Voting Theory Lesson Plan: Voting Systems for Elections
with 3 or More Candidates
Common Core Math 8 Honors Standards
8.H.2 Examine different methods of voting.
a) Understand different voting systems.
b) Determine the results of an election using various voting systems.
Teacher Notes
Voting is at the heart of democracy. The goal of an election is to determine “the will of the
people”. When the choice is between 2 candidates, it’s easy to determine the winner using
majority rule: whoever receives more than 50% of the votes is the winner. But there is no easy
way to choose a winner from 3 or more candidates. It’s not uncommon for none of the
candidates to receive a majority of the votes. People can then reasonably disagree about who
should be declared the winner.
Several different voting systems have been devised and are used for elections with 3 or more
candidates. In this lesson, students will investigate some of these voting systems: plurality,
single and sequential run-offs, point systems (Borda Count), and pairwise comparisons
(Condorcet Method). All of these are good methods for choosing the winner of an election with
3 or more candidates, but each also has problems that in some cases can lead to unfair results.
In fact, in 1952, economist Kenneth Arrow used mathematics to prove a remarkable fact: for
elections of three or more candidates, there is no consistently fair method for choosing a
winner. This is known as Arrow’s Impossibility Theorem.
Lesson Launch
Give the students the election results from the example below. Ask them to decide who they
think the winner of the election should be using at least two different ways of determining the
winner.
EXAMPLE: Imagine a club of 55 people that hold an election among 5 candidates for president.
For simplicity we will call the candidates A, B, C, D, and E. The results of the election are shown
on a kind of ballot called a preference list ballot. (This example is given in Using and
Understanding Mathematics, A Quantitative Reasoning Approach, Fourth Edition, Jeffrey
Bennett and William Briggs.)
Preference List Ballot Results for the Club Election
First
A
Second
D
Third
E
Fourth
C
Fifth
B
B
E
D
C
A
C
B
E
D
A
D
C
E
B
A
E
B
D
C
A
E
C
D
B
A
Number of voters who chose each ordering
18
12
10
9
4
2
Voting Systems Lesson Plan
Following a discussion of the students’ ideas, hand-out the student worksheet and show the
class how the winner would be determined using each of 5 different voting system methods.
Choose a Winner by Plurality.
In plurality voting, only first-place votes are considered. The candidate with the most first-place
votes wins, even though this may be considerably fewer than a majority of the votes. This is
the voting system commonly used in US political elections.
A:
B:
C:
D:
E:
18 first place votes
12 first place votes
10 first place votes
9 first place votes
4 + 2 = 6 first place votes
A is the winner by Plurality.
Choose a Winner by a Single Runoff.
“Not so fast!” yell the supporters of Candidate B. They suggest a runoff between A and B
because they were the top two candidates in first-place votes. Although 18 people voted for A
in first place, the other 37 ranked A in last place. These 37 people would all vote for B over A in
a runoff. Thus, B wins the runoff easily by a vote of 37 to 18. Supporters of Candidate B can
now proclaim victory for their candidate.
B is the winner by a Single
Runoff.
Choose a Winner by Sequential Runoffs.
Now the supporters of Candidate C chime in. They claim that a single runoff is unfair because it
ignores rankings below the top two. They suggest a series of runoffs called a sequential runoff:
the candidate with the fewest first-place votes is eliminated at each stage and all the others
move up in their rankings. Runoffs continue until someone claims a majority.
1. With 6 first-place votes, E is
eliminated in the first runoff.
First
Second
Third
Fourth
# who chose
each ordering
2. With 9 first-place votes, D is
eliminated in the second runoff.
A
D
C
B
B
D
C
A
C
B
D
A
D
C
B
A
B
D
C
A
C
D
B
A
18
12
10
9
4
2
3. With 12 + 4 = 16 first-place votes,
B is now eliminated in the third runoff.
