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Module-1 - Math
Psychology (St. Michael's College (Iligan))
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Contents
Voting Methods and Apportionment Outline
1
Preference Tables and the Plurality Method
2
i
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Voting Methods and Apportionment
CONTENT STANDARD
In this module, the student will able to demonstrates understanding of key concept, uses
and importance of voting and apportionment in real life situation.
Learning Content
Preference Tables and
the Plurality Method
The Borda Count
and the
Plurality-with-Elimination
Method
The Pairwise Comparison
Method and Approval
Voting
Apportionment
Apportion Flaws
Learning Objective
• Interpret the information in a preference table.
• Determine the winner of an election using the plurality
method
• Decide if an election violates the head-to-head comparison
criterion
• Determine the winner of an election using the Borda Count
method
• Decide if an election violates the majority criterion.
• Determine the winner of an election using the the pluralitywith-elimination method
• Decide if an election violates the monotonicity criterion
• Determine the winner of an election using the pairwise
comparison method
• Decide if an election violates the irrelevant alternatives
criterion.
• Describes Arrow’s impossibility theorem.
• Determine the winner of an election approval voting.
• Compute standard divisors and quotas.
• Apportion seats using Hamilton’s method.
• Apportion seats using Jefferson’s method.
• Apportion seats using Adam’s method.
• Apportion seats using Webster’s method.
• Apportion seats using the Huntington-Hill method.
• Illustrate the Alabama Paradox.
• Illustrate the population paradox.
• Illustrate the new states paradox.
• Describe the quota rule.
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Lesson 1: Preference Tables and the Plurality Method
Learning Objectives:
1. Interpret the information in a preference table.
2. Determine the winner of an election using the plurality method.
3. Decide if an election violates the head-to-head comparison criterion.
PRELIMINARIES:
Voting seems like such a simple idea: two candidates both
want a position, and whichever one gets the most votes wins.
But like most things in the modern world, elections rarely
turn out to be as simple as they appear. The most obvious complication arises when there are more than two candidates. Should the winner just be the one who gets the most
votes, even if less than half of the votes want him or her in
office? Maybe voters should rank the candidate in order of
preferences . . . but then how do we decide on the winner?
We will begin our study of voting methods by examining a method for summarizing the
results when candidates are ranked in order of preference by voters. We’ll then study the
simplest of the methods of determining the winner of an election, and begin a study of the
weaknesses inherent in different voting systems.
LESSON DEVELOPMENT:
Preference Tables
Suppose there are three candidates running for club president. We’ll call them A, B and C.
Instead of simply voting for the single candidate of your choice, you are asked to rank each
candidate in order of preference. This type of ballot is called a preference ballot.
In this case, there are six possible ways to rank the candidates, as shown.
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Now, suppose that the 20 club members voted as follows.
Of the 6 possibe rankings, only 4 appear in the 20 ballots. Nine people voted for the
candidates in order of preference ABC, five people voted ACB, four people voted BCA, and
two people voted CBA.
A preference table can be made showing the results.
Number of Votes
First choice
Second choice
Third choice
9 5 4 2
A A B C
B C C B
C B A A
The sum of the numbers in the top row indicates the total number of votes. Also note
that 9 + 5 or 14 voters picked candidate A as their first choice, 4 picked candidates B as
their first choice, and 2 voters picked candidate C as their first choice.
Because no voters cast ballots ranking as BAC or CAB, those possible rankings are not
listed as column in the table.
EXAMPLE 1: Interpreting a Preference Table
Four candidates, W , X, Y , and Z, are running for student government president. The students were asked to rank all candidates in order of preference. The results of the election
are shown in the preference table.
Number of Voters
First Choice
Second Choice
Third Choice
Fourth Choice
a.
b.
c.
d.
86
X
W
Y
Z
42
W
Z
X
Y
19
Y
Z
X
W
13
X
Z
W
Y
40
Y
X
Z
W
How many students voted?
How many people voted for candidates in the order, Y , Z , X, W ?
How many students picked candidate Y as their first choice?
How many students picked candidate W as their first choice?
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Solution:
1. To find the total number of voters, find the sum of the numbers in the top row.
86 + 42 + 19 + 13 + 40 = 200
2. The ranking Y , Z , X, W is in the third column of the table, which is headed by the
number 19. This means that 19 voters chose that order.
3. There were 19 voters whose Y , Z , X, W (third column) and 40 that chose Y , X , Z, W
(fifth column), and those are the only rankings with Y listed first. So, 19 + 40 = 59
voters listed candidate Y first.
