Math for Liberal Arts MAT 110: Chapter 12 Notes Voting: Does the - - PowerPoint PPT Presentation

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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.

Maj ority Rule

The simplest type of voting involves only two choices. With m ajority 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.

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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? What was Gore’s percentage of the popular vote?

50,456,002 0.4787 47.87% 105,405,100   50,999,897 0.4838 48.38% 105,405,100  

Settled by majority rule of electoral votes rather than by popular vote.

Variations on Maj ority 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%

• f the vote, or a unanimous vote. In these cases, a super

m ajority 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.

Maj ority Rule

Example 12.A.1: Use the preference table below to determine who won using majority rule.

Plurality Method

Plurality m ethod: The candidate with the most first- place votes wins.

• Notice it said most, not majority.

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Plurality Method

Example 12.A.2: Use the preference table below to determine who won using plurality method.

S ingle Runoff

Single (top-two) runoff m ethod: 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.

S ingle Runoff

Example 12.A.3: Use the preference table below to determine who won using single runoff method.

S equential Runoff

Sequential runoff m ethod: A series of runoffs is held, eliminating the candidate with the few est first-place votes at each stage. Runoffs continue until one candidate has a m ajority 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.

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S equential Runoff S equential Runoff S equential Runoff

Slide 12-15

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.

1st = 4 points 2nd = 3 points 3rd = 2 points 4th = 1 point

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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 com parisons (Condorcet m ethod): 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

A vs. B A vs. C A vs. D A vs. E B vs. C B vs. D B vs. E C vs. D C vs. E D vs. E

Voting Methods with Three or More Choices

• Plurality m ethod: The candidate with the most first-place

votes wins.

• Single (top-two) runoff m ethod: 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 m ethod: A series of runoffs is held,

eliminating the candidate with the few est first-place votes at each stage. Runoffs continue until one candidate has a m ajority

• f 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 com parisons: The candidate who wins

the most pairwise (one-on-one) contests is the winner of the election.

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Winner

• Everyone
• Is
• A
• Winner