disks freely with prospective users. You are granted permission to make as many copies as you wish. Do not alter the program, data files, or documentation. Please see the LICENSE.DOC text file for additional information. Be sure to include the complete TreePlan package, including program, sample decision tree data files, documentation on disk, TreePlan User's Manual Page iii and all other files listed in the PACKING.LST text file. If any files listed in PACKING.LST, or the PACKING.LST file itself, are missing, then the package is not complete and distribution is forbidden. Please contact Decision Support Services to obtain a complete package suitable for distribution. Disk vendors, bulletin boards, other distributors, and individuals may distribute the complete TreePlan package for a distribution fee not exceeding $12 as long as you clearly explain the shareware concept, the need for users to register the products they use, and the fact that the price of your disks/service is a copying fee only and does not constitute payment for the product. Please see the VENDOR.DOC, SYSOP.DOC, and DESCRIBE.DOC text files for additional information. ASP Ombudsman ------------------------------------------------------------------ This program is produced by a member of the Association of Shareware Professionals (ASP). ASP wants to make sure that the shareware principle works for you. If you are unable to resolve a shareware-related problem with an ASP member by contacting the member directly, ASP may be able to help. The ASP Ombudsman can help you resolve a dispute or problem with an ASP member, but does not provide technical support for members' products. Please write to the ASP Ombudsman at 545 Grover Road, Muskegon, MI 49442-9427 or send a Compuserve message via Compuserve Mail to ASP Ombudsman 70007,3536. TreePlan User's Manual Page iv Table of Contents Chapter 1: Introduction 1 What TreePlan Can Do 1 Necessary Equipment 2 Optional Equipment 2 Installing TreePlan 2 Files on disk 3 Floppy-drive installation 3 Hard-drive installation 4 Reading the README.DOC file 4 Starting and stopping TreePlan 5 Chapter 2: Basic Decision Tree Concepts 7 DriveTek Problem 7 Nodes and Branches 8 Terminal Values 9 DriveTek Unsolved Tree 10 Strategy 11 Payoff Distribution 12 Certainty Equivalent 12 DriveTek Strategies 13 Rollback Method 16 DriveTek Solved Tree 17 Sensitivity Analysis 18 Probability 18 Value 20 Chapter 3: Advanced Decision Tree Concepts 21 Risk Attitude 21 Sensitivity Analysis 27 Lottery Certainty Equivalent 27 Risk Coefficient 29 Expected Value of Perfect Information 30 Valley Problem (A) 30 Valley EVPI Tree 31 Bayesian Revision of Probabilities 34 Valley Problem (B) 38 Valley Solved Tree 43 Expected Value of Sample Information 44 Valley EVSI Tree 46 Risk Attitude and Value of Information 47 TreePlan User's Manual Page v Chapter 4: General Treeplan Features 49 Screen Layout 49 TreePlan Modes 51 Tree Navigation In READY Mode 52 Chapter 5: Building A Tree With TreePlan 55 Step 1: Node Add Option 56 Step 2: Node Event Option 56 Step 3: Node Decision Option 57 Step 4: Node Event Option 58 Step 5: Node Decision Option 59 Step 6: Node Copy Option 59 Chapter 6: TreePlan Menu Options 61 Node Menu Options 61 Node Decision Option 62 Node Event Option 62 Node Add Option 62 Node Copy Option 63 Node Insert Option 63 Node Oops Option 64 Node Shorten Option 64 Node Terminal Option 64 Node Remove Option 65 Node New Option 65 Solve Menu Options 65 Solve Risk Options 66 Solve Risk Change Options 66 Solve Risk Max/Min Option 67 Solve Risk Neutral Option 67 Solve Risk Direct Option 67 Solve View Option 68 Solve Distribution Options 68 Solve Distribution Screen Option 68 Solve Distribution Printer/File Options 69 Solve Print Options 69 Solve Print Screen Option 69 Solve Print Printer/File Options 70 TreePlan User's Manual Page vi Print Menu Options 70 Print Screen Option Print Printer/File Options 71 File Menu Options 71 File Retrieve Option 72 File Save Option 72 File Erase Tree/Print Options 73 File List Tree/Print Options 73 File Directory Option 73 EVPI Menu Options 73 EVPI View Option 74 EVPI Retain Option 75 Bayes Menu Options 75 Bayes Link Options 75 Bayes Input Options 76 Bayes Screen Option 77 Bayes Printer/File Options 77 Bayes Transfer Options 78 Table Value Menu Options 78 Table Value Load Option 78 Table Value View Option 79 Table Value Printer/File Options 79 Table Probability Menu Options 80 Table Probability Load Option 80 Table Probability Tenths Option 81 Table Probability View Option 81 Table Probability Printer/File Options 81 Table Lottery-CE Menu Options 81 Table Lottery-CE Load Option 82 Table Lottery-CE View Option 82 Table Lottery-CE Printer/File Options 83 Table Risk-Coefficient Menu Options 83 Table Risk-Coefficient Load Option 83 Table Risk-Coefficient View Option 84 Table Risk-Coefficient Printer/File Options 84 TreePlan User's Manual Page vii Default Menu Options 84 Default Format Options 85 Default Layout Options 86 Default Setup Option 87 Default Printer Options 87 Default Directory Option 88 Default Clear Option 88 Default Update Option 89 Zoom Menu Options 89 Zoom Screen Option 89 Zoom Printer/File Options 90 Appendix A: References 91 Appendix B: TreePlan Error Messages 92 Index 96 TreePlan User's Manual Page viii Chapter 1: Introduction ****************************************************************** This chapter discusses TreePlan's features, the computer equipment supported by TreePlan, and the installation procedures for preparing TreePlan to run on your system. What TreePlan Can Do ================================================================== TreePlan is a computer program for analyzing decision tree models. Decision trees are especially appropriate for analyzing sequential decision problems under uncertainty. For information about decision tree concepts, basic and advanced, please see Chapter 2 and Chapter 3. TreePlan decision tree software includes the following features: Create and modify decision tree models on screen Save and retrieve tree data files Print the decision tree diagram, unsolved or solved Solve (rollback) using expected value or a utility function Assess risk attitude using an exponential utility function Automatically compute Expected Value of Perfect Information Revise prior probabilities using Bayes rule Perform sensitivity analysis of cash flows, probabilities, or risk attitude Use horizontal-bar menus to choose commands Select extensive on-screen help For information about TreePlan's general features and menu options, please see Chapter 4 and Chapter 6. Chapter 1: Introduction Page 1 Necessary Equipment ================================================================== You need the following equipment to run TreePlan: Computer: TreePlan is designed to run on the IBM PC, XT, AT, PS/2, and fully compatible personal computers. Operating System: TreePlan runs on DOS 2.0 or later (either PC- DOS or MS-DOS). Memory: To run TreePlan, your computer must have a minimum of approximately 384 kilobytes (384 KB) of random access memory (RAM). Disk Drive: TreePlan will run on a minimal system with only a single floppy drive (double-sided 5.25-inch 360 KB). TreePlan also runs on higher density floppy drives (720 KB, 1.2 MB, and 1.44 MB), hard drives, and electronic disks (virtual or RAM disks). Optional Equipment ================================================================== The following equipment is not required to run TreePlan: Printer: You can use a dot matrix, daisy wheel, or laser printer for printing TreePlan's diagrams and tables. The printer interface can be either parallel or serial. PostScript printers are not directly supported. Graphics card: You can run TreePlan on a system with a text-only monochrome adapter card or a graphics card. The monochrome or color monitor attached to the card must support an 80- column display. TreePlan automatically detects the type of graphics card you're using and acts accordingly. TreePlan does not make use of expanded or extended memory, a mouse, or a math coprocessor. Installing TreePlan ================================================================== The following sections discuss the files on the original TreePlan TreePlan User's Manual Page 2 disk (or disks), installation procedures for floppy-drive and hard-drive systems, and methods for reading the README.DOC file. Files on disk ------------------------------------------------------------------ The original TreePlan disk includes the following files: TREEPLAN.EXE TreePlan program DRIVETEK.TRE sample tree data file VALLEY_A.TRE sample tree data file VALLEY_B.TRE sample tree data file README.DOC late-breaking news, if any The following file may be on the same disk (3.5-inch) or on a separate disk (5.25-inch): TREEPLAN.DOC documentation (this User's Manual) Put a write-protect tab on the original TreePlan 5.25-inch disk (or open the write-protect window in the upper right-hand corner of the 3.5-inch disk) before you copy the files to your hard disk or floppy work disk. Then store the original in a safe place free of dirt, moisture, and magnetic fields. Floppy-drive installation ------------------------------------------------------------------ If you're using a floppy-drive system with no hard disk (or if you always want to run TreePlan from a floppy drive), you should make a work disk before using TreePlan. Most floppy-drive computers have one floppy-drive named A and a second floppy-drive named B. If your floppy-drives are named differently, make the appropriate substitutions as you read the instructions below. (1) Be sure the original TreePlan disk is write-protected. (2) Place the original TreePlan disk in drive A and a blank, formatted disk in drive B. (Refer to your DOS manual for instructions for formatting a disk.) (3) Type COPY A:*.* B: and press Enter. This command tells DOS to copy all files on drive A to drive B. Chapter 1: Introduction Page 3 (4) After all files are copied, remove the original TreePlan disk from drive A and store it safely. Affix a label to the disk in drive B to identify it as your TreePlan work disk. Hard-drive installation ------------------------------------------------------------------ If you have a hard-disk system, you probably want to copy the TreePlan files into a directory on the hard disk. In the following procedure, you first create a directory called TREEPLAN off the root directory and then copy your TreePlan files into it. Most hard disk computers have a floppy disk drive named A and a hard disk drive named C. If your drives are named differently, make the appropriate substitutions as you read the instructions below. (1) At the DOS prompt, type C: and press Enter to make drive C the current drive. (2) Type CD C:\ and press Enter to go to the root directory. (3) Make a new directory called TreePlan. Type MD TREEPLAN and press Enter. (4) Change to the new directory. Type CD TREEPLAN and press Enter. (5) Be sure the original TreePlan disk is write-protected. Insert it in floppy drive A and type COPY A:*.* C: and press Enter. (6) After all files are copied, remove the TreePlan disk from drive A and store it safely. It is usually convenient to keep your tree data files in the same directory as the TreePlan program files, but you can put them in a separate directory if you wish. Reading the README.DOC file ------------------------------------------------------------------ Any last-minute changes or additions to the TreePlan program are documented in a file called README.DOC. You should review this file carefully before working with TreePlan and make note of any TreePlan User's Manual Page 4 changes. To display the README.DOC file, go to the directory containing your TreePlan files. At the DOS prompt, type TYPE README.DOC and press Enter to view the file. You can send the README.DOC file to your printer using one of the following DOS commands. PRINT README.DOC COPY README.DOC PRN TYPE README.DOC > PRN Starting and stopping TreePlan ================================================================== If you are using TreePlan on a floppy disk, insert the TreePlan work disk in floppy drive A. At the DOS prompt, type A: and press Enter to make drive A the default drive. To start TreePlan, type TREEPLAN and press Enter. If you using TreePlan on a hard-disk system, move to the directory containing the TreePlan program files. For example, if the program files are in directory TREEPLAN on drive C, at the DOS prompt, first type C: and press Enter to make drive C the default drive, and then type CD TREEPLAN and press Enter to move to the TreePlan directory. To start TreePlan, type TREEPLAN and press Enter. You can also start TreePlan using one or both of the two command- line switches. If you want TreePlan to display a larger cursor that is easier to see (e.g., if you're using a laptop computer), type TreePlan B at the DOS prompt and press Enter. If you're using a somewhat non-standard video display, you may want to override TreePlan's autodetection of your graphics card and force TreePlan to use monochrome "colors"; at the DOS prompt, type TreePlan M and press Enter. If you want both a block cursor and monochrome, type TreePlan BM and press Enter. To stop TreePlan, first press the Escape key repeatedly until the mode indicator in the upper-right corner of the screen shows READY. In READY mode, type /Q to quit using the program. (If you have made changes to your decision tree since you last saved the tree data on disk, you will be asked to confirm that you want to quit without saving.) While selecting menu options or editing an entry, you can always use the Escape key to cancel an operation or back out of a Chapter 1: Introduction Page 5 situation. If you press the Escape key repeatedly, you will always return to READY mode. TreePlan User's Manual Page 6 Chapter 2: Basic Decision Tree Concepts ****************************************************************** This chapter reviews some basic concepts concerning decision tree models, including nodes, branches, terminal values, strategy, payoff distribution, certainty equivalent, and the rollback method. It is especially important to understand the concept of a strategy before using TreePlan. If you are familiar with decision tree terminology and concepts, you may want to skim this chapter and Chapter 3 quickly; then refer to Chapter 4 for TreePlan's general features. If you are not well-acquainted with decision trees, you should study this chapter and Chapter 3 carefully and also consult other references. Decision trees are useful models for analyzing sequential decision problems under uncertainty. A decision tree is a graphic model which describes the decisions to be made, the events that may occur, and the outcomes associated with combinations of decisions and events. Probabilities are assigned to the events, and values are determined for each outcome. A major goal of the analysis is to determine the best strategy. The following problem, adapted from an example in the Spurr text (1973), will be used to illustrate the basic concepts of decision tree modeling. DriveTek Problem ================================================================== The management of DriveTek Research Institute has learned that Artex Computers is interested in developing a tape drive for a proposed new computer system. Artex does not have research people available to develop the new drive itself, so the company is going to subcontract the development to an independent research firm. Artex has offered a fee of $250,000 for developing the new tape drive and has asked for proposals from various research firms. The contract is to be awarded not on the basis of price (set at $250,000) but on the basis of both the technical plan shown in the proposal and the reputed technical competence of the firm submitting the proposal. DriveTek Research Institute is considering submitting a proposal to Artex Computer to develop the new tape drive. DriveTek Research management estimates that it will cost about $50,000 to prepare a proposal; further, they estimate that the chances are Chapter 2: Basic Decision Tree Concepts Page 7 about 50/50 that they will be awarded the contract. However, DriveTek Research engineers are uncertain about how they will develop the tape drive if they are awarded the contract. Three alternative approaches can be tried. The first approach is a mechanical method with a cost of $120,000, and the engineers are certain they can develop a successful model with this approach. A second approach involves the use of electronic components. The engineers estimate that it will cost only $50,000 to develop a model of the tape drive using the electronic approach, but that there is only a 50 percent chance that the results will be satisfactory. A third approach involves the use of magnetic components; the cost of developing a model using this approach will be $80,000, with a 70 percent chance of success. DriveTek Research has sufficient time to try only two approaches. Thus, if they try either the magnetic or electronic method and it fails, the second attempt will have to use the mechanical method in order to guarantee a successful model. The management of DriveTek Research is uncertain about how to take all this information into account in making the immediate decision about whether to spend $50,000 to develop a proposal to send to Artex Computers. Nodes and Branches ================================================================== TreePlan uses three kinds of nodes and two kinds of branches to represent a decision tree. A decision node is a point where a choice must be made, usually shown as a square on hand-drawn decision trees. TreePlan shows a decision node as square brackets, [], sometimes with the letter D between the square brackets to emphasize that it is a decision node, [D]. The branches extending from a decision node are decision branches, each branch representing one of the possible alternatives or courses of action that are available at that point. There are two major decisions in the DriveTek problem. First, they must decide whether to prepare a proposal or not. Second, if they prepare a proposal and are awarded the contract, they must decide which of the three approaches to use to satisfy the contract. An event node is a point where uncertainty is resolved, i.e., a point where the decision maker learns about the occurrence of an event. An event node, sometimes called a "chance node," is usually shown as a circle on decision diagrams. TreePlan shows an TreePlan User's Manual Page 8 event node as curved parentheses, (), sometimes with the letter E between the parentheses to emphasize that it is an event node, (E). The event set consists of the event branches extending from an event node, each branch representing one of the possible events that may occur at that point. There are three sources of uncertainty in the DriveTek problem: whether they are awarded the contract or not, whether the electronic approach is OK or bad, and whether the magnetic approach is OK or bad. In general, decision nodes and branches represent the controllable factors in a decision problem; event nodes and branches represent uncontrollable factors. The third kind of node is a terminal node, representing the final result of a combination of decisions and events. Terminal nodes are the endpoints of a decision tree. TreePlan shows a terminal node as a pair of vertical lines, ||, sometimes with the letter T between the lines to emphasize that it is a terminal node, |T|. The following table summarizes the three kinds of nodes and two kinds of branches that TreePlan uses to represent a decision tree. Type of Written TreePlan Node Symbol Symbol Node Successor -------- -------- --------- -------------- Decision square [] or [D] decision branches Event circle () or (E) event branches Terminal endpoint || or |T| terminal value Terminal Values ================================================================== Each terminal node has an associated terminal value, sometimes called a payoff value, outcome value, or endpoint value. Each terminal value measures the result of a scenario, i.e., the sequence of decisions and events on a unique path leading from the initial decision node to a specific terminal node. TreePlan lets you assign a partial-cash-flow value to each decision branch and event branch, and TreePlan automatically sums the partial-cash-flow values on the branches leading to a terminal node to determine the terminal value. If you want to assign terminal values directly instead of using partial-cash-flows, simply assign zero value to all branches except the branches immediately preceding the terminal nodes. Chapter 2: Basic Decision Tree Concepts Page 9 On an unsolved tree, the branch name, probability, and partial- cash-flow value are arranged as follows. Branch name Decision branch: [D]------------------------... Value Branch name Event branch: (E)------------------------... Value Probability In the DriveTek problem, there are distinct partial-cash-flows associated with many of the decision and event branches. In other decision problems under uncertainty, it may be necessary to use a more elaborate financial model to determine the terminal values. DriveTek Unsolved Tree ================================================================== A condensed version of the unsolved decision tree model for the DriveTek problem is shown below. Partial-cash-flows and terminal values are in thousands of dollars. (The diagram was prepared using TreePlan, printed to a file, and condensed with a word processor so that it would fit on a single page.) TreePlan User's Manual Page 10 Use mech +----------||........................ +80 | -120 | | Elec OK | +----------||............+150 | Try elec | 0 0.5 Awarded +----------() +----------[] -50 | Elec bad Use mech | +250 0.5 | +----------[]----------|| +30 | | 0 0.5 -120 | | | | Mag OK Prepare | | +----------||............+120 +----------() | Try mag | 0 0.7 | -50 | +----------() | | -80 | Mag bad Use mech | | +----------[]----------|| 0 | | 0 0.3 -120 [] | | | No award | +----------||.................................... -50 | 0 0.5 | | Don't +----------||................................................ 0 0 For example, the +30 (thousand dollars) terminal value on the far right of the diagram is associated with the following scenario: Branch type Branch name Partial-cash-flow ----------- ----------- ----------------- Decision Prepare proposal -50 Event Awarded contract +250 Decision Try electronic approach -50 Event Electronic approach bad 0 Decision Use mechanical approach -120 ---- Terminal value +30 Strategy ================================================================== A strategy is a specification of an initial choice and any subsequent choices to be made by the decision maker. The subsequent choices are usually conditional on events. The Chapter 2: Basic Decision Tree Concepts Page 11 specification of a strategy must be comprehensive; if the decision maker gives the strategy to a colleague, the colleague must know exactly which choice to make at each decision node that might be encountered. Most decision problems have many possible strategies, and a goal of the analysis is to determine the optimal strategy, taking into account the decision maker's risk attitude. There are four strategies in the DriveTek problem. For example, one of the strategies could be stated as follows: Prepare the proposal; if not awarded the contract, stop; if awarded the contract, try the magnetic method; if the magnetic method is OK, stop; if the magnetic method is bad, use the mechanical method. The four strategies will be discussed in detail below. Payoff Distribution ================================================================== Each strategy has an associated payoff distribution, sometimes called a risk profile. The payoff distribution of a particular strategy is a probability distribution showing the probability of obtaining each terminal value associated with a particular strategy. In decision tree models, the payoff distribution can be shown as a list of possible payoff values, x, and the discrete probability of obtaining each value, P(X=x), where X represents the uncertain terminal value. Since each strategy can be characterized by its payoff distribution, the goal of selecting the best strategy becomes a problem of choosing the best payoff distribution. One approach for making the choice is to use certainty equivalents, described in the next section. Certainty Equivalent ================================================================== A certainty equivalent is a certain payoff value which is equivalent, for the decision maker, to a particular payoff distribution. If the decision maker can determine his or her certainty equivalent for the payoff distribution of each strategy, then the optimal strategy is the one with the highest certainty equivalent. The certainty equivalent, i.e., the minimum selling price for a payoff distribution, depends on the decision maker's personal TreePlan User's Manual Page 12 attitude toward risk. A decision maker may be risk preferring, risk neutral, or risk avoiding. For additional information, please consult the Risk Attitude section of Chapter 3. If the terminal values are not regarded as extreme (relative to the decision maker's total assets), if the decision maker will encounter other decision problems with similar payoffs, and if the decision maker has the attitude that he or she will "win some and lose some," then the decision maker's attitude toward risk may be described as risk neutral. If the decision maker is risk neutral, the certainty equivalent of a payoff distribution is equal to its expected value. The expected value of a payoff distribution is calculated by multiplying each terminal value by its probability and summing the products. The next section illustrates each of the four strategies of the DriveTek problem. DriveTek Strategies ================================================================== Each strategy is described by a shorthand statement and a more detailed statement. The branches that might be encountered when the decision maker follows the strategy are shown in decision tree form; the appropriate branches are extracted from the unsolved tree diagram without showing partial-cash-flow values. Each payoff distribution is shown as a discrete probability distribution, with value x and probability P(X=x). Since a strategy specifies a choice at each decision node, the uncertainty about terminal values depends only on the occurrence of events. The probability of obtaining a terminal value is calculated as the joint probability of the events on the path leading to the terminal node. If the decision maker is risk neutral, the expected value is the appropriate certainty equivalent for choosing among the strategies. Strategy 1: Prepare; if Awarded, Use mech. Details: Prepare the proposal; if not awarded the contract, stop (payoff = -50); if awarded the contract, use the mechanical method (payoff = +80). Chapter 2: Basic Decision Tree Concepts Page 13 Use mech +----------||........................ +80 | Awarded | +----------[] | 0.5 Prepare | +----------() | | [] | No award +----------||.................................... -50 0.5 Value Prob. ----- ----- +80 .50 -50 .50 ----- ----- (Expected Value = +15) 1.00 Strategy 2: Prepare; if Awarded, Try elec. Details: Prepare the proposal; if not awarded the contract, stop (payoff = -50); if awarded the contract, try the electronic method; if the electronic method is OK, stop (payoff = +150); if the electronic method is bad, use the mechanical method (payoff = +30). Elec OK +----------||............+150 Try elec | 0.5 Awarded +----------() +----------[] | Elec bad Use mech | 0.5 +----------[]----------|| +30 Prepare | 0.5 +----------() | | [] | No award +----------||.................................... -50 0.5 TreePlan User's Manual Page 14 Value Prob. ----- ----- +150 .25 +30 .25 -50 .50 ----- ----- (Expected Value = +20) 1.00 Strategy 3: Prepare; if Awarded, Try mag. Details: Prepare the proposal; if not awarded the contract, stop (payoff = -50); if awarded the contract, try the magnetic method; if the magnetic method is OK, stop (payoff = +120); if the magnetic method is bad, use the mechanical method (payoff = 0). Awarded +----------[] | 0.5 | | | Mag OK Prepare | | +----------||............+120 +----------() | Try mag | 0.7 | | +----------() | | | Mag bad Use mech | | +----------[]----------|| 0 | | 0.3 [] | | No award +----------||.................................... -50 0.5 Value Prob. ----- ----- +120 .35 0 .15 -50 .50 ----- ----- (Expected Value = +17) 1.00 Strategy 4: Don't. Details: Don't prepare the proposal (payoff = 0). Chapter 2: Basic Decision Tree Concepts Page 15 [] | | Don't +----------||................................................ 0 Value Prob. ----- ----- (Expected Value = 0) 0 1.00 Rollback Method ================================================================== If you examine the four strategies in the previous section, you will see that strategy 2 has the payoff distribution with the highest expected value. If we have a method for determining certainty equivalents, e.g., expected values for a risk neutral decision maker, we don't need to examine every possible strategy explicitly. Instead, we can use the rollback method to determine the single best strategy. The rollback algorithm, sometimes called backward induction, starts at the terminal nodes of the tree and works backward to the initial decision node, determining the certainty equivalent for each node. At each event node, the certainty equivalent is determined using expected value if the decision maker is risk neutral; at each decision node, the certainty equivalent is set equal to the highest certainty equivalent on the immediate successor nodes. When the rollback method has finished assigning certainty equivalents to each node, we can work forward through the tree to identify the optimal strategy. TreePlan uses double lines instead of single lines to indicate branches that are part of the optimal strategy. For standard printers without the line-drawing characters, TreePlan uses equal signs for branches of the optimal strategy, =====, and dashes for other branches, -----. Partial-cash-flow values are not shown on a solved tree; the branch name, probability, and rollback certainty equivalent (CE) of the successor node are arranged as follows. TreePlan User's Manual Page 16 Rollback CE Decision branch: [D]========================... Branch name Prob. Rollback CE Event branch: (E)========================... Branch name DriveTek Solved Tree ================================================================== A condensed version of the solved decision tree for the DriveTek problem is shown below. +80 +----------||........................ +80 | Use mech | | 0.5 +150 | +==========||............+150 | +90 | Elec OK 0.5 +90 +==========() +==========[] Try elec | 0.5 +30 +30 | Awarded | +==========[]==========|| +30 | | Elec bad Use mech | | | | 0.7 +120 +20 | | +----------||............+120 +==========() | +84 | Mag OK | Prepare | +----------() | | Try mag | 0.3 0 0 | | +----------[]----------|| 0 | | Mag bad Use mech [] | | | 0.5 -50 | +==========||.................................... -50 | No award | | 0 +----------||................................................ 0 Don't The rollback method has identified strategy 2 as being optimal, and the rollback value on the initial branch of the optimal strategy is +20, the same as the expected value for the payoff distribution of strategy 2. Chapter 2: Basic Decision Tree Concepts Page 17 Sensitivity Analysis ================================================================== A decision maker may not be completely confident about his or her preliminary probability assignments and partial-cash-flow estimates. Sensitivity analysis can be used to determine whether or not the optimal strategy is sensitive to changes in the base- case probabilities and values. If a small change in one of the inputs indicates that a different strategy is optimal, then the decision maker should think very carefully about that probability or value before implementing the strategy. Sensitivity analysis may help the decision maker gain insight into which factors of a decision problem are critical. Probability ------------------------------------------------------------------ The decision maker in the DriveTek problem may not be confident about the probability that the electronic approach will be satisfactory. We refer to the initial probability assignment, 0.5, as the base case. TreePlan can be used to see how the optimal strategy depends on this factor. The following table was prepared using TreePlan, printed to a file, and inserted into this manual with a word processor. Branch name: Electronic success R = 0 Base probability: 0.50000 Number of event sets: 1 ------------------------------------------------------- Temporary Certainty equivalent Same strategy probability of optimal strategy as base case? ----------- -------------------- ------------- 0.00000 17,000.00 No 0.10000 17,000.00 No 0.20000 17,000.00 No 0.30000 17,000.00 No 0.40000 17,000.00 No 0.50000 20,000.00 Yes 0.60000 26,000.00 Yes 0.70000 32,000.00 Yes 0.80000 38,000.00 Yes 0.90000 44,000.00 Yes 1.00000 50,000.00 Yes The first line of the heading shows the branch name (which we have called "Elec OK" on the condensed tree diagrams); "R = 0" means that certainty equivalents are determined using expected value, TreePlan User's Manual Page 18 i.e., the decision maker is risk neutral. The second line shows the base-case probability, 0.5, and notes that there is only one set of events with this branch name in the decision tree. Each row in the body of the table shows the results of the rollback procedure using a different value for the probability of electronic success. For example, the sixth row shows the results for the base case when the probability is 0.5. The fifth row shows the results when the probability of electronic success is temporarily set equal to 0.4; in this case the expected value of the optimal strategy is +17K and the optimal strategy is not the same as the base-case strategy. To more precisely determine the probability where the strategy changes, you can use TreePlan to investigate probabilities from 0.4 to 0.5 in steps of 0.01, as shown below. Branch name: Electronic success R = 0 Base probability: 0.50000 Number of event sets: 1 ------------------------------------------------------- Temporary Certainty equivalent Same strategy probability of optimal strategy as base case? ----------- -------------------- ------------- 0.40000 17,000.00 No 0.41000 17,000.00 No 0.42000 17,000.00 No 0.43000 17,000.00 No 0.44000 17,000.00 No 0.45000 17,000.00 Yes 0.46000 17,600.00 Yes 0.47000 18,200.00 Yes 0.48000 18,800.00 Yes 0.49000 19,400.00 Yes 0.50000 20,000.00 Yes TreePlan does not indicate ties for the optimal strategy, but if you view the solved tree when the probability of electronic success is 0.45, you will see that the rollback certainty equivalents for "Try elec" and "Try mag" are both equal to +84K. This sensitivity analysis indicates that if the probability is below 0.45, the optimal strategy will change. Since 0.45 is very close to the base-case probability 0.5, the decision maker should consider the probability of electronic success as a critical factor in this decision problem. Chapter 2: Basic Decision Tree Concepts Page 19 Value ------------------------------------------------------------------ The decision maker in the DriveTek problem may want to investigate the effect of changes in the cost of the mechanical approach. A sensitivity analysis table using TreePlan is shown below. Branch name: Use mechanical method R = 0 Base value: -120,000.00 Number of branches: 3 ------------------------------------------------------- Certainty equivalent Same strategy Temporary value of optimal strategy as base case? ------------------ -------------------- ------------- -120,000.00 20,000.00 Yes -115,000.00 21,250.00 Yes -110,000.00 22,500.00 Yes -105,000.00 23,750.00 Yes -100,000.00 25,000.00 No -95,000.00 27,500.00 No -90,000.00 30,000.00 No The branch name was called "Use mech" on the condensed tree diagrams; this branch with partial-cash-flow -120K appears at three places in the decision tree. The fifth row in the body of the table shows the results of the rollback method when the cost of the mechanical method is 100K. If you use TreePlan to view the solved tree for this case, you will see that the rollback certainty equivalents for "Use mech" and "Try elec" are equal. Thus, if the cost of the mechanical approach is less than 100K (instead of the base-case 120K), the optimal strategy changes. TreePlan User's Manual Page 20 Chapter 3: Advanced Decision Tree Concepts ****************************************************************** This chapter discusses some advanced concepts concerning decision tree models, including risk attitude, utility functions, Bayesian revision of probabilities, and value of information. Risk Attitude ================================================================== In Chapter 2 we reviewed some of the basic concepts concerning decision tree models, including strategy, payoff distribution, and certainty equivalent. Recall that a certainty equivalent is a certain payoff value which is equivalent, for the decision maker, to a particular payoff distribution. If the decision maker can determine his or her certainty equivalent for the payoff distribution of each strategy in a decision tree model, then the optimal strategy is the one with the highest certainty equivalent. Unfortunately, it can be difficult to determine one's certainty equivalent for a complex payoff distribution. We can aid the decision maker by first determining his or her certainty equivalent for a simple payoff distribution and then using that information to infer the certainty equivalent for more complex payoff distributions. In Chapter 2 we discussed the circumstances when it is appropriate for a decision maker to use expected values as certainty equivalents. If the terminal values in a decision situation are extreme or if the situation is "one-of-a-kind" so that the outcome has major implications for the decision maker, an expected value analysis may not be appropriate. Such situations may require explicit consideration of risk. A utility function, U(X), can be used to represent a decision maker's attitude toward risk. The values or certainty equivalents, X, are plotted on the horizontal axis; utilities or expected utilities, U or U(X), are on the vertical axis. You can use the plot of the function by finding a value on the horizontal axis, scanning up to the plotted curve, and looking left to the vertical axis to determine the utility. A typical utility function might have the general shape shown below if you draw a smooth curve approximately through the points. Chapter 3: Advanced Decision Tree Concepts Page 21 U | ***** | **** Utility | *** | *** or | ** | ** Expected | * Utility | ** | * |* +------------------------- X Value or Certainty Equivalent Since more value means more utility, the inverse, X = X(U), of the utility function is well-defined, where X is the value or certainty equivalent corresponding to a utility or expected utility, U. On the plot of the utility function, you locate a utility on the vertical axis, scan right to the plotted curve, and look down to read the corresponding value. The concept of a payoff distribution or lottery is important for discussing utility functions. Recall that a payoff distribution or lottery is a set of payoffs, e.g., X1, X2, and X3, with corresponding probabilities, P1, P2, and P3. A fundamental property of a utility function is that the utility of the certainty equivalent (CE) of a lottery is equal to the expected utility of the lottery's payoffs, i.e, U(CE)= P1*U(X1) + P2*U(X2) + P3*U(X3). It follows that if you compute the expected utility (EU) of a lottery, EU = P1*U(X1) + P2*U(X2) + P3*U(X3), the certainty equivalent of the lottery can be determined using the inverse of the utility function, CE = X(EU). If a utility function has been determined, you can use this fundamental property to determine the certainty equivalent of any lottery. First, using a plot of the utility function, locate each payoff on the horizontal axis and determine the corresponding utility on the vertical axis. Second, compute the expected utility of the lottery by multiplying each utility by its TreePlan User's Manual Page 22 probability and summing the products. Third, locate the expected utility on the vertical axis and determine the corresponding certainty equivalent on the horizontal axis. Instead of using a plot of a utility function, TreePlan uses an exponential function to represent risk attitude. The general form of the exponential utility function is U(X)= A - B*Exp(-R*X). The parameters A and B determine vertical and horizontal scaling; R, the risk coefficient, determines the curvature and depends on the person's risk attitude. Exp is the standard exponential function, i.e., Exp(Z) represents the value e raised to the power of Z, where e is the base of the natural logarithms. After the parameters A, B, and R have been determined, the exponential utility function and its inverse can be used to determine the certainty equivalent for any lottery. TreePlan uses a simple lottery, called a risk attitude assessment lottery, to determine the decision maker's attitude toward risk. This lottery has equal probability of obtaining each of the two payoffs. It is good practice to use a better payoff at least as large as the highest payoff in the decision problem and a worse payoff as small or smaller than the lowest payoff; the payoffs should be far enough apart that the decision maker perceives a definite difference in the two outcomes. You specify three values for the fifty-fifty lottery: the Better payoff, the Worse payoff, and the Certainty Equivalent, as shown below. +---------------|| Better Certainty | 0.5 payoff Equivalent = () | +---------------|| Worse 0.5 payoff According to the fundamental property of utility functions, the three values are related as follows. U(CE)= 0.5*U(Better) + 0.5*U(Worse) If you use the general form for an exponential utility function with parameters A, B, and R, and if you simplify terms, it follows that R must satisfy the following equation. Exp(-R*CE) = 0.5*Exp(-R*Better) + 0.5*Exp(-R*Worse) Chapter 3: Advanced Decision Tree Concepts Page 23 Given the values for CE, Better, and Worse, you could use trial- and-error to find the value of R that exactly satisfies the equation. TreePlan uses an efficient search procedure (Newton- Raphson) to find R. After R is determined, if you want to plot a utility function so that U(High) = 1.0 and U(Low) = 0.0, you can use the following formulas to determine the scaling parameters A and B. A = Exp(-R*Low) / [Exp(-R*Low) - Exp(-R*High)] B = 1 / [Exp(-R*Low) - Exp(-R*High)] For the general form of an exponential utility function, the inverse function X(U) for finding the certainty equivalent CE corresponding to an expected utility EU is CE = (-1/R)*Log[(-1/B)*(EU-A)], where Log(X) represents the natural logarithm of X. For example, in the DriveTek problem, the maximum and minimum terminal values are $150,000 and $-50,000, respectively. After