Thus far, the “Making Decisions” series has presented tools and processes used primarily for prioritization or single selection decisions. Decision trees, in contrast, can be used to aid strategy decisions by mapping a series of possible events and outcomes. Its graphical format allows a decision tree to present a substantial amount of information, while the logical progression of strategy decisions remains clear and easy to follow. The use of probabilities and monetary values of outcomes provides for a straightforward comparison of strategies. Constructing a Decision Tree In lieu of abstract descriptions and generic procedures, decision tree development will be elucidated by working through examples. To begin, the simple example posed by Magee (1964) of the decision to host a cocktail party indoors or outdoors, given there is a chance of rain, will be explored. In this example, there is one decision with two options (indoors or outdoors) and one chance event with two possible outcomes (rain or no rain, a.k.a. “shine”). This combination yields four possible results, as shown in the payoff table of Exhibit 1. It could also be called an “outcome table,” as it only contains qualitative descriptions; use of the term “payoff table” becomes more intuitive with the introduction of monetary values of outcomes. While the payoff table is useful for organizing one’s thoughts pertaining to the potential results of a decision, it lacks the intuitive nature of a graphical representation. For this, the information contained in the payoff table is transferred to a decision tree, as shown in Exhibit 2. Read from left to right, a decision tree presents a timeline of events. In this case, there is the decision (indoors or outdoors), represented by a square, followed by a chance event (rain or shine), represented by a circle, and the anticipated outcome of each combination (“disaster,” “real comfort,” etc.), represented by triangles. The square and circles are nodes and each triangle is a leaf; each are connected by branches. A summary of decision tree elements is provided in Exhibit 3. The decision tree of Exhibit 2 provides an aesthetically pleasing presentation of the cocktail party host’s dilemma. However, beyond organizing information, its descriptive nature has done little to assist the host in making the required decision. The power of the decision tree becomes evident when the financial implications of decisions are presented for analysis and comparison. To demonstrate this, we modify the cocktail party example to be more than a purely social gathering. Let’s say it is a fundraising event where attendees’ generosity is directly linked to their satisfaction with the event (i.e. comfort level). This hypothetical fundraising scenario results in the payoff table of Exhibit 4 and the decision tree of Exhibit 5 that present anticipated amounts to be raised in each situation. Clearly, this is an important decision; the collection potential ranges from $30,000 to $100,000. However, the host is no closer to a rational decision, left to clutch a rabbit’s foot or make the decision with a dartboard. What information would help the host decide? The probability of rain, naturally! The local meteorologist has forecast a 60% chance of rain during the fundraiser; this information is added to the decision tree, as shown in Exhibit 6. The best decision is still not obvious, but we are getting close now! To compare the relative merits of each option available, we calculate expected values. The expected value (EV) of an outcome is the product of its anticipated monetary value (payoff) and the probability of its occurrence. For example, a rainy outdoor event has an anticipated payoff of $30,000 and a 0.60 probability of occurrence. Therefore, EV = $30,000 x 0.60 = $18,000. Likewise, the expected value of an outdoor event with no rain is EV = $100,000 x 0.40 = $40,000. The EV of a chance node is the sum of its branches’ EVs; thus EVout = $18,000 + $40,000 = $58,000 for an outdoor event. The expected value of an indoor event is calculated in the same way; EVin = $62, 000. Expected value calculations are shown to the right of the decision tree in Exhibit 7. The calculations are not typically displayed; they are included here for instructional purposes. The EV of a decision node, or its position value, is equal to that of the preferred branch – the one with the highest payoff or lowest cost (if all are negative). Hence, the position value of this decision is $62,000 and the event will be held indoors. Two hash marks are placed through the “outdoors” branch to signify that it has not been selected; this is called “pruning” the branch. The completed, pruned decision tree is shown in Exhibit 7. The process of calculating expected values and pruning branches is performed from right to left. Reflecting the reversal of direction, this process is called “rolling back” or “folding back” the decision tree, or simply “rollback.” Use of a decision tree is advantageous for relatively simple situations like the example above. Its value only increases as the decision environment becomes more complex. Again, this is demonstrated by expanding the previous example. In addition to the indoor/outdoor decision, consideration will also be given to a public fundraiser as well as the private, “invitationonly” event previously presented. While public events tend to extract smaller individual donations, increased attendance can offset this, though attendees’ generosity remains linked to their comfort. An expanded payoff table, presented in Exhibit 8, includes fundraising estimates for a public event. (This format was chosen because a threedimensional table is graphically challenging; tabular representation of longer decision chains become confusing and impractical.) The decision tree, shown in Exhibit 9, now presents two decisions and one chance event. The upper decision branch consists of the previous decision tree of Exhibit 7, while the lower branch presents the public event information. In this scenario, the private cocktail party is abandoned (its branch is pruned); instead, a public event will be held. The position value of the root node – the first decision to be made – is equal to the EV of the preferred strategy. The path