First
Second
# who chose
each ordering
A
C
C
A
C
A
C
A
C
A
C
A
18
12
10
9
4
2
First
Second
Third
# how chose
this ordering
A
C
B
B
C
A
C
B
A
C
B
A
B
C
A
C
B
A
18
12
10
9
4
2
With 18 first place votes, A is now
eliminated and C is declared the winner
with 12 + 10 + 9 + 4 + 2 = 37 first place
votes.
C is the winner by sequential runoffs.
Choose a Winner by a Point System, (Borda Count).
Now is the turn of Candidate D’s supporters to argue that the winner should be selected by a
point system. Because there are 5 candidates, first-place votes are worth 5 points, secondplace votes are worth 4 points, and so on, down to 1 point for fifth-place votes. A system that
assigns points for every ranking is called a Borda Count after the French mathematician and
astronomer Jean-Charles de Borda (1733-1799).
A gets (18 × 5) + (12 × 1) + (10 × 1)) + (9 × 1) + (4 × 1) + (2 × 1) = 127 points
B gets (18 × 1) + (12 × 5) + (10 × 4) + (9 × 2) + (4 × 4) + (2 × 2) = 156 points
C gets (18 × 2) + (12 × 2) + (10 × 5) + (9 × 4) + (4 × 2) + (2 × 4) = 162 points
D gets (18 × 4) + (12 × 3) + (10 × 2) + (9 × 5) + (4 × 3) + (2 × 3) = 191 points
E gets (18 × 3) + (12 × 4) + (10 × 3) + (9 × 3) + (4 × 5) + (2 × 5) = 189 points
D wins by the Borda Count.
Choose a Winner by Pairwise Comparisons, (the Concorcet Method).
Now, Candidate E’s supporters point out an important fact about this election’s rankings.
Suppose they say, that the vote had been only between E and A, without the other candidates.
A got 18 votes for those who placed A above E, but E got 37 votes from those who placed E
above A. E wins A, 37 to 18. Now suppose that the vote had been between B and E. B wins 22
votes over E but E wins 33 votes over B. E wins B, 33 to 22. And E wins C, 36 to 19 and E wins
D, 28 to 27. Because Candidate E beats every other candidate in one-on-one contests, E’s
supporters now declare victory for E. This method that analyzes the outcomes of one-on-one
contests is called the Concorcet method because it was invented by French mathematician and
political leader, Marie Jean Antoine Nicholas de Caritat, the Marquis de Concorcet (1743-1794).
E wins by the Concorcet Method.
As you probably have guessed this is no ordinary election. The election results were carefully
created to emphasize that sometimes there is no absolutely fair way of deciding elections
among more than two candidates.
The Approval Voting Method.
There is one more method that needs to be understood. We cannot use our example for this
method because it uses a different type of ballot. Traditionally voting systems have been based
on one person, one vote. However, an alternate voting system called the Approval Voting
Method asks voters to mark whether they approve or disapprove of each candidate. Voters
may approve as many candidates as they like and the candidate with the most approval votes
wins. While approval voting ensures that the winning candidate is acceptable to the largest
number of people, another candidate might be the first choice of the majority.
Work through the first worksheet as a class. The second worksheet is assigned to provide
independent practice.
Name _________________________________ Class Period ______ Seat Number ______
Voting Systems for 3 or More Candidates
When an election is between 2 candidates, whoever receives a majority of the votes is the
winner. Unfortunately, there is no easy way to choose a winner from 3 or more candidates. It’s
not uncommon for none of the candidates to receive a majority of the votes. People can then
reasonably disagree about who should be declared the winner. Several voting systems have
been devised and are used, but in some cases, each has problems that can lead to inconclusive
or unfair results. In fact, in 1952 economist Kenneth Arrow used mathematics to prove a
remarkable fact: for elections involving three or more candidates, there is no consistently fair
method for choosing a winner. This is known as Arrow’s Impossibility Theorem. Today you will
investigate several voting systems: Plurality, Runoffs, A point System (Borda Count), and
Pairwise Comparisons (Concorcet Method).