careful consideration of a fifty-fifty lottery with payoffs of $150,000 and $-50,000, you might decide that you are indifferent between the lottery and $30,000 for certain, as shown below. Certainty Better payoff Equivalent +---------------|| 150,000 | 0.5 30,000 = () | Worse payoff +---------------|| -50,000 0.5 For an exponential utility function to be consistent with this assessment lottery, R must satisfy Exp(-30000*R) = 0.5*Exp(-150000*R) + 0.5*Exp(+50000*R). In this example, TreePlan determines that the equation is satisfied if R is 0.0000041108. If you want to plot a utility function with U(-50000) = 0.0 and U(150000) = 1.0, you can determine the scaling parameters A and B by substituting the values of R = 0.0000041108, Low = -50000, and High = 150000 into the formulas. The resulting values are A = TreePlan User's Manual Page 24 1.784062 and B = 1.4525967, so the appropriate utility function is U(X) = 1.784062 - 1.4525967*Exp(-0.0000041108*X), and the inverse, after some simplification, is CE = -243261.65*Log(1.2281881 - 0.6884223*EU). This utility function and its inverse can be used to determine certainty equivalents in the rollback method. For example, when the DriveTek tree is solved, the event node after the decision branch "Try magnetic method" corresponds to the following lottery. Magnetic success +------------------|| 120,000 | 0.7 () | Magnetic failure +------------------|| 0 0.3 Using the general form of the utility function, the utilities of the payoffs are U(120000)= 1.784062 - 1.4525967*Exp(-0.0000041108*120000) = 0.8970912 and U(0)= 1.784062 - 1.4525967*Exp(-0.0000041108*0) = 0.3314653, and the expected utility of the lottery is EU = 0.7*U(120000) + 0.3*U(0) = 0.7*0.8970912 + 0.3*0.3314653 = 0.6279638 + 0.0994396 = 0.7274034. Using the inverse of the general form of the utility function, we obtain the certainty equivalent of the lottery as follows. CE = -243261.65*Log(1.2281881 - 0.6884223*0.7274034) = -243261.65*Log(0.7274274) = (-243261.65)*(-0.3182411) = 77416 When the DriveTek tree is solved, the rollback method assigns the value $77,416 as the certainty equivalent for the event node after the decision branch "Try magnetic method". TreePlan prints the Chapter 3: Advanced Decision Tree Concepts Page 25 relevant portion of the solved tree as shown below. [] 0.700 +120,000 | +------------------------|| | +77,416 | Magnetic success +------------------------() Try magnetic method | 0.300 0 +------------------------[] Magnetic failure The $77,416 certainty equivalent in this example is consistent with the risk attitude expressed in the assessment lottery, a $30,000 certainty equivalent for a fifty-fifty lottery with payoffs of $150,000 and $-50,000. In both the assessment lottery and the lottery in this example, the certainty equivalent is less than the expected value; this risk attitude is called risk aversion. The table below shows the three categories for risk attitude, the relationship between the certainty equivalent (CE) and expected value (ExpVal) of lotteries, and the sign of the risk coefficient in an exponential utility function. Risk attitude Lottery Risk coefficient, R --------------- ----------- ------------------- risk averse CE < ExpVal positive (R > 0) risk neutral CE = ExpVal zero (R = 0) risk preferring CE > ExpVal negative (R < 0) When TreePlan evaluate certainty equivalents in the rollback method, it ignores the scaling parameters and uses a simplified form of the exponential utility function and its inverse. The built-in functions are U(X) = Exp(-R*X) and X(U) = -Log(U)/R, which yield the same results as the general form. In the risk neutral situation (labeled R = 0), expected values are used instead of utilities for computations. An exponential utility function has a property called "constant risk aversion." If a fixed amount is added to each payoff of a lottery, the certainty equivalent increases by the same amount. (A linear utility function, corresponding to risk neutral attitude and equivalent to using expected values, also has this property.) In the assessment lottery of the example, the decision maker has a $30,000 certainty equivalent for a fifty-fifty lottery with payoffs of $150,000 and $-50,000. If the decision maker's risk attitude over a wide range of payoffs can be described by an exponential utility function, i.e., if the decision maker exhibits constant risk aversion, the decision maker necessarily has a TreePlan User's Manual Page 26 $25,000 certainty equivalent for a fifty-fifty lottery with payoffs of $145,000 and $-55,000 (where $-5,000 has been added to the payoffs of the assessment lottery). Similarly, he or she has a $87,416 certainty equivalent for a lottery with a 0.7 chance of $130,000 and a 0.3 chance of $10,000 (where $10,000 has been added to the payoffs associated with "Try magnetic method"). For additional information about constant risk aversion and techniques for assessing a utility function, please consult the Holloway text (1979). An alternative way to specify the main parameter of an exponential utility function is to use the "risk tolerance," which is the reciprocal of TreePlan's risk coefficient. For additional information concerning the use of risk tolerance to characterize attitude toward risk, please consult the McNamee text (1987). Sensitivity Analysis ================================================================== Just as a decision maker may not be completely confident about preliminary probability assignments and partial-cash-flow estimates, he or she may not be sure about attitude toward risk. You can use TreePlan to perform sensitivity analysis to determine whether or not the optimal strategy is sensitive to changes in risk attitude. You can express the changes by specifying various certainty equivalents for the assessment lottery (keeping the lottery payoffs constant) or by specifying various values for the risk coefficient directly. If a small change in risk attitude indicates that a different strategy is optimal, then the decision maker should think very carefully about risk attitude before implementing the strategy. Lottery Certainty Equivalent ------------------------------------------------------------------ The decision maker in the DriveTek problem may not be confident that expected values should be used for certainty equivalents. We could refer to the risk neutral situation as the base case, using a $50,000 certainty equivalent for a fifty-fifty assessment lottery with payoffs of $150,000 and $-50,000. You can use TreePlan to see how the optimal strategy depends on the certainty equivalent of the assessment lottery. The following table was prepared using TreePlan, printed to a file, condensed with a word processor, and inserted into this manual. Chapter 3: Advanced Decision Tree Concepts Page 27 Base lottery CE: 50,000.00 Better payoff: 150,000.00 Base risk coefficient: 0 Worse payoff: -50,000.00 ---------------------------------------------------------------- Temporary lottery Temporary Cert. equiv. Same strategy cert. equiv. risk coeff. of opt. strat. as base case? ----------------- -------------- -------------- ------------- 50,000.00 + 0.0000000000 20,000.00 Yes 45,000.00 + 0.0000010017 16,709.62 Yes 40,000.00 + 0.0000020135 13,525.43 Yes 35,000.00 + 0.0000030460 10,428.80 Yes 30,000.00 + 0.0000041108 7,402.04 Yes 25,000.00 + 0.0000052213 4,428.02 Yes 20,000.00 + 0.0000063933 1,865.95 No 15,000.00 + 0.0000076463 0.00 No 10,000.00 + 0.0000090054 0.00 No 5,000.00 + 0.0000105038 0.00 No 0.00 + 0.0000121876 0.00 No On the upper-right side of the heading, the first and second lines show the payoffs of the risk attitude assessment lottery, which applies to all cases shown in the table. The upper-left side of the heading shows the certainty equivalent of the assessment lottery that specifies the base case for the table; the heading also shows the associated risk coefficient for the base case. Each row in the body of the table shows the results of the rollback procedure using a different value for the certainty equivalent of the assessment lottery. For example, the first row shows the results for the base case when the certainty equivalent of the assessment lottery is $50,000. The fifth row shows the results when the lottery's certainty equivalent is $30,000, which is the situation described in the Risk Attitude section of this chapter. For both cases, the strategy "Prepare; Try elec" is optimal. The seventh row shows the results when the certainty equivalent is $20,000; in this case the certainty equivalent of the optimal strategy is $1,865.95, and the optimal strategy is not the same as the base-case strategy. Additional sensitivity analyses (not shown) reveal that the strategy "Prepare; Use mech" is optimal if the decision maker's certainty equivalent of the assessment lottery is less than $23,089. Further sensitivity analyses (not shown) indicate that if the decision maker's certainty equivalent of the assessment lottery is less than $16,087, the strategy "Don't prepare proposal" is optimal. To summarize, the strategy "Prepare; Try electronic method" is TreePlan User's Manual Page 28 optimal for a risk neutral decision maker, "Prepare; Use mechanical method" is optimal for a decision maker who is somewhat risk averse, and "Don't prepare proposal" is optimal for a very risk averse decision maker. Risk Coefficient ------------------------------------------------------------------ You can also use TreePlan to perform sensitivity analysis to determine whether or not the optimal strategy is sensitive to changes in risk attitude by specifying various values for the risk coefficient directly. A condensed sensitivity analysis table using TreePlan is shown below. Base lottery CE: 50,000.00 Better payoff: 150,000.00 Base risk coefficient: 0 Worse payoff: -50,000.00 ---------------------------------------------------------------- Temporary lottery Temporary Cert. equiv. Same strategy cert. equiv. risk coeff. of opt. strat. as base case? ----------------- -------------- -------------- ------------- 50,000.00 + 0.0000000000 20,000.00 Yes 45,008.31 + 0.0000010000 16,714.99 Yes 40,065.96 + 0.0000020000 13,566.83 Yes 35,219.74 + 0.0000030000 10,563.30 Yes 30,511.63 + 0.0000040000 7,709.05 Yes 25,977.10 + 0.0000050000 5,005.86 Yes 21,644.12 + 0.0000060000 2,633.85 No 17,532.82 + 0.0000070000 696.22 No 13,655.80 + 0.0000080000 0.00 No 10,018.84 + 0.0000090000 0.00 No 6,621.92 + 0.0000100000 0.00 No The interpretation of the heading and cases in this table is the same as that in the previous table. The base strategy is "Prepare proposal; Try electronic method," corresponding to a zero risk coefficient. Additional sensitivity analyses (not shown) reveal that the strategy "Prepare proposal; Use mechanical method" is optimal if the decision maker's risk coefficient is less than 0.0000056610, and the strategy "Don't prepare proposal" is optimal if the risk coefficient is less than 0.0000073658. Chapter 3: Advanced Decision Tree Concepts Page 29 Expected Value of Perfect Information ================================================================== Although we seldom have the opportunity to obtain perfect information, the expected value of perfect information is a useful measure because it is the upper limit on the value of any additional information in a decision problem. We can determine this measure by comparing the expected value of the optimal strategy when a perfect prediction is available with the expected value without the perfect information; the difference is the value of the information itself. The following definitions and abbreviations are useful for discussing this topic. EVUU = Expected Value Under Uncertainty, the expected value of the optimal strategy without perfect information EVPP = Expected Value with Perfect Prediction, the expected value of the optimal strategy when perfect information is available EVPI = Expected Value of Perfect Information, the difference between EVPP and EVUU EVPI = EVPP - EVUU The following problem, adapted from an example in the Bierman text (1991), will be used to illustrate the basic concepts of value of information and Bayesian revision of probabilities. Valley Problem (A) ================================================================== Valley Products, Inc., has developed a new product and must decide whether to spend $3 million to introduce it. The market potential is uncertain, and Valley has characterized the possible sales as either high or low. If the product is a success, sales will be high, with net $7 million (sales revenues minus all costs except initial introduction); if the product is a flop, sales will be low, with net $1 million. Valley thinks there is a 30% chance of high sales and a 70% chance of low sales. A condensed version of the unsolved decision tree model for the Valley problem (sample tree data file VALLEY_A.TRE), is shown below. Partial-cash-flows and terminal values are in millions of dollars. (The diagram was prepared using TreePlan, printed to a file, and condensed with a word processor.) TreePlan User's Manual Page 30 High sales +------------|| +4.00 Introduce | +7.00 0.30 +------------() | -3.00 | Low sales | +------------|| -2.00 [] +1.00 0.70 | | Don't +------------||.............. 0 0.00 A condensed version of the solved decision tree (rollback certainty equivalents at the left of each node) for the Valley problem is shown below. 0.30 +4.00 +------------|| +4.00 -0.20 | High sales +------------() | Introduce | 0.70 -2.00 | +------------|| -2.00 [] Low sales | | 0.00 +============||.............. 0 Don't The expected value of the strategy "Introduce" is $-0.2 million, and the optimal strategy is "Don't introduce" with zero expected value. In the Valley problem, the expected value under uncertainty (EVUU) is zero. We will compare this expected value (obtained with no additional information) with expected values obtained with perfect and imperfect predictions. Valley EVPI Tree ================================================================== To determine the value of information, TreePlan constructs an expanded EVPI tree. A condensed version of the unsolved EVPI tree, showing partial-cash-flow values below each branch, is shown below. Chapter 3: Advanced Decision Tree Concepts Page 31 High sales +------------||.............. +4.00 Introduce | +7.00 0.30 +------------() | -3.00 | Low sales No info | +------------||.............. -2.00 +------------[] +1.00 0.70 | 0.00 | | | Don't | +------------||............................ 0 | 0.00 | | High sales | +------------|| +4.00 | Introduce | +7.00 1.00 [] +------------() | | -3.00 | Low sales | "High" | +------------|| -2.00 | +------------[] +1.00 0.00 | | 0.00 0.30 | | | | Don't | | +------------||.............. 0 | | 0.00 | Perf. info | +------------() High sales 0.00 | +------------|| +4.00 | Introduce | +7.00 0.00 | +------------() | | -3.00 | Low sales | "Low" | +------------|| -2.00 +------------[] +1.00 1.00 0.00 0.70 | | Don't +------------||.............. 0 0.00 The top section of the tree following the decision branch "No info" shows the original Valley problem when no additional information is available. The major portion of the tree following the decision branch "Perf. info" shows the decision and event branches associated with using a perfect prediction. Since Valley thinks there is a 30% chance of high sales, the probability that the perfect predictor will say "High sales" is 0.3. After a prediction of "High" or "Low" is received, Valley must decide whether to introduce the product. If the perfect prediction was "High," the probability of high sales TreePlan User's Manual Page 32 is 1.0; if the prediction was "Low," the chance of high sales is zero. A condensed version of the solved EVPI tree for the Valley problem is shown below. Rollback certainty equivalents (expected values) are shown on the left side of each node. 0.30 +4.00 +------------||.............. +4.00 -0.20 | High sales +------------() | Introduce | 0.70 -2.00 0.00 | +------------||.............. -2.00 +------------[] Low sales | No info | | | 0.00 | +------------||............................ 0 | Don't | | 1.00 +4.00 | +============|| +4.00 | +4.00 | High sales [] +============() | | Introduce | 0.00 -2.00 | 0.30 +4.00 | +------------|| -2.00 | +============[] Low sales | | "High" | | | | 0.00 | | +------------||.............. 0 | | Don't | +1.20 | +============() 0.00 +4.00 Perf. info | +------------|| +4.00 | -2.00 | High sales | +------------() | | Introduce | 1.00 -2.00 | 0.70 0.00 | +------------|| -2.00 +============[] Low sales "Low" | | 0.00 +============||.............. 0 Don't In the Valley EVPI trees, we have assumed that the perfect prediction is free. The optimal strategy is to obtain the perfect information; if the prediction is "High," introduce the product; if the prediction is "Low," don't introduce. The expected value of the optimal strategy using the perfect prediction is $1.2 Chapter 3: Advanced Decision Tree Concepts Page 33 million. The expected value of perfect information is determined as follows. EVPI = EVPP - EVUU = $1.2 million - $0 = $1.2 million If the cost of the perfect prediction is $1.2 million (partial- cash-flow -1.20 on the branch "Perf. info"), all terminal values in that portion of the tree will be decreased by 1.20. When the tree if solved, all rollback certainty equivalents will also be decreased by 1.20, and the expected value associated with using the perfect prediction will be 0. Therefore, if the cost is $1.2 million, Valley will be indifferent between using the perfect prediction and acting without any additional information. Thus, the maximum that Valley should be willing to pay for information in this problem is $1.2 million. When you use TreePlan to determine EVPI, TreePlan automatically constructs an expanded tree, solves it using the rollback method with expected values, computes EVPI as the difference between the expected value of the best strategy with no additional information (your original problem) and the expected value of the best strategy using a perfect prediction, and displays EVPI in the lower left corner of the screen. Bayesian Revision of Probabilities ================================================================== Decision problems involving the option to gather information usually have the following general structure. +-------- +-------- +-------- +-------- []-------- ()-------- []-------- ()-------- +-------- +-------- +-------- +-------- Info Info Main Main Decision Event Decision Event For example, in the Valley EVPI tree, the info decision is whether to act without additional information or use the perfect prediction; the info event is a prediction of "High" or "Low;" the main decision is whether to introduce the product or not; the main event is actual high sales or low sales. The EVPI tree TreePlan User's Manual Page 34 describes a hypothetical situation; realistic decision problems involve imperfect predictions where the information-gathering options have a similar structure. To analyze a decision problem of this kind, the probabilities required for the decision tree may not be available directly. We may have to perform Bayesian revision of probabilities before using the rollback method to solve the tree. For Bayesian analysis, TreePlan uses the terms shown in the following table. Probability Bayesian Name General Type -------------- ------------- ---------------- P(Main) Prior Simple, Marginal P(Info | Main) Likelihood Conditional P(Main & Info) Joint P(Info) Preposterior Simple, Marginal P(Main | Info) Posterior Conditional "Main" refers to the set of events that directly affect the payoffs, i.e., the major uncertainties that we would like to predict. In Bayesian terms, P(Main) is a "Prior" probability, i.e., the probability of a main event prior to obtaining any additional information. In general terms, P(Main) may be called a simple, marginal, or unconditional probability. In Valley's original problem, P(High) = 0.3 and P(Low) = 0.7 are examples of P(Main). The original problem, before considering any information-gathering options, is sometimes referred to as the "prior problem." "Info" refers to the set of events that are the results of the information-gathering effort. In Bayesian terms, P(Main | Info) is a "Posterior" probability, i.e., the probability of a main event conditional on a result of the information-gathering activity. In general terms, P(Main | Info) is a conditional probability. In Valley's EVPI problem, P(High | "High") = 1.0 is an example of a posterior probability. Bayesian revision involves using the prior probabilities and other probabilities expressing the reliability of the information- gathering activity to obtain the posterior probabilities that are needed for the decision tree. In a sense, the prior probabilities are "revised," after receiving information, to become posterior probabilities. In Bayesian terms, P(Info | Main) is a "Likelihood" probability, i.e., the probability of a info event conditional on a main event. In general terms, P(Info | Main) is a conditional probability. The likelihoods are a way of showing the relationship between the Chapter 3: Advanced Decision Tree Concepts Page 35 info events and the main events, thereby measuring the reliability or accuracy of the information. In other words, likelihoods are a way of expressing how well the info events predict the main events. Bayesian revision uses priors and likelihoods as inputs and produces posteriors as one of the outputs. Priors measure our original state of information; likelihoods measure the accuracy of the information; posteriors measure our state of information after receiving the information. In Bayesian terms, P(Info) is