along the retained branches, from root node to leaf, defines the preferred strategy. Expanding further the fundraising event example provides a more realistic decisionmaking scenario, where decisions and chance events are interspersed. This example posits that the fundraiser is being planned for the political campaign of a candidate that has not yet received a nomination. Pundits have assigned a 50% probability that the candidate will prevail over the primary field to receive the nomination. Given this information, the campaign manager must decide if an event venue and associated materials should be reserved in advance. From experience, the campaign manager believes that costs will double if event planning is postponed until after the nomination is secured. The decision tree for the prenomination reservation decision is presented in Exhibit 10. The format has been changed slightly from previous examples in order to present some common attributes. Larger trees, in particular, benefit from reducing the “clutter” created by the presentation of information along the branches. One way to do this is to align information in “columns,” the headings of which clarifies a more succinct branch label than would otherwise be possible. In this example, the headings provide a series of questions to which “yes” or “no” responses provide all the information required to understand the tree. Reading the tree is easier when it is not crowded by the larger labels of “indoors,” “private,” “nominated,” etc. The upper right portion of Exhibit 10 contains the same tree as Exhibit 9, though it has been rearranged. The tree in Exhibit 10 is arranged such that the topmost branches represent “yes” responses to the heading questions. While this is not required, many find decision trees organized this way easier to read and follow the logic of the chosen strategy. The example of Exhibit 9 utilized a single monetary value, the total funds raised, to make decisions. It could be assumed that this is a net value (expenses had been deducted prior to constructing the tree) or that expenses are negligible. In this regard, Exhibit 10, again, presents a more realistic scenario where expenses are significant and worthy of explicit analysis. The cost of hosting each type of event is presented in its own column and requires an additional calculation during rollback. The position value of the decision node is no longer equal to the EV of the subsequent chance node as in previous examples. To determine the position value of the indoors/outdoors decision, the cost of the preferred venue must be deducted from the EV of its branch. A noteworthy characteristic of this example is the atypical interdependence of the value of a branch. Notice that the cost of the “no event” branch following a decision to reserve a venue is $16,000. This cost was determined by first rolling back the upper portion of the tree. The cost of the preferred branch in the “nominated” scenario is $16,000; therefore, this is the cost to be incurred, whether or not an event is held, because it follows the decision to reserve a venue. The cost of the preferred venue is simply duplicated on the “no event” branch. It is important not to overlook this cost; it reduces the EV of the “Nominated?” chance node by $8,000 ($16,000 cost x 0.50 probability). Another interesting characteristic of the campaign fundraiser decision, as seen in Exhibit 10, is that the position value of the reservation decision is the same, regardless of the decision made. The example was not intentionally designed to achieve this result; values and probabilities were chosen arbitrarily. It is a fortunate coincidence, however, as it provides additional realism to the scenario and allows us to explore more fully the role of this, or any, decisionmaking aid. An inexperienced decisionmaker may, upon seeing equal EVs at the root decision, throw up his/her hands in frustration, assuming that the effort to construct the decision tree was wasted. This, of course, is not true; you may have realized this already. A theme running throughout the “Making Decisions” series is that decisionmakers are aided, not supplanted, by the tools presented. Expected value is only one possible decision criterion; further analysis is needed. Notice that the “no” branch of the “Reserve?” decision has been pruned in Exhibit 10. Why? A thorough answer requires a discussion of risk attitudes, introduced in the next section. For now, we will approach it by posing the following question: Given the expected value of a reservation is equal to that of no reservation, is it advisable to make the reservation anyway? Consider a few reasons why it is:
Advanced Topics in Decision Tree Analysis The discussion of decision tree analysis could be expanded far beyond the scope of this post. Only experienced practitioners should attempt to incorporate many of the advanced elements, lest the analysis go awry. This section serves to expose readers to opportunities for future development; detailed discussions may appear in future installments of “The Third Degree.” Eager readers can also consult references cited below or conduct independent research on topics of interest. Risk attitudes, mentioned in the previous section, influence decisions when analysis extends beyond the decision tree. The strategy described in the example – reserving a venue for an indoor public event (see Exhibit 10) – is a risk averse strategy; it seeks to avoid the risks associated with foregoing a reservation. Some of these risks were mentioned in the justification for the equalEV decision presented and include economic and noneconomic factors. If the campaign manager has a risk seeking attitude, s/he is likely to reserve an outdoor venue for a public event, despite the lower expected value of such a decision. A risk seeking decisionmaker will pursue the highest possible payoff, irrespective of its expected value or the risks involved. In the example, this corresponds to the $140,000 of contributions anticipated at a public outdoor event. Risk neutral decisionmakers accept the strategy with the highest expected value without further consideration of the risks involved. This option does not exist in our example; further consideration was required to differentiate