4. Only one ranking has candidate W first — the one in the second column. There were
42 voters who submitted that order, so 42 people chose candidate W as their first
choice.
TRY THIS ONE!
1
The student Activities Committee at St. Michael’s College is choosing for an end-of-year
banquet, and they ask al members to list the four possible location in order of preference.
The choices are Appetina Restaurant (A), Levan HillTop View Restaurant (B), Sunburst
Restaurant (C), and Am’s Chicken Restaurant (D). The results are shown in the preference
table.
Number of Voters 19 13 12 9 4 2
First Choice
C B C C A B
Second Choice
B C A B C A
A D B D D D
Third Choice
D A D A B C
Fourth Choice
a. How many members voted?
Members Voted:
b. How many listed Sunburst Restaurant as their first choice?
Sunburst Restaurant(first choice):
c. How many members listed Sunburst Restaurant and Am’s Chicken Restaurant in
their top two?
Sunburst Restaurant(top two):
Am’s Chicken Restaurant (top two):
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Math Note: Plurality does not necessarily mean majority; it is simply means more votes
than any other candidate receives. Majority means more than 50 % of the votes cast.
We shall now study four common voting methods.
The Plurality Method
The simplest method of determining a winner in an election with three or more candidates is called the plurality method
In an election with three or more candidates that uses the plurality method to determine a
winner, the candidate with the most first-place votes in the winner
EXAMPLE 2: Using the Plurality Method
The preference table for a club presidential election consisting of three candidates is shown.
Number of Votes
First choice
Second choice
Third choice
4 7 5 4
B A C B
C C A A
A B B C
a. Using the plurality method, determine the winner.
b. Can you make an argument as to why candidate B shouldn’t win the election?
Solution:
a. In this situation, only the first-place votes for each candidate are considered. Candidate A received 7 first-place votes (column 2). Candidate B received 4 + 4 or 8
first-placed votes (columns 1 and 4). Candidate C received 5 first-place votes (column 3). Candidate B is the winner since that candidate received the most first-place
votes.
Candidates
Candidate A
Candidate B
Candidate C
First-place votes
7 votes
4 + 4 = 8 votes
5 votes
b. This is an important question – it’s our first indication of why just calling the person
with the most votes isn’t necessarily the best approach. Look at the bottom row of
the table: of the 20 people that voted, 12 ranked B as their LEAST favorite candidate!
If more than half of those voting really really don’t want that candidate to be club
president, should he or she win?
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TRY THIS ONE!
2
An election was held for the discussion for the the chairperson of the Psychology Department. There were three candidates: Professor Jones (J), Professor Kline (K), and Professor
Lane (L). The preference table for the ballot is shown.
Number of Votes
First choice
Second choice
Third choice
2 4
L J
J K
K L
1 3
K L
L K
J J
1. Who won the election if the plurality method of voting was used?
Candidates
Professor Jones
Professor Kline
Professor Lane
First-place votes
2. Do you think this is correct choice? Why or why not?
In Example 2, the top row consists of the number of voters who ranked the candidates in the
order shown in column. Instead of numbers in the top row, percents can also be used. That allows
us to draw a pie chart illustrating the results, which we’ll do in Example 3.
EXAMPLE 3: Using Percentages to Summarize a Preference Table
For the preference table in Example 2, calculate the percentage of voters that chose each
candidate and rewrite the table with the percentages in place of the number of voters. Then
use the results to draw a pie chart illustrating the first-place votes each candidate.
Solution:
From adding the numbers along the top of the original preference table, we know that
there were 20 votes cast. We can find the percentage for each ballot by dividing the number
of voters by 20 and converting to percent form.
4
= 0.2 = 20%
20
7
Second column:
= 0.35 = 35%
20
First column:
6
5
= 0.25 = 25%
20
4
Fourth column:
= 0.2 = 20%
20
Third column:
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The preference table now looks like this:
Number of Votes
First choice
Second choice
Third choice
20%
B
C
A
35%
A
C
B
25%
C
A
B
20%
B
A
C
There are 360◦ in a full circle, so to find the number of degrees for each portion, we
find the appropriate percentage of 360◦
Candidate A: 35% of 360◦ = 0.35 × 360◦ = 126◦
Candidate B: 40% of 360◦ = 0.40 × 360◦ = 144◦
Candidate C: 25% of 360◦ = 0.25 × 360◦ = 90◦
The pie chart shown
Figure 1.1: Pie Chart of First-place votes
TRY THIS ONE!