a "Preposterior" probability, i.e., the probability of an information-gathering result. In general terms, P(Info) may be called a simple, marginal, or unconditional probability. Preposteriors are another output of Bayesian revision. Preposteriors appear on the decision tree and precede the posterior probabilities, thus the name. A fifth kind of probability involved in Bayesian revision is "Joint" probability, P(Main & Info). Joints are an intermediate step in the revision process; they do not appear on the decision tree. Using Bayesian terms, the following diagram shows how the probabilities are related. Priors --------> +--------+ ---> Preposteriors ---> +------------+ | Joints | | Posteriors | Likelihoods ---> +--------+ ----------------------> +------------+ Joints are calculated from Likelihoods and Priors. Preposteriors are calculated from Joints. Posteriors are calculated from Joints and Preposteriors. Using probability notation, the relationships and formulas are shown below. TreePlan User's Manual Page 36 P(Main) -------> +--------------+ --> P(Info) --> +--------------+ | P(Info&Main) | | P(Main|Info) | P(Info|Main) --> +--------------+ --------------> +--------------+ P(Info & Main) = P(Info | Main) * P(Main) P(Info) = Sum of P(Info & Main) P(Main | Info) = P(Info & Main) / P(Info) The calculations can be organized by substituting into the formulas, setting up a table, or drawing probability trees. Also, you could combine the three formulas into a single formula (not shown here), called "Bayes' rule," where the posteriors P(Main|Info) are expressed as a function of the priors P(Main) and likelihoods P(Info|Main). TreePlan uses probability trees for displaying the inputs, intermediate results, and outputs of Bayesian revision. If you do the computations using probability trees, you can divide the work into eight steps. (1) Construct a probability tree for the inputs. First draw branches for the set of Main events. After each Main event branch, draw a set of branches for the Info events. Label each branch. (2) Write the prior probabilities on the input tree (on the set of branches on the left). (3) Write the likelihoods on the input tree (on each set of branches on the right). (4) Multiply each prior times the following likelihood, and write the resulting joint probability near the far right endpoints of the probability tree. The "Inputs and Joints" probability tree, sometimes called "Nature's tree," resembles the following diagram. +--- P(Info | Main) --- P(Info & Main) +---- P(Main) -----() | +---------------------- () | Priors +---------------------- +------------------() Likelihoods +---------------------- Joints Chapter 3: Advanced Decision Tree Concepts Page 37 (5) Construct a probability tree for the outputs. First draw branches for the set of Info events. After each Info event branch, draw a set of branches for the Main events. Label each branch. (6) Copy each joint probability from the input tree to the appropriate endpoint of the output tree; the order will be different. (7) Compute preposteriors by summing the joint probabilities following each Info event branch. Write a preposterior on each Info event branch. (8) Compute each posterior by dividing a joint probability by the preposterior that precedes it on the tree. Write a posterior on each Main event branch. The "Outputs and Joints" probability tree resembles the following diagram. +--- P(Main | Info) --- P(Info & Main) +---- P(Info) -----() | +---------------------- () | Preposteriors +---------------------- +------------------() Posteriors +--------------------- Joints (In some decision problems, joint probabilities may be used instead of likelihoods to express the accuracy of the information. If the joints and priors are consistent, only steps 5 through 8 of the Bayesian revision process are required. If not, the joints can be used to determine likelihoods, which are combined with priors using Bayesian revision using all eight steps.) Valley Problem (B) ================================================================== Valley has the opportunity to conduct a market survey before deciding whether to introduce the product. The survey will cost $0.2 million, and there are three possible results: a "Success" prediction for the new product, indicating high sales; a "Failure" prediction, indicating low sales; or an "Uncertain" prediction (an inconclusive survey result). However, the survey TreePlan User's Manual Page 38 is imperfect. Even if the survey predicts success, there is a chance that sales will actually be low; if it predicts failure, sales may be high. Based on past experience and modified by personal judgment, Valley has quantified the survey's reliability by assigning the following likelihoods. Likelihoods, P(Info | Main) (Main event) Actual sales (Info event) ------------ Survey prediction High Low ----------------------- ---- --- Survey predicts Success 0.4 0.1 Survey is Uncertain 0.4 0.5 Survey predicts Failure 0.2 0.4 --- --- 1.0 1.0 The table of likelihoods shows two probability distributions, one conditional on High sales and one conditional on Low sales. For example, the 0.2 in the lower left corner of the table is the probability that the survey predicts Failure, conditional on actual sales being High; the 0.1 in the upper right corner is the probability that the survey predicts Success, conditional on actual sales being Low. In an accurate survey, each of these likelihoods is close to zero. Although the survey is imperfect, its cost is less than $1.2 million (EVPI), so further analysis is justified. The following tree diagram, obtained using TreePlan's Zoom File option, shows the structure of the decision problem. Chapter 3: Advanced Decision Tree Concepts Page 39 +-HighSales-|| +-Introduce-() +-No info---[] +-Low sales-|| | | | +-Don't-----|| | | +-HighSales-|| | +-Introduce-() [] +-Success---[] +-Low sales-|| | | | | | +-Don't-----|| | | | | +-HighSales-|| | | +-Introduce-() +-Survey----()-Uncertain-[] +-Low sales-|| | | | +-Don't-----|| | | +-HighSales-|| | +-Introduce-() +-Failure---[] +-Low sales-|| | +-Don't-----|| The tree has the general structure of an information-gathering decision problem. Info Decision No info or Survey Info Event Success, Uncertain, or Failure Main Decision Introduce or Don't Main Event High sales or Low sales Starting with the sample tree data file VALLEY_A.TRE, you can develop the decision tree shown above by using TreePlan's EVPI Retain option and renaming "High sales" to "Success" and "Low sales" to "Uncertain." Then use the Node Add option to obtain the "Failure" branch and the Node Copy option to finish the tree. The result is sample tree data file VALLEY_B.TRE. Using TreePlan, you use the Bayes Link Main-event option to select the Main event node (preceding High sales and Low sales), and you use the Bayes Link Info-event option to select the Info event node (preceding Success, Uncertain, and Failure). Then you use the Bayes Input option to enter the priors (0.3 and 0.7) and likelihoods (from the table above) on a probability tree diagram. The following diagram shows the inputs and the joint probabilities computed by TreePlan. TreePlan User's Manual Page 40 Prior Likelihood Joint ================================================================== Success 0.40000 +-------------------------- 0.12000 | High sales 0.30000 | Uncertain 0.40000 +--------------------------()-------------------------- 0.12000 | | | | Failure 0.20000 | +-------------------------- 0.06000 () | Success 0.10000 | +-------------------------- 0.07000 | | | Low sales 0.70000 | Uncertain 0.50000 +--------------------------()-------------------------- 0.35000 | | Failure 0.40000 +-------------------------- 0.28000 TreePlan also computes the preposteriors and posteriors. You use the Bayes Screen option to view the inputs and outputs of Bayesian revision, shown above and below. These diagrams were obtained using the Bayes File option. Preposterior Posterior Joint ================================================================== High sales 0.63158 Success 0.19000 +-------------------------- 0.12000 +--------------------------() | | Low sales 0.36842 | +-------------------------- 0.07000 | | High sales 0.25532 | Uncertain 0.47000 +-------------------------- 0.12000 ()--------------------------() | | Low sales 0.74468 | +-------------------------- 0.35000 | | High sales 0.17647 | Failure 0.34000 +-------------------------- 0.06000 +--------------------------() | Low sales 0.82353 +-------------------------- 0.28000 Chapter 3: Advanced Decision Tree Concepts Page 41 After Bayesian revision, you use the Bayes Transfer option to transfer the probabilities to the decision tree. Then use the Solve option to determine the optimal strategy. The Valley Solved Tree diagram, shown below, is a condensed version of the solved tree, showing rollback certainty equivalents to the left of each node. In this diagram, the survey partial- cash-flow (-0.20, in millions of dollars) is included in appropriate terminal values. The optimal strategy is: Take the survey; if the survey predicts Success, Introduce the product; if the survey prediction is Uncertain or Failure, Don't introduce. The net gain of the Survey strategy over the No info strategy is 0.14 - 0.00 = 0.14, or $140,000. TreePlan User's Manual Page 42 Valley Solved Tree ================================================================== 0.30 +4.00 +------------||.............. +4.00 -0.20 | High sales +------------() | Introduce | 0.70 -2.00 0.00 | +------------||.............. -2.00 +------------[] Low sales | No info | | | 0.00 | +------------||............................ 0 | Don't | | 0.63 +3.80 | +============|| +3.80 | +1.59 | High sales | +============() | | Introduce | 0.37 -2.20 | 0.19 +1.59 | +============|| -2.20 [] +============[] Low sales | | Success | | | | -0.20 | | +------------||.............. -0.20 | | Don't | | | | 0.26 +3.80 | | +------------|| +3.80 | | -0.67 | High sales | | +------------() | | | Introduce | 0.74 -2.20 | +0.14 | 0.47 -0.20 | +------------|| -2.20 +============()============[] Low sales * Survey | Uncertain | | | -0.20 | +============||.............. -0.20 | Don't | | 0.18 +3.80 | +------------|| +3.80 | -1.14 | High sales | +------------() | | Introduce | 0.82 -2.20 | 0.34 -0.20 | +------------|| -2.20 +============[] Low sales Failure | | -0.20 +============||.............. -0.20 Don't Chapter 3: Advanced Decision Tree Concepts Page 43 Expected Value of Sample Information ================================================================== An alternative approach borrows some terminology from statistical decision theory, where we loosely interpret "sampling" as any information-gathering activity that yields an imperfect prediction. The following definitions and abbreviations are useful for this discussion. EVUU = Expected Value Under Uncertainty, the expected value of the optimal strategy without perfect information EVSP = Expected Value with Sample Prediction, the expected value of the optimal strategy when imperfect information is available, assuming it's free EVSI = Expected Value of Sample Information, the difference between EVSP and EVUU ENGS = Expected Net Gain from Sampling, the difference between EVSI and the cost of sampling (the cost of the imperfect information) EVSI = EVSP - EVUU ENGS = EVSI - Cost EVSI may be interpreted as the maximum amount a risk-neutral decision maker is willing to pay for the imperfect information, and ENGS is the net gain from using the imperfect information over acting without additional information. If ENGS is negative, a risk-neutral decision maker should not use the information- gathering activity; the cost of the information exceeds its value. The Valley EVSI Tree diagram below shows the results for this alternative approach, where the survey is assumed to be free. The Expected Value with Sample Prediction is the rollback value for the Survey($0) option; EVSP = 0.34, in millions of dollars. In this example, we have the following results. EVSI = 0.34 - 0.00 = 0.34 ENGS = 0.34 - 0.20 = 0.14 A risk-neutral decision maker is willing to pay a maximum of TreePlan User's Manual Page 44 $340,000 for the survey. The survey cost is $200,000, so the net gain from using the survey is $140,000. Chapter 3: Advanced Decision Tree Concepts Page 45 Valley EVSI Tree ================================================================== 0.30 +4.00 +------------||.............. +4.00 -0.20 | High sales +------------() | Introduce | 0.70 -2.00 0.00 | +------------||.............. -2.00 +------------[] Low sales | No info | | | 0.00 | +------------||............................ 0 | Don't | | 0.63 +4.00 | +============|| +4.00 | +1.79 | High sales | +============() | | Introduce | 0.37 -2.00 | 0.19 +1.79 | +============|| -2.00 [] +============[] Low sales | | Success | | | | 0.00 | | +------------||.............. 0 | | Don't | | | | 0.26 +4.00 | | +------------|| +4.00 | | -0.47 | High sales | | +------------() | | | Introduce | 0.74 -2.00 | +0.34 | 0.47 0.00 | +------------|| -2.00 +============()============[] Low sales Survey($0) | Uncertain | | | 0.00 | +============||.............. 0 | Don't | | 0.18 +4.00 | +------------|| +4.00 | -0.94 | High sales | +------------() | | Introduce | 0.82 -2.00 | 0.34 0.00 | +------------|| -2.00 +============[] Low sales Failure | | 0.00 +============||.............. 0 Don't TreePlan User's Manual Page 46 Risk Attitude and Value of Information ================================================================== So far we have discussed the value of perfect and imperfect information for a risk neutral decision maker. Determining the value of information for a decision maker who is not risk neutral can be more difficult. If a decision maker has an arbitrary utility function (not exponential) and we want to determine the value of information, we construct a decision tree showing both the original decision problem (with no additional information) and the information- gathering activity. After constructing the tree, complete with probabilities and partial-cash-flow values, we repeatedly use the rollback method to solve the tree using different information- gathering costs. Each time the tree is solved, we compute terminal values including the information-gathering cost, and we use the arbitrary utility function to determine certainty equivalents for the rollback method. We try to find an information-gathering cost so that the certainty equivalent of the optimal strategy of the original problem (with no additional information) equals the certainty equivalent of the optimal strategy for the information-gathering activity. When our search finds such a cost, we have determined the value of information, i.e., the maximum the decision maker should be willing to pay for the information. The unsolved Valley EVPI tree is an example of a tree for determining the value of perfect information, and the unsolved Valley EVSI tree could be used for determining the value of imperfect information. To search for the value of information, we would try various partial-cash-flow values for the "Use perfect prediction" branch or the "Survey" branch. Since TreePlan's built-in utility function is exponential, we would have to use hand calculations for the rollback method with the decision maker's arbitrary utility function. However, if the decision maker's utility function is exponential, TreePlan can be used to determine the value of information using the same approach we used for a risk neutral decision maker. We assume the information is free, solve the tree using the appropriate value for the risk coefficient R, and subtract the certainty equivalent of the no-information strategy from the certainty equivalent of the information-gathering strategy; the difference is the value of information. Chapter 3: Advanced Decision Tree Concepts Page 47 The simplified approach for this special case is appropriate due to the "constant risk aversion" property of exponential utility functions (discussed in the Risk Attitude section of this chapter): If a fixed amount is added to each payoff of a lottery, the certainty equivalent increases by the same amount. Likewise, if the value of information is subtracted from the values of the payoff distribution of the information-gathering strategy, the certainty equivalent will be reduced by the same amount, and the decision maker will be indifferent between the no-information strategy and the information-gathering strategy. TreePlan's EVPI option covers only the risk neutral situation. To determine the value of perfect information using an exponential utility function, follow these steps. (1) Use the Node options to construct the original problem (the prior problem). (2) Use the EVPI Retain option to construct the expanded tree. (3) Use the Solve Risk options to express risk attitude. (4) Use the Solve View option to determine certainty equivalents for the "No additional info" and "Use perfect prediction" strategies. (5) Subtract the certainty equivalents to obtain the value of perfect information. You can use a similar approach to determine the value of imperfect information using an exponential utility function. TreePlan User's Manual Page 48 Chapter 4: General Treeplan Features ****************************************************************** This chapter discusses some general TreePlan features, including the screen layout, modes, and tree navigation. It will be worthwhile to view an existing decision tree on TreePlan's screens while you read the following sections. Follow these steps. (1) Run TreePlan. (If necessary, refer to the section "Starting and stopping TreePlan" at the end of Chapter 1.) (2) Wait for the READY mode indicator to appear in the top right corner of the screen. (3) Type / F R to display a menu of tree files. (Pressing the slash key displays the main menu; F selects the File submenu; and R tells TreePlan you want to Retrieve a file.) (4) Use the arrow keys to highlight the file named DRIVETEK.TRE, and press the RETURN key to select it. If you make a mistake, press the Escape key several times until the READY mode indicator appears; then start over. Screen Layout ================================================================== This section explains TreePlan's screen layout and some of its associated terminology. In READY mode, the screen is divided into several sections as shown below. +------------------------------------------+ | Current field info |Mode| |------------------------------------------| | Previous | +------------ | | branches | | | | | ---- Tree display ----- | | | | | | | +------------ | |------------------------------------------| | Messages |TreePlan| +------------------------------------------+ Chapter 4: General TreePlan Features Page 49 Mode The mode indicator in the upper right corner shows TreePlan's current operating condition. For example, in the READY mode, you can view the entire tree by using the pointer keys; in the MENU mode, you can select options from the menu. The mode often changes when you begin an action, such as executing a command or making an entry. Tree display The main portion of the screen is the tree display which shows two stages of the decision tree in diagram form. Details are shown for all branches on the right side and for one center branch leading to that point. Only branch names are shown for the other center branches. Each stage can have a maximum of five branches. Field A field is a part of the tree that can store data. Each branch has a name field and a partial-cash-flow value field; event branches also have a probability field. The length of the name field is 22 characters, and the length of the probability field is 4 to 7 characters, depending on the format you select. The length of the partial-cash-flow value field, which also depends on the probability format you select, is 14 to 17 characters. A field is uniquely identified by its type and the branch names on the path from the start of the tree to its location. Field pointer The field pointer is a rectangular highlight that appears on one field of the tree display and identifies it as the current field. You can move the field pointer to any field in the tree; your next entry or procedure affects this field. For example, typing an entry or executing certain commands affects the current field. Current field info In READY mode, the top three lines show information about the current field. The first line shows the kind of field, either name, value, or probability, and its contents. If you start typing to replace the contents of a field, the characters you type are shown on the second line, directly below the current contents of the field. If you press F2 to edit a TreePlan User's Manual Page 50 field, the second line initially shows the current contents with the cursor after the last character. The third line uses carets to show the maximum length of the field. Previous branches This section shows a list of names for the branches that precede those currently shown on the tree display. Messages An error message is displayed on the bottom line of the screen when TreePlan has detected a problem. Press Enter or Escape to resume. Menus In MENU mode, the current field info in the top three lines is replaced by menu information. The first line shows the menu path, i.e., the menu and submenu options you have already chosen. The second line shows the current menu with the commands or options currently available for selection. The third line shows either a one-line description of the command currently highlighted on the menu or a list of options available from a submenu. TreePlan Modes ================================================================== The