Question 12.65
37. If the distribution is not unimodal, but at least of the area under the curve separates (on the left) from , and at least separates (on the right) from , is there a 2/3-separation opportunity? (Hint: Start by assuming that the distribution is uniform between and —and hence not unimodal—and that exactly of the voters lie between and . Can always win by taking a position at ? If not, is there a distribution that affords this opportunity?)
37.
Following the hint, will obtain of the vote by taking a position at , as will and , so there will be a threeway tie among the candidates. Because a non-unimodal distribution can be bimodal, with the two modes close to , can win if he or she picks up most of the vote near the two modes, enabling to win with more than of the vote. In this case, Candidate has taken advantage of a -separation opportunity.