Math for Liberal Arts MAT 110: Chapter 12 Notes

Transcription

1 Math for Liberal Arts MAT 110: Chapter 12 Notes Voting Methods David J. Gisch Voting: Does the Majority Always Rule? Choosing a Winner In elections with more then 2 candidates, there are several acceptable ways to choose a winner. Different methods can lead to different winners. Majority Rule The simplest type of voting involves only two choices. With majority rule, the choice receiving more than 50% of the vote is the winner. Some properties of majority rule are listed below. Every vote has the same weight. If a vote for the loser were changed to a vote for the winner, the outcome of the election would not be changed. Majority should not confused with the most votes. Say 12 people voted and the results were 3 Sue 4 Ted 5 Uriel No one received a majority so on one won the contest. 1

2 Mathematics and Politics Table 12.1 shows the official results of the 2000 U.S. presidential election. What was Bush s percentage of the popular vote? 50, 456, % 105,405,100 What was Gore s percentage of the popular vote? 50,999, % 105,405,100 Settled by majority rule of electoral votes rather than by popular vote. Variations on Majority Rule In some cases, a candidate or issue must receive more than a majority of the vote to win such as 60% of the vote, 75% of the vote, or a unanimous vote. In these cases, a super majority is required. A 60% super majority is required to end a filibuster in the U.S. Senate. A 2/3 super majority is required by both the House and Senate to amend the U.S. Constitution. Majority Rule Example 12.A.1: Use the preference table below to determine who won using majority rule. Plurality Method Plurality method: The candidate with the most firstplace votes wins. Notice it said most, not majority. 2

3 Plurality Method Example 12.A.2: Use the preference table below to determine who won using plurality method. Single Runoff Single (top-two) runoff method: The two candidates with the most first-place votes have a runoff. The winner of the runoff is the winner of the election. When using a preference table you do need a runoff. You simply eliminate candidates and move the votes up. Single Runoff Example 12.A.3: Use the preference table below to determine who won using single runoff method. Sequential runoff method: A series of runoffs is held, eliminating the candidate with the fewest first-place votes at each stage. Runoffs continue until one candidate has a majority of the first-place votes and is declared the winner. When using a preference table you do need a runoff. You simply eliminate candidates and move the votes up. 3

4 Slide Borda (points) Point system (Borda count): Points are awarded according to the rank of each candidate on each ballot (first, second, third, ). The candidate with the most points wins. Say we had a ballot with first, second, third, and fourth. 1 st = 4 points 2 nd = 3 points 3 rd = 2 points 4 th = 1 point 4

5 Borda Count (Points) A = 5(18) + 1(12) + 1(10) + 1(9) + 1(4) + 1(2) = 127 B = 1(18) + 5(12) + 4(10) + 2(9) + 4(4) + 2(2) = 156 C = 2(18) + 2(12) + 5(10) + 4(9) + 2(4) + 4(2) = 162 D = 4(18) + 3(12) + 2(10) + 5(9) + 3(4) + 3(2) = 191 E = 3(18) + 4(12) + 3(10) + 3(9) + 5(4) + 5(2) = 189 Condorcet (one-on-one) Method of pairwise comparisons (Condorcet method): The candidate who wins the most pairwise (one-on-one) contests is the winner of the election. You can use the preference ballot to make the comparisons. Pairwise Comparison B vs. D B vs. E C vs. D C vs. E D vs. E A vs. B A vs. C A vs. D A vs. E B vs. C Voting Methods with Three or More Choices Plurality method: The candidate with the most first-place votes wins. Single (top-two) runoff method: The two candidates with the most first-place votes have a runoff. The winner of the runoff is the winner of the election. Sequential runoff method: A series of runoffs is held, eliminating the candidate with the fewest first-place votes at each stage. Runoffs continue until one candidate has a majority of the first-place votes and is declared the winner. Point system (Borda count): Points are awarded according to the rank of each candidate on each ballot (first, second, third, ). The candidate with the most points wins. Method of pairwise comparisons: The candidate who wins the most pairwise (one-on-one) contests is the winner of the election. 5

6 Winner Everyone Is A Winner 6