mode indicator in the upper right corner shows TreePlan's current operating condition. Explanations for TreePlan's eight modes are shown below. Mode Explanation READY TreePlan is ready to do something. Use pointer keys to move the highlighted field pointer; press F2 to edit a field, or simply start typing to replace contents of a field; press / for the main menu. This mode is used to view an unsolved tree and to change branch names, values, and probabilities. MENU Commands or options are displayed on the second line of the display. Use arrow keys to highlight an option and press Enter to select, or type the first letter of the option. Chapter 4: General TreePlan Features Page 51 INSERT While editing an entry, characters you type are inserted at the cursor position. OVERTYPE While editing an entry, characters you type overwrite the contents at the cursor position. Press the Insert key to switch between insert and overtype modes. POINT Use pointer keys to move the highlighted pointer; then press Enter to select. This mode is often used to select a field as part of a command. VIEW Use pointer keys to view; when finished, press Enter or Escape. This mode is often used to view the results of a command operation. WAIT TreePlan is busy performing a task. You may see this mode when TreePlan is accessing a disk or performing calculations. BAYES Use pointer keys to move the highlighted field pointer; press F2 to edit a field, or simply start typing to replace contents of a field; press Enter or Escape when finished. This mode is used only for inputting Bayes probabilities. ERROR TreePlan has detected a problem. An error message is shown on the bottom line of the display. Press Enter or Escape to resume. Tree Navigation In READY Mode ================================================================== This section of the manual explains how to view a tree in READY mode. The commands for building a tree are discussed in a later section. TreePlan uses the following seven special keys as "pointer keys" in READY, VIEW, and POINT modes: Page-Up, Page-Down, Home, Up- arrow, Down-arrow, Left-arrow, and Right-arrow. These keys are usually located on the right side of a computer keyboard It will be worthwhile to view the DriveTek decision tree on TreePlan's screens while you read this section. Follow the steps on the first page of this chapter to run TreePlan and retrieve the DriveTek file (/FR). If you're already viewing the tree, press the Home key to move to the initial branches. A condensed version of the display is shown below. TreePlan User's Manual Page 52 Awarded Previous Branches +---------------[D] ----------------- Prepare | +250,000 0.5 (None) [D]---------------(E) | -50,000 | No award | +---------------|T| +-Don't 0 0.5 To move the highlighted field pointer from field to field, use the four arrow keys: Right-arrow, Left-arrow, Up-arrow, and Down- arrow. You can think of the center and right sections of the tree display as being a window for viewing sections of the complete tree. If you continue to press the Right-arrow key or the Left-arrow key when the pointer is at the boundary of the window, the window shifts to show another section of the tree. For example, if you press the Home key and then press the Right- arrow key two or three times, the window moves to show the branches shown below. Use mech +---------------|T| | -120,000 Previous Branches | ----------------- Awarded | Try elec Prepare (E)---------------[D]---------------(E) | +250,000 0.5 | -50,000 | | +-No award | Try mag +---------------(E) -80,000 If you continue to press the Right-arrow key (starting from Home, press the Right-arrow key four times), you will see the following tree display. Chapter 4: General TreePlan Features Page 53 Previous Branches Terminal ----------------- Value Prepare Use mech Awarded [D]---------------|T| +80,000 | -120,000 | +-Try elec | +-Try mag If you press the Right-arrow key more than four times (starting from Home), you will hear a "bump" sound, indicating that you have reached an extremity of the tree. You cannot move the field pointer to the terminal value. The terminal value is the sum of the partial-cash-flow values on the three branches leading to the terminal node: Prepare, Awarded, and Use mech, with values -50,000, +250,000, and -120,000, whose sum is +80,000. You can use the Page-Up and Page-Down keys to change the branch shown in detail in the center section of the tree display. For example, if you start at Home and press the Right-arrow key four times so that you see the display shown above and then press the Page-Down key once, the display changes to show the details of the "Try elec" branch as shown below. Previous Branches +-Use mech Elec success ----------------- | +---------------|T| Prepare | Try elec | 0 0.5 Awarded [D]---------------(E) | -50,000 | Elec failure | +---------------[D] +-Try mag 0 0.5 The Page-Up and Page-Down keys affect the branches shown in the center section, even if the field pointer is in the right section of the display. If the field pointer is in the center section, you can also use the Up-arrow and Down-arrow keys to change the branch shown in detail in the center section. TreePlan User's Manual Page 54 Chapter 5: Building A Tree With TreePlan ****************************************************************** This chapter is a step-by-step tutorial for building the DriveTek tree with TreePlan. Please see Chapter 2 of this manual for the DriveTek problem and tree diagrams. The figures in this chapter are condensed versions of what you will see on the screen. In these figures, branch names are abbreviated. When you build the tree, you should use more descriptive branch names; for example, type "Prepare proposal" instead of "Prepare". If you make a mistake during this tutorial, you may want to save your work before trying to fix the mistake using the destructive Node options (Oops, Shorten, Terminal, Remove, and New); to save, press /FS in READY mode, and enter a file name other than "DriveTek." Also, you can always start over by using the Node New option. The six steps of the tutorial where you use the Node options are numbered. The tutorial does not cover the details of entering the branch names, partial-cash-flow values, and probabilities; simply use the pointer keys in READY mode to move the field pointer and begin typing or press F2 to edit. When you run TreePlan from the DOS prompt (or when you use the Node New option to erase a tree), the following display appears. Terminal Previous Branches Value ----------------- TreePlan (None) [D]---------------|T| 0 0 The field pointer will initially be on the name field "TreePlan;" change it to "Prepare proposal," and enter the partial-cash-flow value -50,000. The display should appear as follows. Terminal Previous Branches Value ----------------- Prepare... (None) [D]---------------|T| -50,000 -50,000 Chapter 5: Building A Tree With TreePlan Page 55 Step 1: Node Add Option ================================================================== In READY mode, press /NA; point to the initial decision node, and press Enter. The display should appear as follows. +-Prepare... Terminal Previous Branches | Value ----------------- | Decision 2 (None) [D]---------------|T| 0 0 Change the name field from "Decision 2" to "Don't prepare proposal," and press the Up-arrow key twice. The display should appear as follows. Terminal Previous Branches Value ----------------- Prepare... (None) [D]---------------|T| -50,000 | -50,000 | +-Don't... Step 2: Node Event Option ================================================================== In READY mode, press /NE2; point to the terminal node after "Prepare proposal," and press Enter. The display should appear as follows. Event 3 Previous Branches +---------------|T| ----------------- Prepare... | 0 0.0 (None) [D]---------------(E) | -50,000 | Event 4 | +---------------|T| +-Don't... 0 0.0 TreePlan User's Manual Page 56 Change "Event 3" to "Awarded contract," and enter the partial-cash-flow value +250,000. Also, enter the probability 0.5; TreePlan automatically changes the other probability to 0.5. Change "Event 4" to "Not awarded contract." The display should appear as follows. Awarded... Previous Branches +---------------|T| ----------------- Prepare... | +250,000 0.5 (None) [D]---------------(E) | -50,000 | Not... | +---------------|T| +-Don't... 0 0.5 Step 3: Node Decision Option ================================================================== In READY mode, press /ND3; point to the terminal node after "Awarded contract," and press Enter. The display should appear as follows. Decision 5 +---------------|T| | 0 Previous Branches | ----------------- Awarded... | Decision 6 Prepare (E)---------------[D]---------------|T| | +250,000 0.5 | 0 | | +-Not... | Decision 7 +---------------|T| 0 Change "Decision 5" to "Use mechanical method" with partial-cash- flow value -120,000, change "Decision 6" to "Try electronic method" with partial-cash-flow value -50,000, and change "Decision 7" to "Try magnetic method" with partial-cash-flow value -80,000. The display should appear as follows. Chapter 5: Building A Tree With TreePlan Page 57 Use mech... +---------------|T| | -120,000 Previous Branches | ----------------- Awarded... | Try elec... Prepare (E)---------------[D]---------------|T| | +250,000 0.5 | -50,000 | | +-Not... | Try mag... +---------------|T| -80,000 Step 4: Node Event Option ================================================================== In READY mode, press /NE2; point to the terminal node after "Try electronic method," and press Enter. The display should appear as follows. Previous Branches +-Use mech... Event 8 ----------------- | +---------------|T| Prepare... | Try elec... | 0 0.0 Awarded... [D]---------------(E) | -50,000 | Event 9 | +---------------|T| +-Try mag... 0 0.0 Change "Event 8" to "Electronic success" with probability 0.5, and change "Event 9" to "Electronic failure." The display should appear as follows. Previous Branches +-Use mech... Elec.success ----------------- | +---------------|T| Prepare... | Try elec... | 0 0.5 Awarded... [D]---------------(E) | -50,000 | Elec.failure | +---------------|T| +-Try mag... 0 0.5 TreePlan User's Manual Page 58 Step 5: Node Decision Option ================================================================== In READY mode, press /ND1; point to the terminal node after "Electronic failure," and press Enter. The display should appear as follows. Previous Branches ----------------- +-Elec.success Prepare... | Awarded... | Elec.failure Decision 10 Try elec... (E)---------------[D]---------------|T| 0 0.5 0 Change "Decision 10" to "Use mechanical method" with partial-cash- flow value -120,000. Be sure to type this name and value the same as the name and value on the identical branch that follows "Awarded contract." The display should appear as follows. Previous Branches ----------------- +-Elec.success Prepare... | Awarded... | Elec.failure Use mech... Try elec... (E)---------------[D]---------------|T| 0 0.5 -120,000 Before finishing the tree, press /ZS to view a small version of your tree so far, or press /PS, followed by Right-arrow twice, to view the large diagram. When finished, press Escape repeatedly to return to READY mode. The structure of the tree (nodes and branches) following "Try magnetic method" should be the same as the structure following "Try electronic method," but the branch names and probabilities are different. You could construct the remaining portion of the tree using Node options similar to steps 4 and 5 above. An alternative approach is to copy the structure as described below. Step 6: Node Copy Option ================================================================== In READY mode, press /NC. To identify the "Copy from" node, point to the event node after "Try electronic method," and press Enter. Then, to identify the "Copy to" node, point to the terminal node Chapter 5: Building A Tree With TreePlan Page 59 after "Try magnetic method," and press Enter. The display should appear as follows. +-Use mech... | Previous Branches +-Try elec... Elec.success ----------------- | +---------------|T| Prepare... | Try mag... | 0 0.5 Awarded... [D]---------------(E) -80,000 | Elec.failure +---------------[D] 0 0.5 On the event branches after "Try magnetic method," change "Electronic success" to "Magnetic success" with probability 0.7, and change "Electronic failure" to "Magnetic failure" with probability 0.3. The display should appear as follows. +-Use mech... | Previous Branches +-Try elec... Mag.success ----------------- | +---------------|T| Prepare... | Try mag... | 0 0.7 Awarded... [D]---------------(E) -80,000 | Mag.failure +---------------[D] 0 0.3 The DriveTek tree is complete. You can verify your work by using the Zoom Screen and Print Screen options. If you press /SV in READY mode and press Home, the branch "Prepare proposal" should be shown with double lines and certainty equivalent +20,000. If you press /TV in READY mode and point to the partial-cash-flow value -120,000 under one of the branches with name "Use mechanical method," the heading of the sensitivity analysis table should show that there are 3 such branches. If you save a copy of your tree, enter a file name other than DRIVETEK. Then you will always have the original DRIVETEK.TRE file referenced throughout this manual. (The branch names on the DRIVETEK.TRE sample tree data file may differ slightly from those on the tree you created.) TreePlan User's Manual Page 60 Chapter 6: TreePlan Menu Options ****************************************************************** This chapter discusses TreePlan's menu options. When you press the slash key in READY mode, the mode indicator in the top right corner of the screen changes to MENU and the main menu options are shown on the second line of the display. There are eleven options on the main menu: Node, Solve, Print, File, EVPI, Bayes, Table, Default, Zoom, Help, and Quit. When a menu is being displayed, you can select the desired option either by typing the first letter of the option or by using the arrow keys to position the menu pointer on the option and then pressing the Enter key. In addition, the Home and End keys move the menu pointer to the first and last options, respectively. The third line of the control panel either gives a brief description of the menu option that is currently highlighted by the menu pointer or shows the options of a submenu. For example, if the Solve option of the main menu is highlighted, the third line shows "Rollback to find the optimal strategy;" if the File option is highlighted, the third line shows "Retrieve, save, or erase a TreePlan file one disk." Node Menu Options ================================================================== The Node options are used to build a new decision tree model or modify an existing tree by specifying the structure of the tree. You can specify whether a node should be a decision node or an event node, and you can tell TreePlan how many branches should emanate from a node. After a Node option is completed, TreePlan returns to READY mode, and you can enter the appropriate name, value, and probability on each branch. The first five Node options (Decision, Event, Add, Copy, and Insert) increase the number of branches and nodes. Four other Node options (Shorten, Terminal, Remove, and New) decrease the number of branches and nodes; since these are destructive actions, TreePlan asks for confirmation before completing the operation. Chapter 6: TreePlan Menu Options Page 61 Node Decision Option ------------------------------------------------------------------ Use this option to change a terminal node to a decision node with one to five branches. In READY mode, press /ND and select the number of decision branches desired. The mode indicator changes to POINT, and the third line of the display shows "Point to the terminal node to be changed to a decision node." Use the pointer keys to move the node pointer to the appropriate terminal node, and press the Enter key to select it. The tree display shows the new decision node with the new decision branches, and TreePlan returns to READY mode. Each new branch initially has value 0 and an arbitrary name "Decision #," where the highest value of # indicates the total number of branches in the entire tree. Node Event Option ------------------------------------------------------------------ Use this option to change a terminal node to a event node with one to five branches. In READY mode, press /NE and select the number of event branches desired. The mode indicator changes to POINT, and the third line of the display shows "Point to the terminal node to be changed to a event node." Use the pointer keys to move the node pointer to the appropriate terminal node, and press the Enter key to select it. The tree display shows the new event node with the new event branches, and TreePlan returns to READY mode. Each new branch initially has value 0, probability 0, and an arbitrary name "Event #," where the highest value of # indicates the total number of branches in the entire tree. Node Add Option ------------------------------------------------------------------ To add one more branch to an existing decision or event node, press /NA in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to decision or event node where successor branch will be added." Use the pointer keys to move the node pointer to the appropriate node, and press the Enter key to select it. TreePlan adds one decision or event branch parallel to the existing branches and returns to READY mode. The new branch initially has value 0, probability 0 (if it is an event branch), and an arbitrary name indicating the total number of branches in the entire tree. TreePlan User's Manual Page 62 Node Copy Option ------------------------------------------------------------------ To copy a decision or event node with all its successors to a terminal node, press /NC in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to the decision or event node from which successors will be copied." Use the pointer keys to move the node pointer to the appropriate decision or event node, and press the Enter key to select it. As you move the node pointer, the second line of the display shows the kind of node and the number of successor branches. After you select the "from" node, the left portion of the second line of the display shows the kind of node you selected and the number of its successors, and the third line shows "Point to the terminal node to which node and successors will be copied." As you move the node pointer, the right portion of the second line of the display shows the kind of node and the number of successor branches. Use the pointer keys to move the node pointer to the appropriate terminal node, and press the Enter key to select it. When you select the "from" node and its successors, you are selecting a subtree. TreePlan can append the subtree to a terminal node, but not to a decision or event node. Also, TreePlan cannot append the subtree to one of the subtree's own terminal nodes. Node Insert Option ------------------------------------------------------------------ Use this option to insert a new node and branch in the middle of a decision tree. In READY mode, first press /NI, and then select either Decision or Event as the kind of node and branch that will be inserted before an existing node. For example, if you press /NID, the mode indicator changes to POINT, and the third line of the display shows "Point to the node before which a new decision node and branch will be inserted." You can insert the new node and branch in front of (to the left of) any existing decision or event node. The new branch initially has value 0, probability 0 (if it is an event branch), and an arbitrary name indicating the total number of branches in the entire tree. Chapter 6: TreePlan Menu Options Page 63 Node Oops Option ------------------------------------------------------------------ To change a decision node to an event node or to change an event node to a decision node, press /NO in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to decision or event node to be changed." Use the pointer keys to move the node pointer to the appropriate node, and press the Enter key to select it. If you select an event node, TreePlan asks for confirmation before deleting the probabilities. Node Shorten Option ------------------------------------------------------------------ To shorten the tree by removing a node and its single successor branch, press /NS in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to decision or event node with single successor branch to be deleted." Use the pointer keys to move the node pointer to the appropriate node, and press the Enter key to select it. This option can be used only when the branch to be deleted has no parallel branches; if the selected node has two or more successor branches, TreePlan displays an error message. If the selected node has a single successor branch, TreePlan asks for confirmation before deleting the node and branch. If you want to delete an entire set of events or set of decisions in the middle of a tree, first use the Node Remove option on all but one of the branches in the set, and then use the Node Shorten option on the single remaining node and branch. You can also use the Node Shorten option to undo the results of the Node Insert option. Node Terminal Option ------------------------------------------------------------------ To change a decision or event node to a terminal node, type /NT in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to the decision or event node to be changed to a terminal node." Use