3
Rewrite the preference table for the election in Try This One 2, replacing the number of
voters with the percentage of voters for each ballot. Then draw a pie chart illustrating the
first-place votes.
The preference table:
Number of Votes
First choice
Second choice
Third choice
Pie Chart:
%
L
J
K
%
J
K
L
%
K
L
J
%
L
K
J
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The plurality method is a simple way to determine the winner of an election, but it has
some flaws. First, some would suggest that a candidate shouldn’t win an election if less
than half of the voters choose him or her. As we wee from Figure 1.1, candidate B wins the
election in Example 2, even though well less than half of the ballots listed him or her first.
Second, the possibility of a tie exists, and is greater when there are fewer voters. Third,
the method completely ignores information about voters’ preferences except for their firstplace vote. Fourth, this method can sometimes violate what is called the head-to-head
comparison criterion.
A criterion is a way of measuring or evaluating a situation. In this module, we will
discuss various criteria for assessing the fairness of voting systems. The first of theses is
the head-to-head comparison criterion.
DEFINITION: The head-to-head comparison criterion states that if particular candidate
wins all head-to-head comparisons with all other candidates, then that candidate should
win the election
Let’s see if the election in Example 2 violates the head-to-head comparison criterion.
EXAMPLE 4: The Head-to-Head Comparison Criterion
Does the election in Example 2 violate the head-to-head comparison criterion?
Solution:
The idea is to comapre all combinations of two candidates at a time to see which is preferred
in a head-to-head matchup without the third candidate involved.
The preference table for the club president’s election is reprinted here for reference.
Number of Votes
First choice
Second choice
Third choice
4
B
C
A
7 5 4
A C B
C A A
B B C
First compare A with B:
The second the the third preference ballots have candidate A listed higher than candidate
B, and there were 12 voters that chose this order.
The first and fourth have candidate B listed higher, and that order was chosen by 8
candidates.
So candidate A would win a head-to-head matchup with candidate B. That alone doesn’t
mean that the election violates the head-to-head comparison criterion: the criterion doesn’t
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say that the winning candidate has to beat all others in a head-to-head matchup.
First compare A with C:
There were 11 voters who listed candidate A higher that candidate C (the second and fourth
column). There were 9 who listed candidate C higher, so candidate A also wins a head-tohead matchup with candidate C.
Without even comparing B and C, we can see that the head-to-heaad comparison criterion is violated: candidate A defeats both B and C head-to-head, but candidate A didn’t win
the election using the plurality method. (The head-to-head criterion says that any candidate
who defeats all opponents should win the election)
CAUTION: The head-to-head comparison criterion doesn’t say that the winner of an election
has to defeat every opponent head-to-head. It says there is a candidate that does defeat all others
head-to-head, that candidate should win the election
TRY THIS ONE!
4
Does the election in Try This One 2 violate the head-to-head comparison criterion? Why
or why not?
The preference table:
Number of Votes
First choice
Second choice
Third choice
2 4
L J
J K
K L
1 3
K L
L K
J J
Professor Jones Vs. Professor Kline:
Professor Jones Vs. Professor Lane:
Professor Kline Vs. Professor Lane:
The result of Example 4 shows that the plurarity method doesn’t always satisfy the head-tohead comparison criterion. This is not to say that every election conducted by the plurality method
violates the head-to-head comparison criterion. We have simply found that some do, so we say the
method in general doesn’t meet the criterion.
DEFINITION: The head-to-head criterion is called a fairness criterion.
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WORKSHEET 1
Preference Tables and the Plurality Method
Name:
Date:
1. The preference ballots for the election of a CEO by the board of directors are shown.
Make a preference table for the results of the election and answer each questions.
a. How may people voted ?
b. How many people voted for the candidates in the order of preference XZY ?
c. How many people voted for candidate Y as their first choice?
d. Using the plurality method, determined the winner of the election.
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2. The students in Dr. Lee’s math class were asked to vote on the starting time for their
final exam. Their choices were 8:00 A.M., 10:00 A.M., 12:00 P.M., or 2:00 P.M. The results
of the election are shown in the preference table.
Number of Voters
First Choice
Second Choice
Third Choice
Fourth Choice
8 12
8 10
10 8
12 2
2 12
5
3
2
2
12 2 10 8
2 12 12 2
10 8
8 10
8 10 2 12
a. How many students voted?
b. What time was the final exam if the plurality method was used to determined the
winner?
c. Draw a pie chart illustrating the percentage of first-place votes received by each candidate.
d. Using the election results, has the head-to-head comparison criterion been violated?
Explain your answer.
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