## Solving Folding Back the Decision Tree

Once the decision analyst has elicited all the required numerical parameters, he can mathematically "solve'' the decision tree.

Figure 29.6 depicts the solved decision tree. For each of the choices, the decision analyst calculates the expected value according to the principle of maximum expected utility. The expected value of act now is p.cb1 + (1 -p). cb2=0.041 x -\$89,315,780.49 + 0.959 x -\$8,025,000 = -\$11,357,922. The expected value of wait is p. cb3 + (1 -p). cb4=0.041 x -\$115,454,219.5 + 0.959 x \$0 = \$4,733,623 (these equations are identical in form to those we used in analyzing the \$10 bet on a horse race earlier). Keep in mind that this decision model is a very simple one. More elaborate models may have additional chance and decision nodes. In such models, the decision analyst repeats the above calculations for each chance node, starting from the rightmost chance nodes. Once the value of a chance node is calculated, he substitutes the value for the node, allowing us to compute the value of the parent node. The repetitive nature of this right-to-left process is why the process of evaluating a decision tree is sometimes called folding back the tree.

The decision analyst and the expert now have the results of the base-case analysis. When the surveillance data indicate a posterior probability of Cryptosporidium equal to 0.041, the expected cost of issuing an immediate boil-water advisory is -\$11,357,922, whereas the expected cost of waiting for confirmation is -\$4,733,623. Hence, the optimal decision is to wait three more days until laboratory confirmation of the outbreak before issuing a boil-water advisory (under the assumptions of our example analysis, which are not sufficiently realistic to guide real-world decision making without refinement).

The results of the base-case analysis suggesting testing to the point of definitive diagnosis were not surprising.The probability that there was a Cryptosporidium outbreak was low, and the cost of illness was low. We set the cost of issuing a boil-water advisory high (we set the cost of a boil-water advisory \$1 per person per day). figure 29.6 Solved decision tree.

5 The value of \$238 was adjusted from 1993 dollars to 2005 dollars.

6 Please note, that the cost of sickness depends on what day the outbreak is detected.

Cryptosporidium is a mild disease that in many cases does not require a patient to seek medical attention. If we were to conduct a similar analysis for a disease with a high mortality rate, such as anthrax, and assumed a three-day delay for a confirmation, the set of circumstances would be far narrower in which waiting instead of acting to control the disease would be the optimal decision.The maximum time for waiting for confirmatory testing might be measured in hours.We expect decision analyses of such situations to produce recommendations for action that may differ significantly from current practice.