the pointer keys to move the node pointer to the appropriate node, and press the Enter key to select it. Since this command erases all branches following the highlighted node, TreePlan tells you how many branches will be erased and asks for confirmation before completing the operation. TreePlan User's Manual Page 64 Node Remove Option ------------------------------------------------------------------ To remove a node, the previous branch, and any successor nodes and branches, type /NR in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to the node following the branch to be removed." Use the pointer keys to move the node pointer to the appropriate node, and press the Enter key to select it. Since this command erases the highlighted node, the previous branch, and all successor nodes and branches, TreePlan tells you how many branches will be erased and asks for confirmation before completing the operation. Node New Option ------------------------------------------------------------------ To erase the entire tree, type /NN in READY mode. If the tree hasn't been modified since the last time you saved a copy on disk, the entire tree is erase immediately. If you have made changes to the tree since the last time you saved it, TreePlan asks for confirmation before erasing the tree. Solve Menu Options ================================================================== When you press /S in READY mode, TreePlan uses the rollback method to find the optimal strategy. The overall solution procedure has four steps. (1) TreePlan checks the sum of probabilities for each set of event branches. If a sum isn't equal to one, TreePlan returns to READY mode with the field pointer on a probability of the event set so that you can correct the probabilities. If each of the sums equal one, the solution procedure continues. (2) For each terminal node, TreePlan determines the terminal value by summing the partial-cash-flow values on the branches leading from the initial node to the terminal node. (3) The standard rollback method is used to determine a certainty equivalent at each node. At each event node a certainty equivalent is computed (using expected value if the decision maker is risk neutral); at each decision node the highest certainty equivalent of the successor nodes is selected. Chapter 6: TreePlan Menu Options Page 65 When TreePlan uses the exponential utility function, the absolute value of the product of the risk coefficient R times a payoff value or certainty equivalent must be less than 88; if not, R is too extreme relative to the value for computational purposes, and TreePlan displays an error message. (4) TreePlan identifies branches associated with the optimal strategy. After the tree has been solved, the six Solve options are displayed: Risk, View, Distribution, Print, Help, and Quit. Solve Risk Options ------------------------------------------------------------------ To specify risk attitude (utility) for determining certainty equivalents for the rollback method, select Risk from the Solve menu, or press /SR in READY mode. The risk attitude assessment lottery is shown on the main portion of the screen, and the risk coefficient for the exponential utility function is shown in the lower left corner. The six Solve Risk options are displayed: Change, Max/Min, Neutral, Direct, Help, and Quit. Solve Risk Change Options ------------------------------------------------------------------ To change the certainty equivalent (CE), better payoff, and worse payoff of the risk attitude assessment lottery, select Change from the Solve Risk menu, or press /SRC in READY mode. The four Solve Risk Change options are displayed: CE, Better, Worse, and Quit. Use the CE, Better, and Worse options to enter values so that the lottery reflects the decision maker's attitude toward risk. After a value is entered, TreePlan tries to use the lottery inputs in a Newton-Raphson search to determine the risk coefficient parameter R for an exponential function representing the decision maker's risk attitude. If the attempt is successful, the value of R in the lower left corner is updated; if R cannot be determined, the lower left corner is blank. When you finish entering the certainty equivalent and payoff values, select the Quit option to use the current values to determine R and return to the Solve menu. TreePlan checks to see that Worse is less than CE and that CE is less than Better. Also, if the lottery certainty equivalent is too close to one of the TreePlan User's Manual Page 66 payoff values, the search procedure may not be able to determine R. If the three values are not suitable, an error message is displayed; press Escape or Enter to clear the error message, and then enter new values. If you press the Escape key to leave the Solve Risk menu, TreePlan checks the values and tries to determine R. If R cannot be determined, TreePlan reverts to the original values before you selected the Change option. No error message is displayed. Solve Risk Max/Min Option ------------------------------------------------------------------ To use the maximum and minimum terminal values from the decision tree for the assessment lottery's better and worse payoffs, select Max/Min from the Solve Risk menu, or press /SRM in READY mode. TreePlan first searches through the terminal values to find the highest and lowest value. The better payoff in the assessment lottery is set equal to the maximum terminal value, and the worse payoff is set equal to the minimum. Then TreePlan uses the exponential utility function with the current value of the risk coefficient R to determine the certainty equivalent for the assessment lottery; if R is too extreme relative to the new lottery payoffs, TreePlan changes R to zero and uses the expected value of the new payoffs for the certainty equivalent. Solve Risk Neutral Option ------------------------------------------------------------------ To set risk attitude for a risk neutral decision maker so that expected values are used to determine certainty equivalents for the rollback method, select Neutral from the Solve Risk menu, or press /SRN in READY mode. TreePlan changes R to zero and sets the assessment lottery certainty equivalent equal to the expected value of the current lottery's payoffs. Solve Risk Direct Option ------------------------------------------------------------------ To directly specify the value for the risk coefficient R of the exponential utility function, select Direct from the Solve Risk menu, or press /SRD in READY mode. After you enter a new risk coefficient, TreePlan uses the exponential utility function with the new value of R to determine the certainty equivalent for the Chapter 6: TreePlan Menu Options Page 67 assessment lottery; if the new R is too extreme relative to the current lottery payoffs, TreePlan changes R to zero and uses the expected value of the payoffs for the certainty equivalent. Solve View Option ------------------------------------------------------------------ To view the optimal strategy on the screen, select View from the Solve menu, or press /SV in READY mode. The mode indicator changes to VIEW, and a node is highlighted on the tree display. Use the Page-Up, Page-Down, Home, and four arrow keys to move the node pointer to view the entire tree. (Use Page-Up and Page-Down to change the branch shown in detail in the center of the tree display.) Branches associated with the optimal strategy are shown with double lines instead of single lines, and rollback certainty equivalents are shown to the left of each node. Partial-cash-flow values are not shown on the solved tree. When you are finished, press Escape or Enter to return to the Solve menu. Solve Distribution Options ------------------------------------------------------------------ To determine the payoff distribution of the optimal strategy, select Distribution from the Solve menu, or press /SD in READY mode. After the distribution has been determined, the four Solve Distribution options are displayed: Screen, Printer, File, and Quit. Solve Distribution Screen Option ------------------------------------------------------------------ To view the payoff distribution on the screen, select Screen from the Solve Distribution menu, or press /SDS in READY mode. The mode indicator changes to VIEW, and the screen shows terminal values in ascending order with the joint probability that each value will be obtained if the optimal strategy is followed. Terminal values are shown using the current format value setting. Probabilities are displayed with five decimal places; a probability less than 0.000005 is shown as 0.00000. If the same value is associated with two or more terminal nodes of the optimal strategy, the sum of the joint probabilities is shown. The expected value and standard deviation are shown at the bottom of the payoff distribution. Use Up-arrow, Down-arrow, Page-Up, Page- Down, Home, and End to view the entire table. When you are TreePlan User's Manual Page 68 finished, press Escape or Enter to return to the Solve menu. Solve Distribution Printer/File Options ------------------------------------------------------------------ To prepare to send the payoff distribution to the printer or a print-file, select Printer or File from the Solve Distribution menu, or press /SDP or /SDF in READY mode. The top section of the screen shows menu options, the middle section shows applicable current and startup settings, and the bottom section tells how many pages will be required. The body of the payoff distribution always has 45 characters per line, and the number of lines equals the number of unique values of the payoff distribution plus the seven lines for headings and summary measures. The left margin and top margin settings determine where the upper left corner of the payoff distribution will appear on the output page. The bottom margin and page-length settings determine the number of blank lines at the bottom of each page. The number of characters on each line of the print-out or print-file is 45 plus the current left margin, and the number of vertical pages depends on the current settings for top margin, bottom margin, and page length. When the settings are satisfactory, select the Go option. For information about the Layout, Setup, Printer, Extended (a Printer option), and Directory options, please consult the Default Menu Options section of this chapter. Solve Print Options ------------------------------------------------------------------ To prepare to output the solved tree diagram, select Print from the Solve menu, or press /SP in READY mode. The four Solve Print options are displayed: Screen, Printer, File, and Quit. Solve Print Screen Option ------------------------------------------------------------------ To send the solved tree diagram to the screen, select Screen from the Solve Print menu, or press /SPS in READY mode. The mode indicator changes to VIEW, and the solved tree is displayed. Use the Page-Up, Page-Down, Home, End, and four arrow keys to view the entire tree. Use the Home key to move the screen window to the Chapter 6: TreePlan Menu Options Page 69 upper left corner of the tree diagram; if the tree is large, the upper left corner may be blank. Use the End key to move the screen window to the lower right corner of the tree diagram. Use the Left-arrow or Right-arrow keys to move the screen window one branch to the left or right. Branches associated with the optimal strategy are shown with double lines instead of single lines, and rollback certainty equivalents are shown to the left of each node. Partial-cash-flow values are not shown on the solved tree. When you are finished, press Escape or Enter to return to the Solve Print menu. Solve Print Printer/File Options ------------------------------------------------------------------ To prepare to send a large diagram of the solved tree to the printer or a print-file, select Printer or File from the Solve Print menu, or press /SPP or /SPF in READY mode. The top section of the screen shows menu options, the middle section shows applicable current and startup settings, and the bottom section tells how many pages will be required. The menu options, number of lines, number of characters per line, and number of pages are the same for both the solved and unsolved versions of the large diagram. For information about the large diagram, please consult the Print Menu Options section of this chapter. Print Menu Options ================================================================== To prepare to output a large diagram of the unsolved tree, press /P in READY mode. The four Print options are displayed: Screen, Printer, File, and Quit. Print Screen Option ------------------------------------------------------------------ To send the unsolved tree diagram to the screen, select Screen from the Print menu, or press /PS in READY mode. The mode indicator changes to VIEW, and the unsolved tree is displayed. Use the Page-Up, Page-Down, Home, End, and four arrow keys to view the entire tree. Use the Home key to move the screen window to the upper left corner of the tree diagram; if the tree is large, the upper left corner may be blank. Use the End key to move the screen window to the lower right corner of the tree diagram. Use the Left-arrow or Right-arrow keys to move the screen window one branch to the left or right. The branch name is shown above each TreePlan User's Manual Page 70 branch line, and the partial-cash-flow value is shown on the left below the line; for event branches the probability is shown on the right below the line. When you are finished, press Escape or Enter to return to the Solve Print menu. Print Printer/File Options ------------------------------------------------------------------ To prepare to send a large diagram of the unsolved tree to the printer or a print-file, select Printer or File from the Print menu, or press /PP or /PF in READY mode. The top section of the screen shows menu options, the middle section shows applicable current and startup settings, and the bottom section tells how many pages will be required. The size of the body of the large diagram, solved or unsolved, depends on the number of terminal nodes, the number of branches on the longest path of the tree, and the number of characters in the largest terminal value. If T is the number of terminal nodes, the number of lines required is 4*T - 1. If P is the number of branches on the longest path and V is the number of characters in the largest terminal value, the number of characters on each line is 26*P + 3 + V. To print a large diagram, TreePlan divides it into pages that can be joined together after printing. The number of horizontal pages depends on the current settings for left margin and right margin, and the number of vertical pages depends on the current settings for top margin, bottom margin, and page length. If you adjust the current layout settings, TreePlan updates the bottom of the display showing the number of pages required. When the settings are satisfactory, select the Go option. For information about the Format, Layout, Setup, Printer, Extended (a Printer option), and Directory options, please consult the Default Menu Options section of this chapter. File Menu Options ================================================================== To manage files created by TreePlan, press /F in READY mode. Six File options are displayed: Retrieve, Save, Erase, List, Directory, and Help. TreePlan works with two kinds of files: tree data files (identified by a .TRE extension) and print-files (.PRN extension). Chapter 6: TreePlan Menu Options Page 71 Use the Retrieve and Save options to read and write tree data files, which are standard ASCII text files containing branch/node data: node ID number, branch name, partial-cash-flow value, probability, predecessor ID, kind of node, number of successors, and successor IDs. Use various print options to create print-files containing line-oriented tables and diagrams, identical to output that is sent to a printer; these print-files are standard ASCII text files. The File options affect tree data files and print files only in the current directory. Use the Directory option to access files in other directories. File Retrieve Option ------------------------------------------------------------------ To read a tree data file from disk, select Retrieve from the File menu, or press /FR in READY mode. This command replaces the tree in memory with a copy of a tree from a data file on disk. If you have made changes to the tree in memory since the last time you saved it, TreePlan asks for confirmation. The mode indicator changes to POINT, and the display shows a list of tree data files in the current directory. Point to the desired filename, and press Enter to select. TreePlan erases the tree in memory, loads a copy of the tree from the data file, and returns to READY mode. File Save Option ------------------------------------------------------------------ To store a copy of the tree in memory in a tree data file on disk, select Save from the File menu, or press /FS in READY mode. The display shows a list of tree data files in the current directory, and the mode indicator changes to INSERT. Type a file name at the prompt on the second line. Do not use an extension in the filename you type; TreePlan adds the .TRE extension automatically. File names can be up to eight characters long and can include upper-case and lower-case letters, numbers, and the underscore character (_). TreePlan does not accept a period or space character in a file name. If you type a filename that already exists, TreePlan asks for confirmation before it erases the existing file and creates a new tree data file with the same name. After the tree is saved on disk, TreePlan returns to READY mode. TreePlan User's Manual Page 72 File Erase Tree/Print Options ------------------------------------------------------------------ To erase a tree data file or print-file on disk, select Erase from the File menu and then select Tree or Print. Alternatively, press /FET or /FEP in READY mode. The display shows a list of tree data files (.TRE extension) or print-files (.PRN extension) in the current directory, and the mode indicator changes to POINT. Use the pointer keys to move the pointer to the appropriate filename, and press Enter to select it. TreePlan does not ask for confirmation; the file you select is erased immediately. Before selecting a file, press Escape to return to the previous menu without erasing a file. File List Tree/Print Options ------------------------------------------------------------------ To display the names of tree data files or print-files on disk, select List from the File menu and then select Tree or Print. Alternatively, press /FLT or /FLP in READY mode. The display shows a list of tree data files (.TRE extension) or print-files (.PRN extension) in the current directory, and the mode indicator changes to VIEW. When finished, press Enter to return to the File menu, or press Escape to return to the File List menu. File Directory Option ------------------------------------------------------------------ To enter the name of a different disk directory for tree data files and print-files, select Directory from the File menu, or press /FD in READY mode. The mode indicator changes to INSERT with the cursor at the end of the current directory name on the second line. Edit the directory name; press Enter when finished, or press Escape to return to the File menu without changing the current directory. For additional information about using directories, please see the Default Menu Options section of this chapter. EVPI Menu Options ================================================================== To compute Expected Value of Perfect Information (EVPI), press /E in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to event node preceding event set Chapter 6: TreePlan Menu Options Page 73 for EVPI computation." Use the pointer keys to move the node pointer to the appropriate event node, and press the Enter key to select it. After you select an event set for EVPI, TreePlan temporarily expands your decision tree as follows. (1) A decision branch "No additional info" is inserted at the initial node in front of your original decision tree. (2) A second decision branch "Use perfect prediction" is added to the initial node. (3) "Use perfect prediction" is followed by a copy of the event set you selected with the branch names shown in quotes on the expanded tree. (4) A copy of your original decision tree follows each of the perfect prediction event branches. TreePlan automatically assigns conditional probabilities of zero and one to appropriate events. TreePlan solves the expanded tree using the rollback method with expected values, displays the solved tree with the "Use perfect prediction" branch in the center of the display, and shows EVPI in the lower left corner of the screen. EVPI is the difference between the expected value of the best strategy with no additional information (your original problem) and the expected value of the best strategy using a perfect prediction. After the expanded tree has been solved, the mode indicator changes to MENU and the four EVPI options are displayed: View, Retain, Help, and Quit. EVPI View Option ------------------------------------------------------------------ To view the solved expanded tree, select View from the EVPI menu, or press /EV in READY mode. The mode indicator changes to VIEW, and a node is highlighted on the tree display. Use the Page-Up, Page-Down, Home, and four arrow keys to move the node pointer to view the entire tree. (Use Page-Up and Page-Down to change the branch shown in detail in the center of the tree