Activity Sheet 1

The Plurality Method and Other Voting Systems

6. In the first round of voting, skiing gets seventeen votes, rafting gets eleven, and caving gets twelve. Thus, rafting is eliminated. In the follow-up election, skiing gains one of rafting's votes, for a total of eighteen, whereas caving gets its original twelve votes plus ten votes from rafting, for a total of twenty two. Skiing is now eliminated, leaving caving as the winner.

7. Caving gets 2(12) + 17, or 41, points. Rafting gets 2(11) + 18, or 40, points. Skiing gets 2(17) + 5, or 39, points. Caving wins using the Borda count.

8. In the first vote, skiing gets eighteen votes to caving's twenty-two. Thus, skiing is eliminated and caving meets rafting in a head-to-head contest. This time, caving gets nineteen votes, whereas rafting gets twenty-one. Rafting wins.

9. Answers may vary; however, many students will assert that rafting gets an unfair advantage in problem 3.

10. A hint may be needed here. One possible answer is to eliminate skiing, the activity that the greatest number of voters ranked last, and then to hold runoff election between the remaining options.

Activity Sheet 2

Strategic Voting

1. An editor who was voting according to his or her true preferences would probably rank his or her school first and Big City High second, or vice versa. In this problem though, students should discover that another strategy benefits the editor's school. By ranking his or her school's team first and not including Big City High among the top ten, the editor's school gets 9(9) + 10, or 91, points compared with Big City's 9(10) + 0, or 90, points. Give some credit to groups who create a tie, but point out to them that they need not include Big City High in the top ten.

2.a. If the plurality method is used, A wins, with 48 percent of the vote compared with B's 28 percent and C's 24 percent.

2.b. If the voters in this group ranked B ahead of C, then B would win instead of A.

2.c. If the Hare system of voting is used, then C is eliminated first. In the next round, B wins with 52 percent of the vote. The last 10 percent of the voters would be most disappointed with this result. If they submitted a ballot with the ranking C, A, B, then B would be eliminated in the first round and A would beat C.

Activity Sheet 3

Tournament Digraphs and Condorcet Winners

4. The tournament digraph follows:

The exact arrangement of the vertices W, X, Y, and Z is not important. To check students' graphs, verify that exactly one arrow appears between every pair of vertices, three arrows leave W, three arrows point toward Z, and an arrow goes from X to Y.

5. By using the Borda count, W gets 3(3) + 2(2) + 2 = 15 points, X gets 3(4) + 2(3) = 18 points, Y gets 2(2) + 7 = 11 points, and Z gets 3(2) + 2(2) = 10 points, so X wins. Have students verify that they have calculated the right number of total points for each option.

6. One example is given by the following table:

Here, W is a Condorcet winner that gets eliminated in the first round of Hare voting.

7. Since sequential pairwise voting involves only head-to-head contests, a Condorcet winner will win every contest it is in and hence wins the election.