Who is the winner? Imagine a club of 55 people that hold an election among 5 candidates for
president. For simplicity we will call the candidates A, B, C, D, and E. The results of the election
are shown on a kind of ballot called a preference list ballot.
Preference List Ballot Results for the Club Election
First
Second
Third
Fourth
Fifth
Number of voters who chose each ordering
A
D
E
C
B
18
B
E
D
C
A
12
C
B
E
D
A
10
D
C
E
B
A
9
E
B
D
C
A
4
E
C
D
B
A
2
Use the given voting method to determine the winner of the club election. Show the
numbers that justify your choice of a winner.
1.
Choose a Winner by Plurality In plurality voting, only first-place votes are considered.
The candidate with the most first-place votes wins, even through this may be considerably
less than a majority. This is the most common voting system in use today.
2.
Choose a Winner by a Single Runoff. In a single runoff, only the two candidates with the
most first-place votes are compared. How many voters chose the candidate receiving the
most first-place votes over the candidate receiving the second most, first-place votes. And
how many voters chose the candidate receiving the second most, first-place votes over the
candidate receiving the most first-place votes.
3.
Choose a Winner by Sequential Runoffs Using a sequence of runoffs, the candidate with
the least number of first-place votes is eliminated at each stage and all the remaining
candidates move up in their rankings. Runoffs continue until someone claims a majority.
1. First Runoff: _____ is eliminated.
First
Second
Third
Fourth
# who chose
each ordering
2. Second Runoff: _____ is eliminated.
First
Second
Third
# who chose
this ordering
18
12
10
9
4
18
12
10
9
4
2
2
3. Third Runoff: _____ is eliminated.
First
Second
# who chose
each ordering
18
12
10
9
4
2
4.
Choose a Winner by a Point System, (Borda Count) A system that assigns points for every
ranking is called a Borda Count after the French mathematician and astronomer JeanCharles de Borda (1733-1799). Because there are 5 candidates, first-place votes are worth
5 points, second-place votes are worth 4 points, and so on, down to 1 point for fifth-place
votes. The candidate with the most points wins.
5.
Choose a Winner by Pairwise Comparisons, (the Concorcet Method) The Concorcet
method analyzes the outcomes of one-on-one contests between each pair of the
candidates. How many voters chose A over E? How many chose E over A? Who won this
contest? Now how many voters chose B over E? How many chose E or B? Who won this
contest? Whoever beats all the other candidates in one-on-one contests is the winner.
Named after the French mathematician and political leader, Marie Jean Antoine Nicholas
de Caritat, the Marquis de Concorcet (1743-1794), the Concorcet Method does not always
determine a winner.
Name ___________________________________ Class Period ______ Seat Number ______
Independent Practice with Voting Systems
Ballot
Rank the activities in
order of preference.
Caving, C, Rafting, R,
and Water Skiing, S.
1st Choice _____
2nd Choice _____
3rd Choice _____
Who is the Winner? The 40 members of your school adventure club are
trying to decide what type of trip to take. They use a preference list
ballot to choose between a caving, rafting, or water-skiing trip. For
simplicity we will call these choices C, R, and S. The results are shown on
the table below.
First Choice
Second Choice
Third Choice
Number Who Chose Each Ordering
S
R
C
10
S
C
R
7
R
S
C
1
R
C
S
10
C
S
R
4
C
R
S
8
Use the given voting method to determine the winner of the club election. Show the
numbers that justify your choice of a winner.
1.
Choose a winner by plurality.
2.
Choose a winner by a single runoff.
3.
Choose a winner by sequential runoffs.
1st Runoff: First-Place Votes:
S ____ R ____ C ____ Eliminate ______.
2nd runoff: First-Place Votes:
S ____ R ____ C ____ Eliminate ______.