display.) Branches associated with the optimal strategy are shown with double lines instead of single lines, and rollback certainty equivalents are shown to the left of each node. Partial-cash-flow values are not shown on the solved tree. When you are finished, press Escape or Enter to return to the EVPI menu. TreePlan User's Manual Page 74 EVPI Retain Option ------------------------------------------------------------------ To return to READY mode with the expanded EVPI tree, select Retain from the EVPI menu, or press /ER in READY mode. For example, use this option if you want to modify the expanded tree, print the expanded tree diagram, or solve the expanded tree using an exponential utility function to determine value of information. To return to READY mode with only the original tree, select Quit from the EVPI menu or press Escape. Bayes Menu Options ================================================================== To perform Bayesian revision of probabilities, first use the Node options to build a decision tree containing the main events that directly affect the payoffs and the information events that are the results of the information-gathering effort. After you have a tree with the main events and information events, press /B in READY mode to perform Bayesian revision. Eight Bayes options are displayed: Link, Input, Screen, Printer, File, Transfer, Help, and Quit. You must use the Link Main-event and Link Info-event options before using the Input, Screen, Printer, File, and Transfer options. If you use the Link options, return to READY mode, and make any modifications to the tree, you must use the Link options again before using the other five options. Bayes Link Options ------------------------------------------------------------------ To identify the main events and information events on the decision tree for Bayesian revision, select Link from the Bayes menu, or press /BL in READY mode. Two Bayes Link options are displayed: Main-event and Info-event. To identify the main events on the decision tree for Bayesian revision, select Main-event from the Bayes Link options. Then point to a node that precedes the main events, and press Enter to select it. The main event branches usually appear at several places on the decision tree. One set of main events usually has Chapter 6: TreePlan Menu Options Page 75 prior probabilities, and other sets of main events will have posterior probabilities after Bayesian revision. The branch names must be identical for each set of main events. To identify the information events on the decision tree for Bayesian revision, select Info-event from the Bayes Link options. Then point to the node that precedes the information events, and press Enter to select it. The information event branches usually appear at only one place on the decision tree. The set of information events will have preposterior probabilities after Bayesian revision. Bayes Input Options ------------------------------------------------------------------ After you select the main event and information event nodes, to specify input probabilities for Bayesian revision, select Input from the Bayes options, or press /BI in READY mode. Two Bayes Input options are displayed: Prior/Likelihood and Joint. To change the prior and likelihood input probabilities for Bayesian revision, select Prior/Likelihood from the Bayes Input options, or press /BIP in READY mode. The mode indicator changes to BAYES, and a probability tree shows the main events with prior probabilities on the left and information events with likelihood probabilities on the right. Use the pointer keys to move the highlighted field pointer among the prior and likelihood probabilities. To change the joint probabilities for Bayesian revision, select Joint from the Bayes Input options, or press /BIJ in READY mode. The mode indicator changes to BAYES, and a probability tree shows the main events on the left, information events in the center, and joint probabilities on the right. Use the pointer keys to move the highlighted field pointer among the joint probabilities. In BAYES mode, press F2 to edit a probability, or simply start typing to replace the contents of a field with a new probability. While editing a probability, press the Insert key to toggle between INSERT and OVERTYPE modes; press Escape to restore the original probability; press Enter to have TreePlan accept the new probability. When you finish modifying the input probabilities and want to leave the BAYES input mode, press Enter to return to the Bayes menu, or press Escape to return to the Bayes Input options. After you use the Bayes Link options to select the main event and information event nodes, the initial joint probabilities for the TreePlan User's Manual Page 76 two event sets are computed assuming statistical independence. The prior probabilities on the main event branches may be correct, but usually you will have to modify the likelihood or joint probabilities to reflect the correct inputs for Bayesian revision. Bayes Screen Option ------------------------------------------------------------------ After you select the main event and information event nodes, to view all Bayesian probabilities, select Screen from the Bayes options, or press /BS in READY mode. The top probability tree shows inputs (priors and likelihoods) and joint probabilities; the bottom probability tree shows outputs (preposteriors and posteriors) and joint probabilities. All probabilities are shown using five decimal places. Use Up-arrow, Down-arrow, Page-up, Page-down, Home, and End keys to view the probability trees. When finished, press Enter or Escape to return to the Bayes menu. Bayes Printer/File Options ------------------------------------------------------------------ After you select the main event and information event nodes, to prepare to send the Bayes probability trees to the printer or a print-file, select Printer or File from the Bayes menu, or press /BP or /BF in READY mode. The top section of the screen shows menu options, the middle section shows applicable current and startup settings, and the bottom section tells how many pages will be required. The body of the output of the Bayes probability trees always has 78 characters per line, and the number of lines depends on the number of branches plus headings. If M is the number of branches for the main events and I is the number of branches for the information events, the number of lines required is 6 + 6*M*I. The left margin and top margin settings determine where the upper left corner of the top heading will appear on the output page. The bottom margin and page-length settings determine the number of blank lines at the bottom of each page. The number of characters on each line of the print-out or print-file is 78 plus the current left margin, and the number of vertical pages depends on the current settings for top margin, bottom margin, and page length. When the settings are satisfactory, select Go from the menu. For information about the Layout, Setup, Printer, Extended (a Printer option), and Directory options, please consult the Default Chapter 6: TreePlan Menu Options Page 77 Menu Options section of this chapter. Bayes Transfer Options ------------------------------------------------------------------ After you select the main event and information event nodes, to prepare to transfer the Bayesian probabilities to the decision tree, select Transfer from the Bayes menu, or press /BT in READY mode. Since existing probabilities will be modified, TreePlan asks for confirmation before completing the operation. This option changes the probability on every branch of the decision tree with a name identical to the main event and information event branches. If a main event branch on the decision tree is preceded by an information event branch, a posterior probability P(Main | Info) is assigned; if a main event branch has no such predecessor, a prior probability P(Main) is assigned. Likelihood probabilities do not normally appear on a decision tree, but if an information event branch on the decision tree is preceded by a main event branch, a likelihood probability P(Info | Main) is assigned; if an information event branch has no such predecessor, a preposterior probability P(Info) is assigned. Table Value Menu Options ================================================================== To create a table for sensitivity analysis of a partial-cash-flow value, press /TV in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to value for sensitivity analysis." Use the pointer keys to move the field pointer to the value of interest, and press the Enter key to select it. After you select the base-case value, the mode indicator changes to MENU, and the six Table Value options are displayed: Load, View, Printer, File, Help, and Quit. TreePlan also displays the heading of the table, showing the value you selected, the branch name, the risk coefficient R, and the number of branches that have the same name and value. Table Value Load Option ------------------------------------------------------------------ To load a set of temporary values into the table by specifying start, step, and stop values, select Load from the Table Value TreePlan User's Manual Page 78 menu. You can use a positive or negative step value to fill the table with values in ascending or descending order. The table holds a maximum of thirteen temporary values. After you specify the start, step, and stop values, TreePlan determines each temporary partial-cash-flow value, temporarily changes the value on all branches having the same name and base-case value, solves the tree using the rollback method, and displays the results in a table. Each row of the table shows results for a case that can be compared to the base case. The table shows each temporary value, the certainty equivalent of the optimal strategy when that temporary value is used, and whether the optimal strategy using the temporary value is the same as the optimal strategy using the base-case value. After the results are displayed, the mode indicator changes from WAIT to MENU, and you can select other Table Value options. Table Value View Option ------------------------------------------------------------------ To view the optimal strategy for a case, select View from the Table Value menu. The mode indicator changes to POINT, and an entire row of the table is highlighted. Use the pointer keys to highlight a case of interest, and press Enter to select it. The mode indicator changes to VIEW, and the solved tree using the temporary values for the selected case is displayed. Use the pointer keys to view the solved tree; when finished, press Enter or Escape to return to the table with the highlighted row in POINT mode. Select another case, or press Escape to return to the Table Value menu. Table Value Printer/File Options ------------------------------------------------------------------ To prepare to send the value sensitivity analysis table to the printer or a print-file, select Printer or File from the Table Value menu. The top section of the screen shows menu options, the middle section shows applicable current and startup settings, and the bottom section tells how many pages will be required. The body of the table always has 55 characters per line, and the number of lines equals the number of temporary values (a maximum of thirteen cases) plus six lines for the table heading. The left margin and top margin settings determine where the upper left corner of the table will appear on the output page. The bottom margin and page-length settings determine the number of blank lines at the bottom of each page. The number of characters on Chapter 6: TreePlan Menu Options Page 79 each line of the print-out or print-file is 55 plus the current left margin. Values and certainty equivalents are always shown using two decimal places, and the table is always a single page. When the settings are satisfactory, select Go from the menu. For information about the Layout, Setup, Printer, Extended (a Printer option), and Directory options, please consult the Default Menu Options section of this chapter. Table Probability Menu Options ================================================================== To create a table for sensitivity analysis of a probability, press /TP in READY mode. The mode indicator changes to POINT, and the third line of the display shows "Point to probability for sensitivity analysis." Use the pointer keys to move the field pointer to the value of interest, and press the Enter key to select it. After you select the base-case probability, the mode indicator changes to MENU, and the seven Table Probability options are displayed: Load, Tenths, View, Printer, File, Help, and Quit. TreePlan also displays the heading of the table, showing the probability you selected, the branch name, the risk coefficient R, and the number of identical event sets. Event sets are identical if the branches following an event node have the same branch names and the same base-case probabilities. Table Probability Load Option ------------------------------------------------------------------ To load a set of temporary probabilities into the table by specifying start, step, and stop values, select Load from the Table Probability menu. TreePlan accepts only a positive step value, so the table will be filled with probabilities in ascending order. The table holds a maximum of thirteen temporary values. After you specify the start, step, and stop values, TreePlan determines each temporary probability, temporarily changes the probabilities on identical event sets, solves the tree using the rollback method, and displays the results in a table. Each row of the table shows results for a case that can be compared to the base case. The table shows each temporary probability, the certainty equivalent of the optimal strategy when that temporary probability is used, and whether the optimal strategy using the temporary probability is the same as the optimal strategy using the base-case probability. Probabilities are always shown using five decimal places. After the results are displayed, the mode indicator changes from WAIT to MENU, and you can select other TreePlan User's Manual Page 80 Table Probability options. When the base-case probability is changed, probabilities on other branches in the event set are changed so that they retain the same proportional relationship as the base case. For example, assume events Low, Medium, High have probabilities 0.1, 0.3, and 0.6, respectively. If you select probability of Low, 0.1, for sensitivity analysis, the probabilities for Medium and High will have the ratio 0.3 to 0.6 for each case you analyze. If the probability of Low is changed temporarily to 0.2, the remaining 0.8 probability will be allocated according to the base-case ratio, so the temporary probabilities for Medium and High will be 0.26667 and 0.53333. Table Probability Tenths Option ------------------------------------------------------------------ To load the table with temporary probabilities 0.0, 0.1, ..., 1.0, select Tenths from the Table Probability menu. The results are the same as using the Load option with start = 0.0, step = 0.1, and stop = 1.0. Table Probability View Option ------------------------------------------------------------------ The View option of the Table Probability menu works the same as that of the Table Value menu. For information, please consult the Table Value View Option section of this chapter. Table Probability Printer/File Options ------------------------------------------------------------------ The Printer and File options of the Table Probability menu work the same as those of the Table Value menu. For information, please consult the Table Value Printer/File Options section of this chapter. Table Lottery-CE Menu Options ================================================================== If you want to characterize attitude toward risk using the certainty equivalent (CE) of the assessment lottery, first specify Chapter 6: TreePlan Menu Options Page 81 the base-case values for sensitivity analysis. In READY mode, press /SRC, and use the Solve Risk Change options to specify the better payoff, worse payoff, and certainty equivalent for the assessment lottery. After specifying the base case, to create a table for sensitivity analysis of the lottery certainty equivalent, press /TL in READY mode. The mode indicator changes to MENU and the six Table Lottery-CE options are displayed: Load, View, Printer, File, Help, and Quit. TreePlan also displays the heading of the table, showing the base-case lottery certainty equivalent, the base-case risk coefficient R, and the assessment lottery's better and worse payoffs, which apply to all cases that will be shown in the table. Table Lottery-CE Load Option ------------------------------------------------------------------ To load a set of temporary lottery certainty equivalents into the table by specifying start, step, and stop values, select Load from the Table Lottery-CE menu. You can use a positive or negative step value to fill the table with certainty equivalents in ascending or descending order. The table holds a maximum of thirteen temporary certainty equivalents. After you specify the start, step, and stop values, TreePlan determines each temporary certainty equivalent, finds the associated risk coefficient R, solves the tree using the rollback method with the exponential utility function, and displays the results in a table. Each row of the table shows results for a case that can be compared to the base case; the better payoff and worse payoff of the assessment lottery are the same for each case. The table shows each temporary certainty equivalent of the assessment lottery, the associated risk coefficient R, the certainty equivalent of the optimal strategy when that risk coefficient is used in the exponential utility function, and whether the optimal strategy using the temporary risk coefficient is the same as the optimal strategy using the base risk coefficient. After the results are displayed, the mode indicator changes from WAIT to MENU, and you can select other Table Lottery-CE options. Table Lottery-CE View Option ------------------------------------------------------------------ The View option of the Table Lottery-CE menu works the same as that of the Table Value menu. For information, please consult the Table Value View Option section of this chapter. TreePlan User's Manual Page 82 Table Lottery-CE Printer/File Options ------------------------------------------------------------------ The Printer and File options of the Table Lottery-CE menu work the same as those of the Table Value menu, except that the body of the table has 75 characters per line instead of 55. For information, please consult the Table Value Printer/File Options section of this chapter. Table Risk-Coefficient Menu Options ================================================================== If you want to characterize attitude toward risk using the risk coefficient directly, first specify the base-case risk coefficient for sensitivity analysis. In READY mode, press /SRD (Solve Risk Direct) and enter the base-case risk coefficient. If you also want to interpret the associated certainty equivalents of the assessment lottery, press /SRC in READY mode, and use the Solve Risk Change options to specify the better and worse payoffs of the assessment lottery. After specifying the base case, to create a table for sensitivity analysis of the risk coefficient, press /TR in READY mode. The mode indicator changes to MENU and the six Table Risk-Coefficient options are displayed: Load, View, Printer, File, Help, and Quit. TreePlan also displays the heading of the table, showing the base- case lottery certainty equivalent, the base-case risk coefficient R, and the assessment lottery's better and worse payoffs, which apply to all cases that will be shown in the table. Table Risk-Coefficient Load Option ------------------------------------------------------------------ To load a set of temporary risk coefficients into the table by specifying start, step, and stop values, select Load from the Table Risk-Coefficient menu. You can use a positive or negative step value to fill the table with risk coefficients in ascending or descending order. The table holds a maximum of thirteen temporary risk coefficients. After you specify the start, step, and stop values, TreePlan determines each temporary risk coefficient, finds the associated certainty equivalent for the assessment lottery, solves the tree using the rollback method with Chapter 6: TreePlan Menu Options Page 83 the exponential utility function, and displays the results in a table. Each row of the table shows results for a case that can be compared to the base case; the better payoff and worse payoff of the assessment lottery are the same for each case. The table shows each temporary risk coefficient R, the associated certainty equivalent of the assessment lottery, the certainty equivalent of the optimal strategy when the temporary risk coefficient is used in the exponential utility function, and whether the optimal strategy using the temporary risk coefficient is the same as the optimal strategy using the base risk coefficient. After the results are displayed, the mode indicator changes from WAIT to MENU, and you can select other Table Risk-Coefficient options. Table Risk-Coefficient View Option ------------------------------------------------------------------ The View option of the Table Risk-Coefficient menu works the same as that of the Table Value menu. For information, please consult the Table Value View Option section of this chapter. Table Risk-Coefficient Printer/File