Voting Methods

#MathematicsInTheModernWorld
Contents:
• Basic Elements of an Election
• Plurality Method
• Borda Count Method
• Plurality-with-Elimination Method
• Pairwise Comparisons
• Fairness Criteria
Basic Elements of an Election
Math Club Election

Ballot
Voting Method
The Method that we use

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Choose one to tabulate the ballot and
Candidate:
decide a winner
Anna

Bob
Candidates
Cora

Dan
Math Club Election
Single Choice Ballot Preference
List Ballot
all candidates Truncated Preference Ballot
in order of
List all candidates
preference
in order of
Choose one
preference
Candidate:
1st
Anna _____________________ 1st
Dan Cor
2 ___________________
nd
_____________________
Bob Bob a
Bob
__ 2nd___________________
Cora Cor Ana
3rd
__
Ana
a
Dan _____________________ 3rd
4th _____________________
Winner-only Winner-only
_____________________ Winner-only
Full Ranking Partial Ranking
 14 10 8 4 1

Number 14 10 8 4 1
of Voters
Preferen 1st A C D B C

ce 2nd C B B D D
3rd B D C C B
Schedule
4th D A A A A
Numbe 14 10 8 4 1 Preference
r of Schedule
Voters
1st A C D B C
• What can we tell from a preference
2nd C B B D D schedule?
• Candidates: 4 candidates (A,B,C,D)
3rd B D C C B • Voters: 37 voters (14 + 10 + 8
+ 4 + 1)
4th D A A A A • Balloting: 5 piles
Preference Table
Numbe 93 44 10 30 42 81
r of
• How many candidates are Votes
there? 1st A B C C D E
• How many voters submitted a
ballot? 2nd B D A E C D
• How many piles were the 3rd C E E B E C
ballots organized into?
4th D C B A A B
5th E A D D B A
Plurality Method
 Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Plurality Method
The candidate with the most 1st Place votes is the winner.
 Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Using the Plurality Method, which candidate is the winner of the elecyion.
A: 14
B: 4
C: 10 + 1 = 11
D: 8

Anna is the winner!

Plurality VS Majority

Majority –
Plurality -
More than
Most Votes
half
 Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Using the Plurality Method, which candidate is the winner of the elecyion.
A: 14
B: 4
Majority is
C: 10 + 1 = 11 more than
D: 8 18.5 in this
problem.
Anna obtained the
plurality but not the
majority.
 Number of Voters 14 19 10 5 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Using the Plurality Method, which candidate is the winner of the elecyion.
A: 14
B: 4
Majority is
C: 10 + 1 = 11 more than
D: 8 18.5 in this
problem.
Anna obtained the
plurality AND the
majority.
Borda Count Method
 Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Borda Count
• Each place on a ballot is assigned points.
• The candidate with the highest total is the winner.
 Number of Voters 14 10 8 4 1
1st (4 pts) A C D B C
2nd (3 pts) C B B D D
3rd (2 pts) B D C C B
4th (1 pt) D A A A A

Using the Borda Count Method, which candidate is the winner of the elecyion.
A: 4 (14) + 1 (10) + 1 (8) + 1 (4) + 1 (1) = 79
B: 2 (14) + 3 (10) + 3 (8) + 4 (4) + 2(1) = 100
C: 3 (14) + 4 (10) + 2 (8) + 2 (4) + 4 (1) = 110
D: 1 (14) + 2 (10) + 4 (8) + 3 (4) + 3(1) = 81

Corra is the winner!

Plurality – with Elimination
Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Plurality-with-Elimination
• Count 1st place votes.
• Eliminate the candidate with the fewest votes
Repeat rounds until a candidate obtains a majority vote.
 Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Using the Plurality-with-Elimination Method, which candidate is the winner of the

election.

Count 1st place votes.

A B C D Eliminate the candidate with
R1 14 4 11 8 the fewest votes
R2 14 0 11 12 Repeat rounds until a
candidate obtains a majority
R3 14 0 0 24 vote.

Dan is the winner!

Top Two Instant Run-Off Voting Method
Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Top Two Instant Runoff Voting Method

• Eliminate all but the top two candidates.
• The candidate with more votes is the winner.
 Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Using the top two Instant-Runoff Voting Method, which candidate is the winner of the
election.
A: 14
A: 14 C: 10 + 8 + 4 + 1 = 23
B: 4
C: 11
D: 8

Corra is the winner!

