The analyst then represents the choices, uncertainties, and outcomes that the expert identified in a graphical structure by using a decision tree (Figure 29.3) or some other structure described below. By using the standard symbols for building decision trees (Raiffa, 1968), the decision analyst draws a rectangle, called a decision node, to represent the decision (the node issue BWA in Figure 29.3) and two branches (lines) to represent the two possible choices: act now and issue a
Wo: 1-p figure 29.3 Decision tree model.
boil-water advisory, or wait 72 hours for the results of testing of sick individuals and testing of water.2
The decision analyst draws circles, called chance nodes, to represent uncertain propositions (the chance node Crypto outbreak represents the proposition that there is a Cryptosporidium outbreak). The decision analyst draws two chance nodes because both choices are affected by uncertainty. These chance nodes are identical because both choices are affected by the same uncertainty. Two branches emanate from each chance node to represent the two values of the proposition, i.e., yes, it is true that there is a Cryptosporidium outbreak, and no, it is not true. This decision tree has four outcomes. Each outcome represents a different future scenario. For example, the topmost outcome, described by the variable cb1, represents the outcome of acting now and subsequently discovering that a Cryptosporidium outbreak indeed was in progress.
One can understand a decision tree as a model of what might happen in the future based on a decision that we make in the present. The chance nodes represent fate. No matter what the decision maker decides, only fate will determine whether he is a hero or a goat. If the decision maker decides to issue a boil-water advisory and is right (i.e., three days later the laboratory confirms his wisdom), he is a hero. If he decides to issue a boil-water advisory and he is wrong (the second end point of the model), he is a goat. He could decide to await further testing and also either be a hero or a goat (the fourth and the third end points of the decision tree). However, it is important to note that a bad outcome (goat) does not necessarily mean a bad decision. As we will show when we fold back the decision tree, the model will recommend a decision that in the long run (i.e., if, hypothetically, the decision maker makes the same decision the next 1000 times he faces this situation), will be most beneficial to him (or the organization that he represents).
It is worth emphasizing that a key benefit of a decision analysis is that it establishes rational criteria for taking difficult decisions under uncertainty. If a decision maker makes the decision recommended by the model, he can defend his decision by reference to the principle of maximum expected utility. The psychology of decision making is such that decision makers tend to be overly conservative about taking actions for which they may have remorse (Tversky and Kahneman, 1974). If the decision maker makes a decision that has been carefully analyzed in advance, taking into account the values of his community as expressed by the costs used in the model, as well as all available information expressed as chance nodes in the model, then it is more likely he will feel comfortable that he will be perceived as a responsible decision maker, whatever the ultimate outcome of a particular decision is.
For completeness, we shall briefly digress in this paragraph to explain that decision analysts may use alternative graphical models, such as influence diagrams (Howard and Matheson, 1984b). Readers simply need to be aware that there are graphical models other than decision trees. Figure 29.4 is an influence diagram representation of the boil-water decision. A decision analyst may use influence diagrams when modeling decision problems with more complex probabilistic structure. He may use dynamic influence diagrams (Tatman and Shachter, 1990) or the influence view formalism (Leong, 1994) when modeling a decision that is revised frequently as new information accumulates. The decision analyst may use a Markov decision process (Beck and Pauker, 1983; Dean and Wellmann, 1991), or combination of Markov decision process and the influence view formalism (Magni et al., 2000) to model a series of different decisions that are made over time (e.g., decisions made during an outbreak investigation).
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