In question 19, since I.M. Hogg is risk neutral, this implies that his objective is to determine the number of negative campaign ads which will maximize the expected number of voters. The number of negative campaign ads is represented by n, where n is an integer value ranging from 0 to 4.
In order to solve this problem, the first step involves setting up an equation for the expected number of voters (E(voters)). One hundred percent of the time, I.M. Hogg gets 500,000 votes; this represents his “core” base of voters. The probability that the ads backfire is 0.2n, in which case he fails to obtain any more votes beyond his base of 500,000 voters. However, the probability that the ads don’t backfire is 1-0.2n, in which case he gets an additional 100,000 + 40,000n votes (over and above his base of 500,000 voters). Anyway, once you have your equation for E(voters), you can maximize it by differentiating it with respect to n, setting the resulting equation equal to 0, and solving for n. Since n will not likely be an integer, simply round it to the nearest integer value.