# Question

Chelsea Bush is an emerging candidate for her party’s nomination for President of the United States. She now is considering whether to run in the high-stakes Super Tuesday primaries. If she enters the Super Tuesday (S.T.) primaries, she and her advisers believe that she will either do well (finish first or second) or do poorly (finish third or worse) with probabilities 0.4 and 0.6, respectively. Doing well on Super Tuesday will net the candidate’s campaign approximately $16 million in new contributions, whereas a poor showing will mean a loss of $10 million after numerous TV ads are paid for. Alternatively, she may choose not to run at all on Super Tuesday and incur no costs.

Chelsea’s advisers realize that her chances of success on Super Tuesday may be affected by the outcome of the smaller New Hampshire (N.H.) primary occurring three weeks before Super Tuesday. Political analysts feel that the results of New Hampshire’s primary are correct two-thirds of the time in predicting the results of the Super Tuesday primaries. Among Chelsea’s advisers is a decision analysis expert who uses this information to calculate the following probabilities:

P {Chelsea does well in S.T. primaries, given she does well in N.H.} = 4/7

P {Chelsea does well in S.T. primaries, given she does poorly in N.H.} = 1/4

P {Chelsea does well in N.H. primary} = 7/15

The cost of entering and campaigning in the New Hampshire primary is estimated to be $1.6 million.

Chelsea feels that her chance of winning the nomination depends largely on having substantial funds available after the Super Tuesday primaries to carry on a vigorous campaign the rest of the way. Therefore, she wants to choose the strategy (whether to run in the New Hampshire primary and then whether to run in the Super Tuesday primaries) that will maximize her expected funds after these primaries.

(a) Construct and solve the decision tree for this problem.

(b) Perform sensitivity analysis systematically by generating a data table that shows Chelsea’s optimal policy and expected payoff when the prior probability that she will do well in the New Hampshire primary is each of the following multiples of 1/15: 0, 1, 2, . , 15.

(c) Assume now that the prior probability that Chelsea will do well in the New Hampshire primary is indeed 7/15. However, there is some uncertainty in the estimates of a gain of $16 million or a loss of $10 million depending on the showing on Super Tuesday. Either amount could differ from this estimate by as much as 25 percent in either direction. For each of these two financial figures, perform sensitivity analysis to check how the results in part (a) would change if the value of the financial figure were at either end of this range of variability (without any change in the value of the other financial figure). Then do the same for the four cases where both financial figures are at one end or the other of their ranges of variability.

Chelsea’s advisers realize that her chances of success on Super Tuesday may be affected by the outcome of the smaller New Hampshire (N.H.) primary occurring three weeks before Super Tuesday. Political analysts feel that the results of New Hampshire’s primary are correct two-thirds of the time in predicting the results of the Super Tuesday primaries. Among Chelsea’s advisers is a decision analysis expert who uses this information to calculate the following probabilities:

P {Chelsea does well in S.T. primaries, given she does well in N.H.} = 4/7

P {Chelsea does well in S.T. primaries, given she does poorly in N.H.} = 1/4

P {Chelsea does well in N.H. primary} = 7/15

The cost of entering and campaigning in the New Hampshire primary is estimated to be $1.6 million.

Chelsea feels that her chance of winning the nomination depends largely on having substantial funds available after the Super Tuesday primaries to carry on a vigorous campaign the rest of the way. Therefore, she wants to choose the strategy (whether to run in the New Hampshire primary and then whether to run in the Super Tuesday primaries) that will maximize her expected funds after these primaries.

(a) Construct and solve the decision tree for this problem.

(b) Perform sensitivity analysis systematically by generating a data table that shows Chelsea’s optimal policy and expected payoff when the prior probability that she will do well in the New Hampshire primary is each of the following multiples of 1/15: 0, 1, 2, . , 15.

(c) Assume now that the prior probability that Chelsea will do well in the New Hampshire primary is indeed 7/15. However, there is some uncertainty in the estimates of a gain of $16 million or a loss of $10 million depending on the showing on Super Tuesday. Either amount could differ from this estimate by as much as 25 percent in either direction. For each of these two financial figures, perform sensitivity analysis to check how the results in part (a) would change if the value of the financial figure were at either end of this range of variability (without any change in the value of the other financial figure). Then do the same for the four cases where both financial figures are at one end or the other of their ranges of variability.

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