Unit 1­Sec1.1­1.4.

notebook August 31, 2023

MATH 120
Contemporary Applications
of Mathematics
Course Content

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Unit 1­Sec1.1­1.4.notebook August 31, 2023

Unit 1
Election and Voting

In this unit we will study the mathematics of Elections.

We will cover the following 4 sections:

1.1 Preference Ballots and Preference Schedules

1.2 The Plurality Method

1.3 The Borda Count Method

1.4 The Plurality­with­elimination Method

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Section 1.1
Preference Ballots and Preference Schedules
• Preference ballot
A ballot in which the voters are asked to rank the
candidates in order of preference.

• Linear ballot
A ballot in which ties are not allowed.

Not a Linear Ballot

Not Allowed

• Preference ballots (37 Ballots)

• Preference Schedule

• Preference Schedule in Table Form

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• The first is that a voter’s preference are transitive,

i.e.,
that a voter who prefers candidate A over candidate B
and prefers candidate B over candidate C
automatically
prefers candidate A over C.

• Secondly, that the relative preferences of a voter are not

affected by the elimination of one or more of the
candidates.
Candidate B is eliminated

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Section 1.2: The Plurality Method

The Plurality Method declares the winner to be the candidate with the
most first place votes.
For example, if the preference schedule of a voting is

# Voters 12 14 10 8 2
1st A B C D E
2nd D E D A B
3rd B D A E A
4th C C B C D
5th E A E B C

The winner is candidate B because he got 14 first­place votes,

the most among all candidates.

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Terminologies and Some Definitions

• Plurality method
Election of 1st place votes

• Plurality candidate
The Candidate with the
most 1st place votes

• Majority rule
The candidate with a more
than half the votes should
be the winner.

• Majority candidate
The candidate with the
majority of 1st place votes

• Condercet candidate
The Candidate who win
everyone else head­to­
head

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Fairness Criterions

1) The Majority Criterion

The Majority Criterion
If candidate X has a majority of the 1st place votes,
then candidate X should be the winner of the election.

2) The Condorcet Criterion

The Condorcet Criterion

If candidate X is preferred by the voters over each of
the other candidates in a head­to­head comparison,
then candidate X should be the winner of the election.

Note: The plurality method satisfies the majority criterion!

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Examples
Example 1 The election ballots of Example 1.1 page 4 have been collected into piles as shown:

a) Construct a preference schedule in table form

Voters 14 10 8 4 1
1st A C D B C

2nd B B C D D

3rd C D B C B

4th D A A A A

b) Who is the winner of the election according to the Plurality method?

c) Is there a majority candidate?

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d) Is there a condercet candidate?

e) Does the Plurality method satisfy the condercet criterion, in

this example?

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Example 2 The election ballots produced the following preference schedule:

a) Who is the winner of the election according to the Plurality

method?

b) Is there a majority candidate?

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c) Is there a condercet candidate?

d) Does the Plurality method satisfy the condercet criterion, in this

example?

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Example 3

An election is held to choose a Chair of the Mathematics Department. The candidates are A, B, C, D.
The preference schedule produced by the election is

# Voters 5 3 5 3 2 3
1st A A C D D B
2nd B D E C C E
3rd C B D B B A
4th D C A E A C
5th E E B A E D
a) How many people voted in this election?

b) Who is the winner of the election, by the Plurality Method?

c) Is there a condercet candidate? If yes, who?

d) If it came down to a choice between A and D, which one would win?

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Section 1.3: The Borda Count Method

In the Borda Count Method each place on a ballot is assigned points.

In an election with N candidates we give 1 point for last place, 2 points
for second from last place, and so on.

The winner will be the one who gets more points.

Example
Consider the following preference schedule of a certain election.

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Example
Consider the following preference schedule of a certain election.
# Voters 8 7 6 2 1
1st A D D C E
2nd B B B A A
3rd C A E B D
4th D C C D B
5th E E A E C
a) Using the Borda Count Method, find the winner of the election.

b) Did the Borda count method, in this election, violate any fairness
criterion?

General Rule:
If X is a majority candidate

X is a condercet candidate.

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Example:
In an election, we have 4 candidates (A, B, C and D) and the
total number of votes are 110. We are using the Borda Count
method for this election. A has 320 points, B has 290 points
and C has 180 points. How many points did D get?

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Example: (Critical thinking 1)

In an election with four candidates (A,B,C,D) and 110 voters, using

the Borda Count Method, A gets 230 points, B gets 300 points and D
gets 350 points, who won the election?

Example: (Critical thinking 2)

In an election with five candidates (A,B,C,D,E), using the Borda
Count Method, A gets 110 points, B gets 150 points,C gets 200
points,D gets 150 points, and E got 600 points. How many people
voted in this election?

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Example: (Critical thinking 2)

In an election with 40 voters, using the Borda Count Method, the total
Borda points is 840 points. Find how many candidates in this
election?

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Section 1.4: The Plurality­with­elimination Method

Round 1. Count the first­place votes for each candidate, just as you would
in the plurality method. If a candidate has a majority of first­place votes,
that
candidate is the winner. Otherwise, eliminate the candidate (or candidates
if there is a tie) with the fewest first­place votes.

Subsequent Rounds. Continue the process of eliminating the candidate

with the fewest first­place votes until we get a majority winner.
Example
Consider the following preference schedule of a certain election. Total # of Voters
37

Using the Plurality­with­elimination method, determine the winner.

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Using the Plurality­with­elimination method, determine the winner.

Round 1.
A has 14 first­place votes
No Majority
B has 4 first­place votes
C has 11 first­place votes
Eliminate B
Winner
D has 8 first­place votes

Round 2.
# Voters 14 10 8 4 1
1st A C D D C
New Schedule 2nd C D C C D
3rd D A A A A

A has 14 first­place votes No Majority

C has 11 first­place votes Eliminate C
D has 12 first­place votes Winner

Round 3.
# Voters 14 10 8 4 1
New Schedule 1st A D D D D
2nd D A A A A

A has 14 first­place votes Majority

D has 23 first­place votes Winner is D
Winner

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Exercise 1

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Exercise 2

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Sample MC Questions

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