Pairwise Comparison Method
 Number of 14 10 8 4 1
Voters

1st A C D B C

2nd C B B D D

3rd B D C C B

4th D A A A A

Using the Pairwise Comparison Method, which candidate is the winner of the election.

Comparisons Votes Winner

AvB A = 14 B -> 1 pt
B = 10 + 8 + 4 + 1 = 23
AvC A = 14 C -> 1 pt
C = 10 + 8 + 4 + 1 = 23
AvD A = 14 D -> 1 pt
D = 23
BvC B = 8 + 4 = 12 C -> 1 pt
C = 14 + 10 + 1 = 25
BvD B = 14 + 10 + 4 = 28 B -> 1 pt
D= 8+1=9
CvD C = 14 + 10 + 1 = 25 C -> 1 pt
D = 8 + 4 = 12

Corra is the winner!

 Condorcet Candidate

A candidate preferred by a majority when the The Pairwise Comparison chooses the
candidates are compared in a head-to-head
comparison is called a Condorcet candidate.
Condorcet candidate.
 Review
Method How a Winner is Determined
(Winner only elections)
Plurality The candidate with the most 1st place votes is the winner.
Borda Count 1. Each place on a ballot is assigned points.
2. The candidate with the highest total is the winner.
Plurality-with- For each round,
Elimination 1. Count 1st place votes.
2. Eliminate the candidate with the fewest votes.
Repeat rounds until a candidate obtains a majority vote.
Top-Two IRV 1. Eliminate all but the top two candidates
2. The candidate with more votes is the winner.
Pairwise Comparisons 1. Determine all possible comparisons between any two
candidates
2. Determine the winner in each comparison
3. The candidate with the most wins is the winner.
Fairness Criteria & Arrow’s Impossibility
Theorem
 Number of Voters 14 10 8 4 1
1st A C D B C
2nd C B B D D
3rd B D C C B
4th D A A A A

Under the Plurality Method, A is the winner with the most 1st place votes (14).

However, 23 voters had ranked as their least preferred candidate.

 Arrow’s
Impossibility
Theorem
There are no voting method that
satisfy certain reasonable
assumptions regarding voter
preference.
Fairness Criteria 1

If the majority candidate, the that

Majority Criterion candidate should be the winner if
the election.
Fairness Criteria 2

If there is a Condorcet candidate,

Condorcet Criterion then that candidate should be the
winner of the election.
Fairness Criteria 3

If Voters change their votes to increase the

Monotonicity Criterion preference for a candidate, it should not
harm that candidate’s chances of winning.
Fairness Criteria 4

If a non-winning
choice is
Independence-of- removed from
Irrelevant- the ballot, it
Alternatives (IIA) should not
Criterion change the
winner of the
election.
Though no voting system can satisfy all of the fairness criteria,
some voting systems are better than others in some situation.
We hope that those in power will pick the voting method that is
most fair based on the type of election.
Voting System in the
Philippines
The voting system in the Philippines follows a democratic
process where citizens elect public officials through periodic
elections. Here’s an overview of how it works:
Types of Elections: Elections in the Philippines include:
• National Elections for positions like the President, Vice
President, Senators, and party-list representatives.
• Local Elections for Governors, Mayors, Councilors, and
other local officials.
• Barangay Elections for the smallest administrative units,
called barangays.
Voting System in the
Philippines
Voter Eligibility: Filipino citizens aged 18 and above who are registered voters can
participate in elections. Voter registration is managed by the Commission on Elections
(COMELEC), and re-registration is required if a voter has been inactive for multiple election
cycles.

Voting Process:
• Automated System: The Philippines uses an automated election system where voters
mark their ballots and submit them into a machine that automatically counts votes.
This system has been in use since 2010 for faster and more transparent counting.
• Election Day: Elections are held on a single day, typically in May for national and local
elections, and in October for barangay elections. Election Day is a public holiday to
encourage voter participation.
Voting System in the
Philippines

•Vote Counting and Results: After polls close, vote-counting

machines transmit results to central servers where COMELEC
consolidates and verifies them. Results are typically available within
a few hours to days after the polls close.

•Election Laws and Regulations: Various laws govern elections,

including restrictions on campaigning, spending limits, and
prohibitions on activities such as vote-buying, which are enforced by
COMELEC and supported by law enforcement.