Options ------------------------------------------------------------------ The Printer and File options of the Table Risk-Coefficient menu work the same as those of the Table Value menu, except that the body of the table has 75 characters per line instead of 55. For information, please consult the Table Value Printer/File Options section of this chapter. Default Menu Options ================================================================== To verify or set TreePlan's global settings, press /D in READY mode. Nine Default options are displayed: Format, Layout, Setup, Printer, Directory, Clear, Update, Help, and Quit. There are two sets of default settings: current and startup. The current settings are those in effect when you are using TreePlan. You can change the current settings from the Default menu and from various printer and print-file menus. The startup settings are the settings used by TreePlan when you start the program from the DOS prompt. Initially, TreePlan looks for the configuration file (TREEPLAN.CFG) containing the settings. TreePlan User's Manual Page 84 If the file is found, the startup settings are set equal to those stored in the configuration file. If TreePlan can't find the file or has a problem reading the file, TreePlan sets the startup settings equal to settings that are a permanent part of the TreePlan program. After the startup settings have been determined, TreePlan sets the current settings equal to the startup settings. If you run TreePlan from the A: drive without a configuration file and press /D in READY mode, you will see the following settings. Current Startup Setting Setting ------- ------- Format + Value decimal places 0 0 + Probability decimal places 3 3 Layout + Left margin 0 0 | Right margin 80 80 | Top margin 0 0 | Bottom margin 0 0 + Page-Length 66 66 Setup string for printer (None) (None) Printer + Interface LPT1 LPT1 | Extended characters No No | Automatic Line Feed No No + Paper continuous feed Yes Yes Directory for files A:\ (None) The startup settings shown above are TreePlan's built-in settings; the current settings are the same, except for the directory for files, which TreePlan has set to the drive/directory where you started the program. Default Format Options ------------------------------------------------------------------ To select decimal places for value and probability formats, select Format from the Default menu or from various print-printer and print-file menus. The format options determine how values (partial-cash-flows, certainty equivalents, and terminal values) and probabilities will be formatted for display when viewing or printing the large tree diagram. The number of decimal places for values can be zero (integer) or two (dollars and cents), and the number of decimal places for probabilities can be two, three, four, or five. Some program Chapter 6: TreePlan Menu Options Page 85 options override the format settings; for example, probabilities of a payoff distribution are always shown with five decimal places. Default Layout Options ------------------------------------------------------------------ To enter margins and page length for printer and print-file output, select Layout from the Default menu or from various print- printer and print-file menus. The layout options determine how output will be placed on a page; these settings apply to both printer and print-file output. The left margin setting specifies the number of space characters (blanks) that TreePlan sends to the printer at the start of each line on a page. The left margin must be between 0 and 40. The right margin setting specifies the total number of characters on each line that TreePlan sends to the printer. The right margin must be between 41 and 240. Some program options override this setting; for example, Bayes probability output always sends the current left margin plus 78 characters. If you set the right margin too high, TreePlan may send more characters per line than your printer can accept; in this case, your printer may lose the extra characters or print them on the next line, sometimes with unpredictable results. Some printers can be set to print in a condensed or compressed mode, in which case the printer will be able to accept more characters per line. The top margin setting specifies the number of blank lines at the top of each page of output. The top margin must be between 0 and 10. The bottom margin setting specifies the number of blank lines at the bottom of each page. The bottom margin must be between 0 and 10. The page length setting specifies the total number of lines on each page that TreePlan sends to the printer. The page length must be between 21 and 999. Some of these lines, including the top and bottom margins, may be blank. Most printers print six lines per inch, so standard 11-inch paper will have 66 lines per page. Some printers can be set to print eight lines per inch, in which case you could set the page length to 88. TreePlan User's Manual Page 86 Default Setup Option ------------------------------------------------------------------ To enter a setup string for the printer, select Setup from the Default menu or from various print-printer menus. A setup string is a string of characters sent to the printer to activate specific printer functions (e.g., compressed print). Refer to your printer manual for the appropriate setup string characters, translate each character into its three-digit numerical ASCII code, and precede each three-digit code with a backslash when specifying the setup string. For example, \015 sends a special character that tells most IBM and Epson printers to use condensed print, approximately 16.67 characters per inch. In this case, you may be able to print approximately 140 characters per line on standard 8.5-inch paper by using a setup string and adjusting the layout right margin setting. Since special characters may cause unpredictable results in ASCII text files, TreePlan disregards the setup string when sending output to a print-file. Default Printer Options ------------------------------------------------------------------ To select printer hardware settings, select Printer from the Default menu or from various print-printer menus. There are four options: Interface, Extended, Auto-LF, and Paper. (The Extended option is also available from various print-file menus.) These settings specify hardware characteristics of your printer and its interface. The interface setting specifies the printer port: LPT1, LPT2, LPT3, COM1, or COM2. A typical system uses LPT1, sometimes called PRN. If you plan to use COM1 or COM2 for a serial printer, first use the DOS MODE command to specify baud rate, parity, and stop bits before running TreePlan. The extended characters setting specifies whether the printer can print the line drawing characters of the IBM extended character set. If not, TreePlan sends only standard ASCII characters to the printer or print-file. The automatic line feed setting specifies whether the printer performs an automatic line feed after each carriage return. If so, TreePlan ends each line with only a carriage return; if not, TreePlan ends each line with a carriage return and a line feed. Chapter 6: TreePlan Menu Options Page 87 If your printer is overprinting your output on a single line or double-spacing between lines, change this setting. This setting does not affect lines sent to a print-file, which always end with a carriage return and line feed. The paper continuous feed setting specifies whether the printer uses continuous-feed paper. If not, TreePlan pauses after each page of a multi-page print-out so you can change single sheets before resuming printing. This setting does not affect print-file output. Default Directory Option ------------------------------------------------------------------ To enter the name of a directory for tree data files and print- files, select Directory from the Default menu, the File menu, or various print-file menus. TreePlan uses the current directory for saving and retrieving tree data files and for writing print-files to disk. When you run the program by typing TreePlan at the DOS prompt, TreePlan looks for its configuration file in the initial directory. If TreePlan can't find the configuration file, it uses the disk drive and/or directory you ran the program from (the load directory) as the current directory. If the configuration file is found, TreePlan reads it, uses the configuration settings, and checks to see whether the directory specified in the file exists. If the directory exists, TreePlan uses it as the current directory. If TreePlan can't access the directory specified in the configuration file, Treeplan shows an error message. After you press Escape or Enter to clear the error message, TreePlan uses the load directory as the current directory. In this case, the directory specified in the configuration file is shown as the startup directory, even though it may not exist. The Directory option does not create a directory; it only allows TreePlan to make use of files in a directory that already exists. Before running TreePlan, use the DOS command MkDir (or MD) to create a directory. Default Clear Option ------------------------------------------------------------------ To erase all current settings and establish new current settings equal to the startup settings, select Clear from the Default menu or from various print-printer and print-file menus. TreePlan User's Manual Page 88 Default Update Option ------------------------------------------------------------------ To save the current settings as new startup settings in a configuration file (TREEPLAN.CFG), select Update from the Default menu. The original TreePlan disk does not have a configuration file. If the built-in values are appropriate for your computer system and printer, you may continue to use the program without ever creating a configuration file. On the other hand, if you want to customize the settings and retain the new values so that they are used automatically each time you start the program, use this option. If you run TreePlan from a floppy disk, be sure that the disk containing the TreePlan program file TREEPLAN.EXE is in the drive before you execute the Update command. Also, be sure that the floppy disk is not write-protected; remove the write-protect tab on a 5.25-inch disk; close the write-protect window on a 3.5-inch disk. Zoom Menu Options ================================================================== To prepare to output a small diagram of the unsolved tree, press /Z in READY mode. The four Zoom options are displayed: Screen, Printer, File, and Quit. You may want to use this option to view the overall structure of a large tree. Zoom Screen Option ------------------------------------------------------------------ To send a small diagram of the unsolved tree to the screen, select Screen from the Zoom menu, or press /ZS in READY mode. The mode indicator changes to VIEW, and a small diagram of the unsolved tree is displayed. Branch names are shortened by eliminating spaces and vowels or by truncating the name; probabilities, partial-cash-flow values, and terminal values are not shown. Use the Page-Up, Page-Down, Home, End, and four arrow keys to view the entire tree. Use the Home key to move the screen window to the upper left corner of the tree diagram. Use the End key to move the screen window to the lower right corner of the tree diagram. Use the Left-arrow or Right-arrow keys to move the screen window Chapter 6: TreePlan Menu Options Page 89 one branch to the left or right. When you are finished, press Escape or Enter to return to the Zoom menu. Zoom Printer/File Options ------------------------------------------------------------------ To prepare to send a small diagram of the unsolved tree to the printer or a print-file, select Printer or File from the Zoom menu, or press /ZP or /ZF in READY mode. The top section of the screen shows menu options, the middle section shows applicable current and startup settings, and the bottom section tells how many pages will be required. The size of the body of the small diagram of the unsolved tree depends on the number of terminal nodes and the number of branches on the longest path of the tree. If T is the number of terminal nodes, the number of lines required is 2*T - 1. If P is the number of branches on the longest path, the number of characters on each line is 13*P + 2. To print the small diagram, TreePlan divides it into pages that can be joined together after printing. The number of horizontal pages depends on the current settings for left margin and right margin, and the number of vertical pages depends on the current settings for top margin, bottom margin, and page length. If you adjust the current layout settings, TreePlan updates the bottom of the display showing the number of pages required. For information about the Layout, Setup, Printer, Extended (a Printer option), and Directory options, please consult the Default Menu Options section of this chapter. An example of Zoom output is included in the Valley Problem (B) section of Chapter 3. TreePlan User's Manual Page 90 Appendix A: References ****************************************************************** This first four books listed below were specifically referenced in this User's Manual. There are many other books and articles on decision analysis, and decision trees in particular, that you might want to investigate. Bierman, Harold, Jr., Charles P. Bonini, and Warren H. Hausman, _Quantitative Analysis for Business Decisions_, Eighth Edition, Homewood, IL: Richard D. Irwin, Inc., 1991. Holloway, Charles A., _Decision Making Under Uncertainty: Models and Choices_, Englewood Cliffs, NJ: Prentice-Hall, Inc., 1979. McNamee, Peter, and John Celona, _Decision Analysis for the Professional with Supertree_, Redwood City, CA: The Scientific Press, 1987. Spurr, William A., and Charles P. Bonini, _Statistical Analysis for Business Decisions_, Revised Edition, Homewood, IL: Richard D. Irwin, Inc., 1973. The following book is highly recommended for its practical examples and complete coverage of all techniques used in modern decision decision analysis, including influence diagrams, decision trees, and Monte Carlo simulation. Clemen, Robert T., _Making Hard Decisions: An Introduction to Decision Analysis_, Boston, MA: PWS-Kent, 1991. Appendix A: References Page 91 Appendix B: TreePlan Error Messages ****************************************************************** This appendix lists error messages that may appear on the bottom line of the screen as you work with TreePlan. Messages are listed alphabetically. Pressing Escape or Enter clears the error message from the screen but does not correct the problem. Base probability cannot be 1.0: The probability you select for sensitivity analysis cannot be 1.0. (Table Probability Options) Can copy only to a terminal node (Node Copy Option) Cannot copy from a terminal node (Node Copy Option) Cannot copy from the initial node (Node Copy Option) Cannot copy to a successor of the "copy from" node (Node Copy Option) Cannot create file: Be sure to enter a valid file name. Cannot open file: The file you specified either doesn't exist or is in a different disk or directory. Cannot read tree data file from disk: The file is probably corrupted. Cannot write file to disk: The disk you are trying to write to is full, or there are too many entries in the directory. Press the Escape key repeatedly to return to READY mode; change the disk or erase unneeded files; retry the option. Certainty equivalent too close to lottery payoff value: With the current inputs, TreePlan cannot determine the risk coefficient for an exponential utility function. Try a certainty equivalent closer to the expected value of the risk attitude assessment lottery. (Solve Risk Options) Directory does not exist: Make sure you type the exact path name of the directory. Disk drive not ready: The disk drive specified in the directory setting is not ready for input and output. Make sure you have fully inserted a disk in the proper floppy drive; on a 5.25-inch drive make sure the drive door or lever is closed. TreePlan User's Manual Page 92 Disk is write-protected: If you want to erase a file or write a file to disk, remove the write-protect tab on a 5.25-inch disk or close the write-protect window on a 3.5-inch disk. Each temporary certainty equivalent must be greater than worse payoff: Use the Load option to enter appropriate stop and start values. (Table Lottery-CE Load Option) Each temporary certainty equivalent must be less than better payoff: Use the Load option to enter appropriate stop and start values. (Table Lottery-CE Load Option) Enter Worse < CE and CE < Better: The three values you specify for the risk attitude assessment lottery must be chosen so that the worse payoff is strictly less than the certainty equivalent and the certainty equivalent is strictly less than the better payoff. (Solve Risk Change Options) Event probability for EVPI cannot be 1: The event node you select cannot have a branch with probability 1. (EVPI Menu Options) Info-event node must have at least two successor branches (Bayes Link Options) Initial node must be decision node with at least one branch: You cannot change or eliminate the initial decision node. Link to info-event first: Although you have selected a main-event node, you must also select an info-event node before trying to use the Input, Screen, Printer, File, or Transfer options of the Bayes menu. Link to main-event and info-event first: You must select both a main-event node and an info-event node before trying to use the Input, Screen, Printer, File, or Transfer options of the Bayes menu. Link to main-event first: Although you have selected an info- event node, you must also select a main-event node before trying to use the Input, Screen, Printer, File, or Transfer options of the Bayes menu. Load table first: You must use the Load option before trying to use the View, Printer, or File options of a sensitivity analysis table. Main-event node and info-event node cannot be the same (Bayes Link Options) Appendix B: TreePlan Error Messages Page 93 Main-event node must have at least two successor branches (Bayes Link Options) Maximum and minimum terminal values must be farther apart (Solve Risk Max/Min Option) Maximum number of successors is five (Node Add Option) Must be between 0 and 10 (Bottom margin and top margin) Must be between 0 and 40 (Left margin) Must be between 21 and 999 (Page length) Must be between 41 and 240 (Right margin) Must be greater than -100 and less than +100 (Risk coefficient) Must be \nnn...\nnn with each nnn between 000 and 255 (Setup string) No print-files found in current directory (*.PRN) No tree data files found in current directory (*.TRE) Not a TreePlan tree data file: Although the file you selected has the .TRE extension, its internal format indicates that it was not created by TreePlan. (File Retrieve Option) Not enough disk space: The disk you are trying to write to is nearly full. Press the Escape key repeatedly to return to READY mode; change the disk or erase unneeded files; retry the option. Not enough memory: Although your computer has enough memory to start TreePlan, you need more memory to use some of TreePlan's options. Printer error: The printer is unable to print your table or tree diagram. Make sure the printer is turned on, securely connected to your computer, and online. Printer out of paper Risk attitude too extreme: The absolute value of the product of the risk coefficient R times a value or certainty equivalent cannot exceed 88. Use the Solve Risk Options to change R. Select node with only one successor branch (Node Shorten Option) TreePlan User's Manual Page 94 Sum of likelihoods must equal 1.00000: Specify inputs for Bayesian revision so that the sum of probabilities on each set of info-event branches equals 1.00000. Sum of priors must equal 1.00000 (Bayes Input Options) Sum of probabilities must equal 1.00000 Tree holds only 500 branches: The option you selected would require a tree with more than 500 branches. Appendix B: TreePlan Error Messages Page 95 Index ****************************************************************** Action, 8 Name field, 50 Alternative, 8 Nature's tree, 37 Attitude to risk, 21 Node menu options, 61 Backward induction, 16 Partial cash flow, 9 Bayes menu options, 75 Payoff distribution, 12 Bayesian revision, 34 Perfect prediction, 30 Bayes' rule, 37 Posterior probability, 35 Building a tree, 55 Preposterior probability, 36 Print menu options, 70 Certainty equivalent, 12 Prior probability, 35 Chance node, 8 Probability field, 50 Conditional probability, 35 Probability revision, 34 Configuration file, 89 R, 23 Decision node, 8 Revised prior probability, 35 Default menu options, 84 Risk attitude, 21 DriveTek problem, 7 Risk averse, 26 Risk coefficient, 23 Endpoint, 9 Risk neutral, 13 ENGS, 44 Risk profile, 12 Error messages, 92 Risk seeking, 26 Escape key, 5 Risk tolerance, 27 Event node, 8 Rollback method, 16 Event set, 8 EVPI, 30 Sample, 44 EVPI menu options, 73 Sensitivity analysis, 18, 27 EVSI, 44 Simple probability, 35 Expected utility, 22 Solve menu options, 65 Expected value, 13 Standard deviation, 69 Exponential utility, 23 State of information, 36 Extreme R, 66 Strategy, 11 File menu options, 71 Table menu options, 78 Terminal node, 9 Imperfect prediction, 44 Terminal value, 9 Info event, 35 Utility curve, 21 Joint probability, 36 Utility function, 21 Likelihood probability, 35 Valley problem, 30, 38 Lottery, 22 Value field, 50 Main event, 35 Zoom menu options, 89 Marginal probability, 35 Minimum selling price, 12 Modes, 51 TreePlan User's Manual Page 96