First
Second
Third
# who chose
this ordering
S
R
C
10
S
C
R
7
R
S
C
1
R
C
S
10
C
S
R
4
C
R
S
8
4.
Choose a winner by a point system, (the Borda Count)
5. Choose a winner by pairwise comparisons, (the Concorcet Method).
Who is the Winner? Given the preferences of 100 voters in an election of three candidates, A,
B, and C, who is the winner?
First
Second
Third
Number Who Chose Each Ordering
A
B
C
30
A
C
B
5
B
A
C
20
B
C
A
5
C
A
B
10
C
B
A
30
Use the given method to determine the winner of the election. Show the numbers that
justify your choice of the winner.
6.
Choose a winner by plurality.
7.
Choose a winner by a single runoff.
8.
Choose a winner by a sequential runoff.
1st Runoff: First-Place Votes:
A ____ B ____ C ____ Eliminate ____
First
Second
Third
2nd Runoff: First-Place Votes:
A ____ B ____ C ____ Eliminate ____
9.
Choose a winner by a point system, (the Borda Count).
A
B
C
30
A
C
B
5
B
A
C
20
B
C
A
5
C
A
B
10
C
B
A
30
10. Choose a winner by pairwise comparisons, (The Concorcet Method).
Name ___________________________________ Class Period ______ Seat Number ______
Independent Practice with Voting Systems - KEY
Ballot
Rank the activities in
order of preference.
Caving, C, Rafting, R,
and Water Skiing, S.
1st Choice _____
2nd Choice _____
3rd Choice _____
Who is the Winner? The 40 members of your school adventure club are
trying to decide what type of trip to take. They use a preference list
ballot to choose between a caving, rafting, or water-skiing trip. For
simplicity we will call these choices C, R, and S. The results are shown on
the table below.
First Choice
Second Choice
Third Choice
Number Who Chose Each Ordering
S
R
C
10
S
C
R
7
R
S
C
1
R
C
S
10
C
S
R
4
C
R
S
8
Use the given voting method to determine the winner of the club election. Show the
numbers that justify your choice of a winner.
1.
Choose a winner by plurality.
First-Place Votes: S has 𝟏𝟎 + 𝟕 = 𝟏𝟕; R has 𝟏 + 𝟏𝟎 = 𝟏𝟏; C has 𝟒 + 𝟖 = 𝟏𝟐. S wins by plurality but
not by majority.
2.
Choose a winner by a single runoff.
The 2 candidates with the top first-place votes are S with 17 and C with 12.
In a single runoff with S vs. C, S gets 𝟏𝟎 + 𝟕 + 𝟏 = 𝟏𝟖 votes. C gets 𝟏𝟎 + 𝟒 + 𝟖 = 𝟐𝟐. C wins by a
Single runoff.
3.
Choose a winner by sequential runoffs.
1st Runoff: First-Place Votes:
S _17_ R _11_ C _12_ Eliminate __R__.
2nd runoff: First-Place Votes:
S _18_ R _0_ C _22_ Eliminate __S__.
C is the winner by a sequential runoffs
First
Second
Third
# who chose
this ordering
S
R
C
10
S
C
R
7
R
S
C
1
R
C
S
10
C
S
R
4
C
R
S
8
4.
Choose a winner by a point system, (the Borda Count)
S gets (𝟏𝟎 × 𝟑) + (𝟕 × 𝟑) + (𝟏 × 𝟐) + (𝟏𝟎 × 𝟏) + (𝟒 × 𝟐) + (𝟖 × 𝟏) = 𝟕𝟗
R gets (𝟏𝟎 × 𝟐) + (𝟕 × 𝟏) + (𝟏 × 𝟑) + (𝟏𝟎 × 𝟑) + (𝟒 × 𝟏) + (𝟖 × 𝟐) = 𝟖𝟎
C gets (𝟏𝟎 × 𝟏) + (𝟕 × 𝟐) + (𝟏 × 𝟏) + (𝟏𝟎 × 𝟐) + (𝟒 × 𝟑) + (𝟖 × 𝟑) = 𝟖𝟏
C wins by the Borda Count.
