Voting Methods

Introduction

What we will learnThe 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

The Candidates

The voters

I’m picking Beatrice!

Anyone but Carl!

Beatrice is ok, but Carl gets my vote.I like Carl.

Ewww boys!Go Bea!

Amy’s my girl.Carl is the best!

Carl is NOT qualified.

I like Amy.

Just NOT Carl.

Preference BallotsAlthough 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.

Preference BallotsThis voter’s ballot may look like this.

Ballot

First Choice Amy

Second Choice Beatrice

Third (last) Choice Carl

Preference Ballots

Ballot

First Choice Amy

Second Choice Beatrice

Third (last) Choice Carl

A shorter form of this ballot may look like this.

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.

Preference BallotsWith 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!

Counting Principle

Factorial

So, for 3 candidates:

Organizing the Ballots

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).

Preference Ballots

Organizing the Ballots

Organizing the Ballots

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

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

AssumptionsA 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.

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

Example

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

Solution

ABC

BAC

CBA

C??

BAC

ABC

C??

ExampleUse the preference table to answer the questions.

1. How many people voted?2. How many unique ballots were cast?3. With 5 candidates, how many possible unique

ballots could have been cast in the election?

10 4 8 6 2 1

1st B C A D A A2nd D B D B D C3rd C D B C B B4th E E E E C D5th A A C A E E

31

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