for the situation in which B sets the agenda and wants to
ensure that Sociology wins.
to find a Borda weighting system in which candidate B
Every year, college football’s Heisman Trophy is awarded
by means of a Borda-count voting system. Each voter submits first-, second-,
and third-place votes, worth 3 points, 2 points, and 1 point, respectively.
Thus, the Borda-count point scheme used may be called (3-2-1), where the first
digit is the point value of a first-place vote, the second digit denotes the
value of a second-place vote, and the third digit gives the point value of a
third-place vote. In 2004, the vote totals for the top five under the Borda
system were as follows:
(a) Compare the Borda point scores of Leinhart and
Peterson. By what margin of Borda points did Leinhart win?
(b) It seems only fair that a point scheme should give a
first-place vote at least as much weight as a second-place vote and a
second-place vote at least as much weight as a third-place vote. That is, for a
point scheme (x-y-z), we should have x $ y $ z. Given this “fairness”
restriction, is there any point scheme under which Leinhart would have lost? If
so, provide such a scheme. If not, explain why not.
(c) Even though White had more first-place votes than
Peterson, Peterson had a higher Borda-count total. If first-place votes were
weighted enough, White’s edge in first-place votes could give him a higher
Borda count. Assume that second-place votes are worth 2 points and third-place
votes are worth 1 point, so that the point scheme is (x-2- 1). What is the lowest
integer value of x such that White gets a higher Borda count than Peterson?
(d) Suppose that the above vote data represent truthful
voting. For simplicity, let’s suppose that the election were a simple plurality
vote instead of a Borda count. Note that Leinhart and Bush are both from USC,
whereas Peterson and White are both from Oklahoma. Suppose that, due to
Oklahoma loyalty, those voters who prefer White all have Peterson as their
second choice. If these voters were to vote strategically in a plurality
election, could they change the outcome of the election? Explain.
(e) Similarly, suppose that due to USC loyalty, those
voters who prefer Bush all have Leinhart as their second choice. If all four
voting groups (Leinhart, Peterson, White, Bush) were to vote strategically in a
plurality election, who would be the winner of the Heisman Trophy?
(f) In 2004, there were 923 Heisman voters. Under the
actual (3-2-1) system, what is the minimum integer number of first-place votes
that it would have taken to guarantee victory (that is, without the help of any
second- or third-place votes)? Note that a player’s name may appear on a ballot