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Plurality Method in Elections

The document discusses the plurality method for elections. It states that with the plurality method, the candidate with the most first-place votes wins, regardless of other preferences. It provides examples of how to determine the winner using the plurality method when there are multiple candidates. It also discusses fairness criteria like the majority and Condorcet criteria that the plurality method can violate.

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John Carldel Vivo
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136 views2 pages

Plurality Method in Elections

The document discusses the plurality method for elections. It states that with the plurality method, the candidate with the most first-place votes wins, regardless of other preferences. It provides examples of how to determine the winner using the plurality method when there are multiple candidates. It also discusses fairness criteria like the majority and Condorcet criteria that the plurality method can violate.

Uploaded by

John Carldel Vivo
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MATH 11008: Plurality Method

Section 1.2

The Plurality Method: With the Plurality Method all we care


about is first-place votes. The candidate with the most first-place
votes, the plurality candidate, wins the election.

◦ With the Plurality Method, we do not need the entire preference ballot since
only the first choice matters.
◦ When there are three or more candidates, plurality differs from majority.
Plurality is the greatest number of votes. Majority means more than half of
the votes. (A candidate that receives a majority of first-place votes is called
the majority candidate.) However, for two candidates, the plurality of
votes is the majority of votes.

Example 1: The Math Appreciation Society (MAS) is dedicated to the fostering of enjoyment
and appreciation of mathematics among college students. During a recent meeting, they have
four candidates running for president: Alisha (A), Boris (B), Carmen (C), and Dave (D).
Each of the 37 members of the club votes by preference ballot indicating their choices. Below
is the preference schedule for this election.

Number of voters 14 10 8 4 1
1st choice A C D B C
2nd choice B B C D D
3rd choice C D B C B
4th choice D A A A A

Use the Plurality Method to determine the president.


MATH 11008: PLURALITY METHOD
2 SECTION 1.2
Example 2: A 13-member committee is selecting a chairperson. The 3 candidates are Albert
(a), Barbara (b), and Charles (c). Each committee member completely ranked the candidates
on a separate ballot. The preference schedule is listed below.
Number of voters Ranking
4 a>b>c
2 b>c>a
4 b>a>c
3 c>a>b
Use the Plurality Method to determine the chairperson.

Example 3: Consider an election with 1025 voters.


(a) If there are 4 candidates, at least x votes are needed to have a plurality of the votes.
find x.

(b) Suppose that at least 129 votes are needed to have a plurality of the votes. What is the
number of candidates in the election?

(c) Suppose that at least 206 votes are needed to have a plurality of the votes. What is the
number of candidates in the election?

• Fairness Criterion
◦ The Majority Criterion: If candidate X has a majority of the first-place votes,
then candidate X should be the winner of the election.
∗ Note that the Plurality Method satisfies the Majority Criterion.
◦ The Condorcet Criterion: If candidate X is preferred by the voters over each
of the other candidates in a head-to-head comparison, then candidate X should be
the winner of the election.
∗ A candidate that can win all head-to-head matchups is called the Condorcet
candidate.
∗ Note that Example 1 illustrates that the Plurality Method violates the Con-
dorcet Criterion.
• Be careful! When we say that a voting method violates a fairness criterion it only means
that violations CAN happen, not that they must happen.

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