=9.18. You are given the following payoff table (in units of dollars):

State of Nature Alternative S1 S2 A1 400 100 A1 0 100 Prior probability 0.4 0.6 You have the option of paying $100 to have research done to better predict which state of nature will occur. When the true state of nature is S1, the research will accurately predict S1 60 percent of the time (but will inaccurately predict S2 40 percent of the time). When the true state of nature is S2, the research will accurately predict S2 80 percent of the time (but will inaccurately predict S1 20 percent of the time).

a. Given that the research is not done, use Bayes’ decision rule to determine which decision alternative should be chosen.
A

b. Use a decision tree to help find the expected value of perfect information. Does this answer indicate that it might be worthwhile to do the research?

c. Given that the research is done, find the joint probability of each of the following pairs of outcomes:
(i) the state of nature is S1 and the research predicts S1, (ii) the state of nature is S1 and the research predicts S2, (iii) the state of nature is S2 and the research predicts S1, and (iv) the state of nature is S2 and the research predicts S2.

d. Find the unconditional probability that the research predicts S1. Also find the unconditional probability that the research predicts S2.

e. Given that the research is done, use your answers in parts c and d to determine the posterior probabilities of the states of nature for each of the two possible predictions of the research.
T

f. Use the corresponding Excel template to obtain the answers for part e.
g. Given that the research predicts S1, use Bayes’ decision rule to determine which decision alternative should be chosen and the resulting expected payoff.
h. Repeat part g when the research predicts S2.
i. Given that research is done, what is the expected payoff when using Bayes’ decision rule?
j. Use the preceding results to determine the optimal policy regarding whether to do the research and the choice of the decision alternative.
A k. Construct and solve the decision tree to show the analysis for the entire problem. (Using TreePlan is optional.)