1. Voting Methods
Introduction

2. The Mathematics of Voting
What we will learn
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The Mathematics of Voting
Preference Ballots and Preference Schedules
The Plurality Method
The Borda Count Method
The Plurality with Elimination Method
The Method of Pairwise Comparisons

3. The Candidates
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Introducing,
the candidates!

4. The voters I’m picking Beatrice! Anyone but Carl!
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I’m picking Beatrice!
Anyone but Carl!
Beatrice is ok, but Carl gets my vote.
Amy’s my girl.
Carl is the best!
Carl is NOT qualified.
I like Amy.
Just NOT Carl.
I like Carl.
Ewww boys!
Go Bea!

5. Preference Ballots
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Although most people in a democratic society believe in the “one man, one vote” system, sometimes it is convenient to know more than a voter’s top pick.
Preference ballots are used to give us more information. A preference ballot puts candidates in order of preference rather than declaring a voter’s top preference only.

6. Preference Ballots This voter’s ballot may look like this. Ballot
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This voter’s ballot may look like this.
Ballot
First Choice
Amy
Second Choice
Beatrice
Third (last) Choice
Carl

7. Preference Ballots A shorter form of this ballot may look like this.
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A shorter form of this ballot may look like this.
Ballot
First Choice
Amy
Second Choice
Beatrice
Third (last) Choice
Carl
Ballot
1st A
2nd B
3rd C
The only problem with preference ballots is that there can be many different (unique) ballots for only a few candidates.

8. Preference Ballots Multiply!
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With 3 candidates, there are 6 possible unique ballots.
With more than 3 candidates, it may get difficult to list every possible unique ballot. It will be easier to use the following method to count the possible unique ballots.
Multiply!
Start with 3
choices for
first pick.
which leaves 2
choices for
second pick
which leaves 1
choice left
for last.
This makes 6
possible
unique ballots. X X =

9. Counting Principle So, for 3 candidates: Factorial Counting Principle:
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Counting Principle:
With n candidates there are n!
possible unique ballots.
Factorial
Factorial: A symbol in mathematics denoted
with and exclamation point (!)
n! = (n) x (n-1) x (n-2) x (n-3) x ... x (1)
So, for 3 candidates:
3! = 3 x 2 x 1 = 6

10. Organizing the Ballots
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Preference
Ballots
The Mathematics Anxiety Club (MAC)
The 37 members of the MAC are electing a new president.
Their preference ballots are as follows:

11. Organizing the Ballots
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With so many possible unique ballots, it becomes important to organize (or stack) them.
Ballots that are exactly alike get stacked on top of each other and we use the stacked ballots to make a preference table (or preference schedule).

12. Organizing the Ballots
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If we were to organize the 37 MAC preference ballots by stacking like ballots, our stacks would look like this.

13. Organizing the Ballots
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We use our sorted (stacked) ballots to make a preference table (or preference schedule).

14. Assumptions If a voter prefers C to B and prefers B to D,
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A couple of assumptions we will make while studying voting theory.
If a voter prefers C to B
and prefers B to D,
then the voter will
also prefer C to D.
If a candidate is eliminated, then a voter’s ballot will change as shown.

15. Example
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Remember our voters? Try to guess what each voter’s preference ballot would look like based on what they are saying and thinking.

16. Example
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Were there any problems? If you found that two of the voters’ 2nd and 3rd preferences could not be determined, you were correct.

17. Solution
T. Serino A B C B A C C B A C ? A B C C ? B A C

18. Example Use the preference table to answer the questions.
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Use the preference table to answer the questions.
1st B C A D A A
2nd D B D B D C
3rd C D B C B B
4th E E E E C D
5th A A C A E E
1. How many people voted?
2. How many unique ballots were cast?
With 5 candidates, how many possible unique ballots could have been cast in the election? 31 6
120

19. athematical M D
ecision
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