Section 15.1
Voting
Methods
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
INB Table of Contents
2.3-2
Date
Topic
June 24, 2015
Section 15.1 Examples
38
June 24, 2015
Section 15.1 Notes
39
June 24, 2015
Section 15.2 Examples
40
June 24, 2015
Section 15.2 Notes
41
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Page #
What You Will Learn

Plurality Method

Borda Count Method

Plurality with Elimination

Pairwise Comparison Method

Tie Breaking
15.1-3
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Example 1: Voting for the Honor
Society President
Four students are running for president of the Honor
Society: Antoine (A), Betty (B), Camille (C), and Don
(D). The club members were asked to rank all
candidates.
15.1-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Voting for the Honor
Society President
a)
b)
c)
15.1-6
How many students voted in the election?
How many students selected the candidates in this
order: C, A, D, B?
How many students selected A as their first choice?
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plurality Method

This is the most commonly used method, and
it is the easiest method to use when there are
more than two candidates.

Each voter votes for one candidate.

The candidate receiving the most votes is
declared the winner.
15.1-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Electing the Honor
Society President by the Plurality
Method
Consider the Honor Society election given in
Example 2. Who is elected president using the
plurality method?
15.1-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Borda Count Method

Voters rank candidates from the most favorable
to the least favorable.

Each last-place vote is awarded one point, each
next-to-last-place vote is awarded two points,
each third-from-last-place vote is awarded
three points, and so forth.

The candidate receiving the most points is the
winner of the election.
15.1-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Electing the Honor
Society President Using the
Borda Count Method
Use the Borda count method to determine the winner of the election
for president of the Honor Society discussed in Example 2. Recall that
the candidates are Antoine (A), Betty (B), Camille (C), and Don (D).
15.1-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plurality with Elimination

Each voter votes for one candidate.

If a candidate receives a majority of
votes, that candidate is declared the
winner.
15.1-21
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plurality with Elimination

If no candidate receives a majority, eliminate
the candidate with the fewest votes and hold
another election. (If there is a tie for the
fewest votes, eliminate all candidates tied for
the fewest votes.)

Repeat this process until a candidate receives
a majority.
15.1-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 4: Electing the Honor
Society President Using the
Plurality with Elimination Method
Use the plurality with elimination method to determine the
winner of the election for president of the Honor Society
from Example 2.
15.1-23
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Pairwise Comparison Method

Voters rank the candidates.

A series of comparisons in which each
candidate is compared with each of the other
candidates follows.

If candidate A is preferred to candidate B, A
receives one point. If candidate B is preferred
to candidate A, B receives 1 point. If the
candidates tie, each receives ½ point.
15.1-33
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Pairwise Comparison Method
After making all comparisons among
the candidates, the candidate receiving
the most points is declared the winner.
15.1-34
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Number of Comparison
The number of comparisons, c, needed
when using the pairwise comparison
method when there are n candidates is
n(n  1)
c
2
15.1-35
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: Electing the Honor Society President
Using the Pairwise Comparison Method
Use the pairwise comparison method to determine the
winner of the election for president of the Honor Society
that was originally discussed in Example 2.
15.1-36
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Tie Breaking

Breaking a tie can be achieved by either
making an arbitrary choice, such as
flipping a coin, or by bringing in an
additional voter.

Robert’s Rule of Order: president of group
votes only to break a tie or create a tie.

Borda method: could choose person with
most 1st place votes.
15.1-46
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Tie Breaking

Pairwise comparison method: could
choose the winner of a one-to-one
comparison between the two candidates
involved in the tie.

Different tie-breaking methods could
produce different winners.

To remain fair, the method should be
chosen in advance.
15.1-47
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 15.2
Flaws of
Voting
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn

Fairness Criteria

Majority Criterion

Head-to-Head Criterion

Monotonicity Criterion

Irrelevant Alternative Criterion
15.2-49
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Fairness Criteria
Mathematicians and political scientists have
agreed that a voting method should meet the
following four criteria in order for the voting
method to be considered fair.

Majority Criterion

Head-to-head Criterion

Monotonicity Criterion

Irrelevant Alternatives Criterion
15.2-50
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Majority Criterion
If a candidate receives a majority
(more than 50%) of the first-place
votes, that candidate should be
declared the winner.
15.2-51
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Head-to-Head Criterion
If a candidate is favored when
compared head-to-head with every
other candidate, that candidate should
be declared the winner.
15.2-52
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Monotonicity Criterion
A candidate who wins a first election
and then gains additional support
without losing any of the original
support should also win a second
election.
15.2-53
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Irrelevant Alternatives Criterion
If a candidate is declared the winner of
an election and in a second election
one or more of the other candidates is
removed, the previous winner should
still be declared the winner.
15.2-54
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Summary of the Voting Methods and
Whether They Satisfy the Fairness Criteria
Borda
count
Plurality
Pairwise
with
comparison
elimination
Always
satisfies
May not
satisfy
Always
satisfies
Always
satisfies
May not
satisfy
May not
satisfy
May not
satisfy
Always
satisfies
Monotonicity Always
satisfies
May not
satisfy
May not
satisfy
May not
satisfy
Irrelevant
May not
alternatives satisfy
May not
satisfy
May not
satisfy
May not
satisfy
Method Plurality
Criteria
Majority
Head-tohead
15.2-55
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Arrow’s Impossibility Theorem
It is mathematically impossible for any
democratic voting method to
simultaneously satisfy each of the
fairness criteria:
• The majority criterion
• The head-to-head criterion
• The monotonicity criterion
• The irrevelant alternative criterion
15.2-56
Copyright 2013, 2010, 2007, Pearson, Education, Inc.