Section 14'1 Voting Methods

1
Section 14.1Voting Methods

• Objective
• Understand and use preference tables.
• Use the plurality method to determine an
  elections winner.
• Use the Borda count method to determine an
  elections winner.
• Use the plurality-with-elimination method to
  determine an elections winner.
• Use the pairwise comparison method to determine
  an elections winner.

2
Preference Tables

• Preference ballots are ballots in which a voter
  is asked to rank all the candidates in order of
  preference.
• Example Four candidates are running for
  president of the Student Film Institute Paul
  (P), Rita (R), Sarah (S), and Tim (T). Each of
  the clubs members submits a secret ballot
  indicating his or her first, second, third, and
  fourth choice for president.

The 37 ballots are shown to the right.
3
Preference Tables

• A preference table shows how often each
  particular outcome occurred.
• Example

The election ballots are placed into identical
stacks.
Then the identical stacks are placed into a
preference table.
4
Preference TablesUnderstanding a Preference
Table

• Example How many people selected Rita (R) as
  their second choice?
• Solution We find the number of people who voted
  for R as their second choice by reading across
  the row that says Second Choice. When you see a R
  in this row, write the number above it. Then find
  the sum of the numbers
• 14 10 24
• Thus, 24 people selected Rita as their second
  choice.

5
The Plurality Method

• The candidates (or candidates, if there is more
  than one) with the most first-place votes is the
  winner.
• Example For the previous preference table, who
  is declared the winner using the plurality
  method?
• Solution The candidate with the most first-place
  votes is the winner. When using the preference
  table, we only need to look at the row indicating
  the number of first-choice votes.
• Thus, P (Paul) is the winner and the new
  president of the Student Film Institute.

6
The Borda Count Method

• Each voter ranks the candidates from the most
  favorable to the least favorable.
• Each last-place vote is given 1 point, each
  next-to-last-place vote is given 2 points, each
  third-from-last-place vote is given 3 points, and
  so on.
• The points are totaled for each candidate
  separately.
• The candidate with the most points is the winner.

7
The Borda Count MethodUsing the Borda Count
Method

• Example In the previous table, who is the winner
  using the Borda method?
• Solution Because there are four candidates, a
  first-place vote is worth 4 points, a second
  place vote is worth 3 points, a third-place vote
  is worth 2 points, and a fourth-place vote is
  worth 1 point.

8
The Borda Count MethodUsing the Borda Count
Method

• Now, we read down each column and total the
  points for each candidate separately
• P 56 10 8 4 1 79 points
• R 42 30 16 16 2 106 points
• S 28 40 24 8 4 104 points
• T 14 20 32 12 3 81 points
• Because Rita (R) has received the most points,
  she is the winner and the new president of the
  Student Film Institute.

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The Plurality-with-Elimination Method

• The candidate with the majority of first-place
  votes is the winner.
• If no candidate receives a majority of
  first-place votes, eliminate the candidate
  (candidates, if there is a tie) with the fewest
  first-place votes from the preference table.
• Move the candidates in each column below the
  eliminated candidate up one place.
• The candidate with the majority of first-place
  votes in the new preference table is the winner.
• If no candidate receives a majority of
  first-place votes, repeat this process until a
  candidate receives a majority.

10
The Plurality-with-Elimination MethodUsing the
Plurality-with-Elimination Method

• Example In the previous table, who is declared
  the winner using the plurality-with-elimination
  method?

Solution Recall that there are 37 people voting.
In order to receive a majority, a candidate must
receive more than 50 of the first-place votes,
i.e., meaning 19 or more votes.
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The Plurality-with-Elimination MethodUsing the
Plurality-with-Elimination Method

• The number of first-place votes for each
  candidate is
• P(Paul) 14 S(Sarah) 10 1 11 T(Tim)
  8 R(Rita) 4
• We see that no candidate receives a majority
  first-place votes. Because Rita received the
  fewest first-place votes, she is eliminated in
  the first round. The new preference table is

12
The Plurality-with-Elimination MethodUsing the
Plurality-with-Elimination Method

• We repeat the process. So, the number of
  first-place votes for each candidate is
• P(Paul) 14 S(Sarah) 10 1 11 T(Tim)
  8 4 12.
• Once again, no candidate receives a majority
  first-place votes. Because Sarah received the
  fewest first-place votes, she is eliminated from
  the second round. The new preference table is

13
The Plurality-with-Elimination MethodUsing the
Plurality-with-Elimination Method

• The number of first-place votes for each
  candidate is now
• P(Paul) 14 T(Tim) 10 8 4 1 23.
• Because T(Tim) has received the majority of
  first-place votes, i.e., Tim received more than
  19 votes, he is the winner and the new president
  of the Student Film Institute.

14
The Pairwise Comparison Method

• Using the pairwise comparison method, every
  candidate is compared one-on-one with every other
  candidate.
• The number of comparisons made using the pairwise
  comparison method is given by
• where n is the number of candidates, and C is
  the number of comparisons that must be made.

15
The Pairwise Comparison Method

• Voters rank all the candidates and the results
  are summarized in a preference table.
• The table is used to make a series of comparisons
  in which each candidate is compared to each of
  the other candidates.
• For each pair of candidates, X and Y, use the
  table to determine how many voters prefer X to Y
  and vice versa.
• If a majority prefer X to Y, then X receives 1
  point.
• If a majority prefers Y to X, then Y receives 1
  point.
• If the candidates tie, then each receives half a
  point.
• After all comparisons have been made, the
  candidate receiving the most points is the winner.

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The Pairwise Comparison MethodUsing the Pairwise
Comparison Method

• Example For the previous table, who is declared
  the winner using the pairwise comparison method?
• Solution We first find how many comparisons that
  must be made. Since there are 4 candidates, then
  n4 and

17
The Pairwise Comparison MethodUsing the Pairwise
Comparison Method

• With P, R, S, T, the comparisons are P vs. R, P
  vs. S, P vs. T, R vs. S, R vs. T, and S vs. T.

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The Pairwise Comparison MethodUsing the Pairwise
Comparison Method
Using the six comparisons and conclusion to add
points we get P no points S 1 1 1 3
points R 1 1 2
points T 1 point After all the
comparisons have been made, the candidate
receiving the most points is S(Sarah). Sarah is
the winner and new president of the Student Film
Institute.