Reconsider Problem 9.4. Warren Buffy decides that Bayes' decision rule is his most reliable decision criterion. He believes that 0.1 is just about right as the prior probability of an improving economy, but is quite uncertain about how to split the remaining probabilities between a stable economy and a worsening economy. Therefore, he now wishes to do sensitivity analysis with respect to these latter two prior probabilities.

a. Reapply Bayes' decision rule when the prior probability of a stable economy is 0.3 and the prior probability of a worsening economy is 0.6.

b. Reapply Bayes' decision rule when the prior probability of a stable economy is 0.7 and the prior probability of a worsening economy is 0.2.

c. Construct a decision tree by hand for this problem with the original prior probabilities.

d. Use RSPE to construct and solve a decision tree for this problem with the original prior probabilities.

e. In preparation for performing sensitivity analysis, consolidate the data and results on the same spreadsheet as the decision tree constructed in part d (as was done in Figure 9.7 for the case study).

f. Use the spreadsheet (including the decision tree) obtained in parts d and e to do parts a and b.

g. Expanding the spreadsheet as needed, generate a data table that shows which investment Warren should make and the resulting expected profit for the following prior probabilities of a stable economy: 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9.

h. For each of the three investments, find the expected profit when the prior probability of a stable economy is 0 and then when it is 0.9 (with the prior probability of an improving economy fixed at 0.1). Plot these expected profits on a single graph that has expected profit as the vertical axis and the prior probability of a stable economy as the horizontal axis. For each of the three investments, draw a line segment connecting its two points on this graph to show how its expected profit would vary with the prior probability of a stable economy.

Use this graph to describe how the choice of the investment depends on the prior probability of a stable economy.