between two strategies with equal expected values. If a decisionmaker is accustomed to using utility functions, expected values can be replaced with expected utilities (EU) for purposes of comparing strategies. A concept related to both risk attitude and expected utility is the certainty equivalent (CE). A decision’s CE is the “price” for which a decisionmaker would transfer the opportunity represented by the decision to another party, foregoing the upside potential to avoid the downside risk. The examples presented have assumed a short time horizon; the time that elapses between decisions and/or chance events is not sufficient to influence the strategy decision. Decisions made on longer time horizons are influenced by the time value of money (TVM). The concept of TVM is summarized in the following way: an amount of money received today is worth more than the same amount received on any future date. Thus, expected values change over long time horizons; the extent of this change may be sufficient to alter the preferred strategy. There are numerous software tools available to construct decision trees; many are available online and some are free. The output of some will closely resemble the decision trees presented here, using conventional symbols and layout. Others differ and may be more difficult to read and analyze. While softwaregenerated decision trees are often aesthetically pleasing, suitable for presentation, those drawn freehand while discussing alternatives with colleagues may be the most useful. Use of the venerable spreadsheet is also a viable option. Generating a graphical presentation of a decision tree in a spreadsheet can be tedious, but it offers advantages unavailable in other tools. First and foremost, the ubiquity of spreadsheets ensures a high probability that anyone choosing to engage in decision tree analysis already has access to and familiarity with at least one spreadsheet program. With sufficient experience, one can forego the tedium of generating graphics until a formal presentation is required. Such a spreadsheet might look like that in Exhibit 11, which shows the campaign fundraiser example in a nongraphical format. Each branch of the decision tree is shown between horizontal lines; pruned branches are shaded. Expected values of chance nodes are labeled “EV” and those of decision nodes are labeled “position value” to differentiate them at a glance without use of graphic symbols. Calculations and branch selections/pruning can be automated in the spreadsheet. A significant advantage of the spreadsheet is the ability to quickly perform simple sensitivity analysis. Several scenarios can be rapidly explored by adjusting payoffs and probabilities and allowing the spreadsheet to recalculate. More sophisticated analysis tools are also available in most spreadsheet programs; only experienced users should attempt to utilize them for this purpose, however.
Decision Tree Tips in Summary To conclude this introduction, the final section consists of brief reminders and tips that facilitate effective decision tree analysis. The example decision trees presented are symmetrical, but this need not be the case. Likewise, decisions and chance events need not be binary; three, four, or more options or potential outcomes may be considered. However, the rule remains: the potential outcomes of a chance event must be mutually exclusive (only one can occur) and collectively exhaustive (probabilities sum to 1, or 100%). The examples presented considered positive position values. However, EVs and position values could be negative. For example, there may be multiple options for facility upgrades to maintain regulatory compliance. None will increase revenue, so all EVs are negative. In such case, the preferred branch would be the one reflecting the lowest cost. All relevant costs and only relevant costs should be included in EV calculations. Sunk costs and any others that occur regardless of the strategy selected should be excluded. Taxes and interest charges may be relevant, as are opportunity costs and noneconomic factors. Construct the decision tree from left to right; roll back from right to left. Format it to be easily expanded when new alternatives are identified or new events or decisions need to be included in the analysis. To compact a decision tree for presentation, branches can be hidden, allowing the position value of the origin decision to represent the hidden branches. Also, consider combining branches when events occur together or are otherwise logically coupled. Branches should only be combined when doing so does not eliminate a viable alternative strategy. Reviews of decision trees should take place as events unfold and after the final outcome has occurred. Adjustments may not be possible to the strategy under review, but estimates of probabilities and payoffs may be improved for future decisions. Use of decision trees for organizational decisionmaking is consistent, analytical, and transparent. However, decision tree analysis exhibits the same weakness as all other decisionmaking aids – it is dependent upon reliable estimates and sound judgment to be effective. Review of past decisions and outcomes is critical to increasing the quality of an organization’s decisions. For additional guidance or assistance with decisionmaking or other Operations challenges, feel free to leave a comment, contact JayWink Solutions, or schedule an appointment with a “strategy arborist.” For a directory of “Making Decisions” volumes on “The Third Degree,” see Vol. I: Introduction and Terminology. References [Link] “Decision Trees for Decision Making.” John F. Magee. Harvard Business Review, July, 1964. [Link] “Decision Tree Primer.” Craig W. Kirkwood. Arizona State University, 2002. [Link] “Decision Tree Analysis: Choosing by Projecting ‘Expected Outcomes.’” Mind Tools Content Team. [Link] “Decision Tree: Definition and Examples.” Stephanie Glen, September 3, 2015. [Link] “What is a Decision Tree Diagram.” LucidChart. [Link] Mathematical Decision Making: Predictive Models and Optimization. Scott P. Stevens. The Great Courses, 2015. Jody W. Phelps, MSc, PMP®, MBA Principal Consultant JayWink Solutions, LLC jody@jaywink.com
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