MMW | Module 6.1 & 6.2 MATHEMATICS OF VOTING, and THE MATHEMATICS OF APPORTIONMENT

Plurality Winner

Candidate with more first place votes than the other

Plurality Method

The most votes

Majority Winner

Candidate with more than 50% of the first-place votes

Majority Method

Most of the Votes

Not Every Plurality is Majority

Every majority is plurality but?

Voting Method

Mathematical procedure that uses data from the preference table to determine a winner

Majority Method

It is a good method of voting but only guarantees a winner if there are two candidates and an odd number of voters.

Majority Method

If there are more than two candidates, it is possible that none of the candidates receives a majority and thus no winner could be determined by majority rules

Plurality Method

This is by far the simplest and widely-used voting method.

Plurality Method

It may require a tie-breaker though.

Plurality Method

Voting system here in the Philippines uses this method in barangay, local, and national elections.

Plurality Method

Voters simply choose candidates without ranking them.

Plurality Method

Pres. Duterte wins the 2016 election using the?

Majority Method

Pres. Bongbong Marcos wins the 2022 election using the?

Borda Count Method

Each voter ranks all the candidates; that is, each voter selects his or her first choice, second choice, and third choice, and so on. If there are n candidates, each candidate receives n point for each first-place vote, (n-1) points for each second-choice vote, (n-2) points for each third-choice, and so on. The total points received for each candidate from all voters are added.

Borda Count Method

The candidate with the most total points (referred to as a candidate's Borda score) is declared the winner.

Plurality with Elimination Method

Each person votes for his or her favorite candidate receives a majority of votes, that candidate is declared the winner. If no candidate receives a majority, then the candidate with the fewest votes is eliminated and a new election is held. The process continues until a candidate receives a majority of the votes.

Pairwise Comparison Method

This method is otherwise known as Copeland's method.

Copeland's Method

This method is otherwise known as Pairwise Comparison Method

Pairwise Comparison Method

It is sometimes referred to as the head-to-head method or one-to-one comparison.

Pairwise Comparison Method

Each voter ranks all of the candidates; that is, each voter selects his or her first-choice, second choice, third choice, and so on. For each possible pairing of candidates, the candidate with the most votes receives 1 point; if there is a tie, each candidate receives 0.5 point. The candidate who receives the most points is declared The winner.

1

PAIRWISE COMPARISON METHOD: The candidate with the most votes receives __ point

0.5

PAIRWISE COMPARISON METHOD: If there is a tie, each candidate receives __ point.

Weighted Voting System

This arises among shareholders, where each voters controls a number of votes

Players

The voters are called the _______, denoted by Pₙ

Weight

The number of votes held by a voted is called a ______ and is a positive number, denoted by wₙ

Quota

This is the minimum number of votes in order to pass a motion, denoted by q. This must be over half the total weigthts and cannot be more that total weight

One Person, One Vote

Example: (5: 1, 1, 1, 1, 1, 1, 1)

Each person has one vote and five votes, a majority, are required to pass a measure.

Dictartorship

Example: (20, 21, 6, 5, 4, 3)

The person with 21 votes can pass a measure. Even if the remaining four people get together, their votes do not total the quota of 20.

Null System

Example: (28: 6, 3, 5 2)

If all the members of this system vote for a measure, the total number of votes is 16, which is less than the quota. Therefore, no measure can be passed.

Veto Power System

Example: (21: 6, 5, 4, 3, 2, 1)

The sum of all the votes is 21, the quota. Therefore, if any of one voter does not vote for the measure, it will fail. Each voter is said to have a veto power.

Dictator

A player will be a ______ if their weight is greater than or equal to the quota.

Dictator

This can also block any proposal from passing; the other players cannot reach the quota without the dictator.

Veto Power

A player has __________ if their support is necessary for the quota to be reached.

Veto Power

It is possible for more than one player to have veto power, or for no player to have veto power.

Dummy

A player is a _____ if their vote is never essential for a group to reach quota.

Coalition

This is any group of players voting the same way.

Weight

A coalition is a winning coalition if the coalition has enough _____ to meet the quota.

Critical Player

A player is ________ in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition.

61

Based on the 1987 Constitution Filipino voters will pick groups that would comprise 20% of the House of Representatives, or ___ seats, with the remaining 80% elected per legislative district.

2

There will be __ rounds of seat allocation for the party-list group.

Party-List

The COMELEC will rank them according to the total number of votes.

2%

A party-list group that secures at least ___ of the total votes in the party-list race is entitled to at least 1 slot in the House during the first round of seat allocation.

Apportionment

A method of dividing a group of people (or other resources) and assigning them to different places.

Seats

The "_____" are the people or items that are to be shared equally.

States

The "____" are the parties that will receive a proportional share of the "seats".

Standard Divisor

This is the ratio of the total population to the number of seats.

Standard Divisor

It tell us how many people are represented by each seat

SD = total population/number of seats

FORMULA FOR STANDARD DIVISOR

Standard Quota

This is the exact number of seats that should be allocated to each state if decimal values were possible.

SQ = state population/standard divisor

FORMULA FOR STANDARD QUOTA

Lower Quota

It is the standard quota rounded down.

Higher Quota

It is the standard quota rounded up

Hamilton’s Method

Alexander Hamilton proposed this method. He determines how many representatives each state should get by following some steps.

Alexander Hamilton

Who proposed Hamilton’s method?

Hamilton’s Method

This method was approved by Congress in 1791 but was vetoed by President Washington.

Hamilton’s Method

It was later adopted in 1852 and used through 1911.

Jefferson’s Method

Thomas Jefferson proposed a different method of apportionment.

Thomas Jefferson

He proposed a different method of apportionment.

Jefferson’s Method

After Washington vetoed Hamilton's method, this method was adopted and used in Congress from 1791 through 1842.

Jefferson’s Method

This method favors larger states.

Thomas Jefferson

He had political reasons for wanting this method to be used rather than Hamilton's.

Thomas Jefferson

He would also argue that it's the ratio of people to representatives that is the critical thing, and apportionment methods should be based on that.