5. Choose a winner by pairwise comparisons, (the Concorcet Method).
S vs R: S gets 𝟏𝟎 + 𝟕 + 𝟒 = 𝟐𝟏 votes. R gets 𝟏 + 𝟏𝟎 + 𝟖 = 𝟏𝟗 votes. S wins R in a pairwise
comparison.
S vs C: S gets 𝟏𝟎 + 𝟕 + 𝟏 = 𝟏𝟖 votes. C gets 𝟏𝟎 + 𝟒 + 𝟖 = 𝟐𝟐 votes. S wins C. in a pairwise
comparison.
S is the winner by the Concorcet Method.
Who is the Winner? Given the preferences of 100 voters in an election of three candidates, A,
B, and C, who is the winner?
First
Second
Third
Number Who Chose Each Ordering
A
B
C
30
A
C
B
5
B
A
C
20
B
C
A
5
C
A
B
10
C
B
A
30
Use the given method to determine the winner of the election. Show the numbers that
justify your choice of the winner.
6.
Choose a winner by plurality.
A: 𝟑𝟎 + 𝟓 = 𝟑𝟓 first-place votes.
B: 𝟐𝟎 + 𝟓 = 𝟐𝟓 first-place votes.
C: 𝟏𝟎 + 𝟑𝟎 = 𝟒𝟎 first-place votes.
7.
C is the winner by plurality, but not by majority.
Choose a winner by a single runoff.
A and C are the top two first-place vote getters.
A gets 𝟑𝟎 + 𝟓 + 𝟐𝟎 = 𝟓𝟓 votes and C get 𝟓 + 𝟏𝟎 + 𝟑𝟎 = 𝟒𝟓
A is the winner by a single runoff.
8.
Choose a winner by a sequential runoffs.
1st Runoff: First-Place Votes:
A _35_ B _25_ C _40_ Eliminate _C_
First
Second
Third
A
B
C
30
A
C
B
5
B
A
C
20
2nd Runoff: First-Place Votes:
A _45_ B _55_ C _0_ Eliminate _A_
B is the winner by sequential runoffs.
9.
Choose a winner by a point system, (the Borda Count).
A gets (𝟑𝟎 × 𝟑) + (𝟓 × 𝟑) + +(𝟐𝟎 × 𝟐) + (𝟓 × 𝟏) + (𝟏𝟎 × 𝟐) + (𝟑𝟎 × 𝟏) = 𝟐𝟎𝟎 votes.
B gets (𝟑𝟎 × 𝟐) + (𝟓 × 𝟏) + (𝟐𝟎 × 𝟑) + (𝟓 × 𝟑) + (𝟏𝟎 × 𝟏) + (𝟑𝟎 × 𝟐) = 𝟐𝟏𝟎 votes.
B
C
A
5
C
A
B
10
C
B
A
30
C gets (𝟑𝟎 × 𝟏) + (𝟓 × 𝟐) + (𝟐𝟎 × 𝟏) + (𝟓 × 𝟐) + (𝟏𝟎 × 𝟑) + (𝟑𝟎 × 𝟑) = 𝟏𝟗𝟎 votes.
B is the winner by the Borda Count.
10. Choose a winner by pairwise comparisons, (The Concorcet Method).
A vs B: A gets 𝟑𝟎 + 𝟓 + 𝟏𝟎 = 𝟒𝟓. B gets 𝟐𝟎 + 𝟓 + 𝟑𝟎 = 𝟓𝟓. B wins A in a pairwise comparison.
B vs C: B gets 𝟑𝟎 + 𝟐𝟎 + 𝟓 = 𝟓𝟓. C gets 𝟓 + 𝟏𝟎 + 𝟑𝟎 = 𝟒𝟓. B wins C in a pairwise comparison.
B is the winner by pairwise comparisons.