Plurality vs. Majority Vote

Common Core Standards and Mathematical Practices

7th Grade: Number Systems
- 7.NS.A.1 Apply and extend previous understandings of addition and subtraction to add and subtract integers and other rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Mathematical Practices
- MP.2 Reason abstractly and quantitatively.
- MP.3 Construct viable arguments and critique the reasoning of others.
- MP.4 Modeling with Mathematics.

Learning Objectives

Students will be able to...

Calculate the outcome of elections using the plurality and majority voting methods.
Describe the difference between plurality and majority voting.
Describe the difference between majority and supermajority voting.
Identify problems with plurality voting, including limited expression, spoiler effect, and favorite betrayal.
Explain how a candidate can win the majority of votes but lose an election.
Use historical examples to explain how vote splitting and spoiler effect can occur.

Vocabulary

Simple Majority

Type of voting method where the candidate with the majority of the votes wins the election. Satisfies anonymity, neutrality, and monotonicity criterions.

Plurality

This voting method is a generalization of majority. In plurality voting, the candidate with the most votes wins the election. Sometimes this is called "first past the post" or "winner take all." Satisfies anonymity, neutrality, and monotonicity criterions.

Anonymity Criterion

Desirable voting method property; if two voters exchange ballots, the outcome of the election stays the same. In other words, all voters are treated equally.

Neutrality Criterion

Desirable voting method property; if all voters switch the positions of two candidates, then the group rankings of those candidates switch as well. In other words, all candidates are treated equally.

Monotonicity Criterion

Desirable voting method property; if candidate A is a winner and someone changes their ballot by moving A up (switching A with a candidate ranked above A), A is still the winner.

Near-decisiveness

An election is nearly decisive if if the only situation in which a tie can occur, is when both candidates receive the same number of votes.

Quota

A number q votes, or quota, a candidate must receive to win an election.

May's Theorem

Theorem saying that the simple majority voting method is the only voting method for a two-candidate election that satisfies the anonymity, neutrality, and monotonicity criterions while also being nearly decisive.

See link for information.

Supermajority (with parameter 'p')

Let p be a number such that 1/2 < p ≤ 1 and let t be the number of voters. The social choice function where the candidate who gets at least ⌈pt⌉ votes is declared the winner is called the supermajority method with parameter p.

May's Theorem says that the best method for two candidate elections is simple majority. But what if we have more than two candidates? The obvious generalization of majority is plurality.

What's wrong with plurality?

You can't express yourself

In plurality voting, we get a list of candidates but we can only express our opinion about one. In other voting methods, such as ranked choice voting, voters can provide more information about their thoughts and express their opinion about all candidates on the ballot.

Spoiler Effect

Plurality is sensitive to the spoiler effect. A "spoiler" candidate can enter the race and lose but take enough votes from another candidate to change the winner. Read about a famous example of the spoiler effect, the 2000 US Presidential Election, below.

Favorite Betrayal

In plurality, voters must choose between 1) voting for their first choice candidate at the risk that another more popular candidate will win or 2) voting between two front-runner candidates and abandoning their favorite. This means voters can be punished for voting for their first choice candidate!

Election Examples

Plurality vs. Majority

Consider an election with three candidates. Candidate A receives 40% of the votes, Candidate B receives 30% of the votes, and Candidate C receives 30% of the votes.

Candidate A wins even though the majority of voters (60%) did not choose this candidate.

2. Plurality vs. Majority (with possible Spoiler Effect)

A group of 4th graders are choosing a movie to watch in class. 8 students vote for Bolt, 3 students vote for Coco, and 6 students vote for The Incredibles. Based on these results, the class watches the movie Bolt.

8 votes

Bolt was the plurality winner, but not the majority winner.

3 votes

6 votes

Because the class was choosing between three movies, the plurality winner Bolt was not necessarily the movie with more than half the votes (majority winner.) The majority of students did not vote for Bolt, but it still won!

Since Coco and The Incredibles are both Pixar movies and Bolt is not, Coco might have be acting as a spoiler for The Incredibles. It is possible that Coco voters would have preferred a Pixar movie (The Incredibles) to a Walt Disney Animation Studios movie (Bolt.) If this is the case, then Coco took enough votes from the Incredibles to cost The Incredibles the election.

Consider what would have happened if the teacher held a run-off. If Coco voters were given the opportunity to vote for the Incredibles after Coco was eliminated, The Incredibles might have had 9 votes compared to Bolt's 8 votes. The Incredibles would have won the plurality vote and the majority vote (9/17 or 52%.) In elections between 2 candidates, the majority winner is also the plurality winner.

Runoffs don't always prevent a spoiler effect, but in this case it might have.

3. Vote Splitting

Vote splitting is a bad byproduct of the plurality voting method. Vote splitting brings unpopular or fringe candidates to the top.

Consider an election with 11 candidates, including one fringe candidate with very extreme political views. If 10 regular candidates each receive 9% of the vote and the fringe candidate receives 10% of the vote, the votes are split and the fringe candidate wins with only 10% of the vote. 90% of voters did not vote for the winning candidate!

Some Real World Examples of Issues with Plurality

1998: Minnesota Gubernatorial Election

The 1998 Minnesota gubernatorial race had three main candidates:

Jesse “The Body” Ventura (Reform): 37%
Norm Coleman (Republican): 34%
Skip Humphrey (Democrat): 28%

In this election, Ventura won because he had the plurality of votes. However, almost everyone who voted for Humphrey had Coleman as second choice. This means 62% of voter preferred Coleman over Ventura. Coleman and Humphrey had 'split' the non-reform voters between the two of them.

2000: Gore vs. Bush

Due to the use of the Electoral College, the result of the 2000 Presidential Election depended on the winner Florida's popular vote. As shown in the chart on the left, Gore lost to Bush by 1,725 votes in Florida. However, most Nader supporters would have (probably) voted for Gore had Nader not been in the election. Even though Nader had no chance of winning, he influenced the election via the spoiler effect.

2016: Republican Primary

In the 2016 Republican Party Primary, Trump won the plurality (but not majority) vote with 44.9%. However, polling suggests that Trump would have lost to Cruz, Rubio, and Kasich in one-on-one elections. This would suggest that generally, non-Trump voters (55.1%) would have preferred another non-Trump candidate. The non-Trump candidates split the non-Trump voters, allowing what was considered a fringe candidate to win.

Who uses each system?

It varies! Here are all the examples of different voting methods used around the world.

Here are a few examples of different systems!

United States Elections

Most elections in the United States use plurality. The one exception is presidential election, which uses the Electoral College.

Senegal National Assembly

Senegal's has a mixed member majoritarian system. As part of this election process, a simple majority party block vote (party list who wins majority list takes all seats in the district) gives some of the legislature's seats.

San Marino (European Micro State)

Elections for San Marino's Grand and General Counsel require a single coalition to have a majority of the popular vote. If not, a subsequent runoff election continues to determine which coalition will receive the majority of seats. (The chart above shows parties, not the coalitions.)

Mexico

Mexico has a multi and single winner electoral system for its legislative branch. In the lower house, 300 are voted by plurality and 200 are voted by proportional representation.

China

Through a tiered system of elections, members are indirectly elected by their respective municipal, provincial, and regional congresses.

Other Voting Systems:

Cardinal Methods

Cardinal voting is when a voting system in which each candidate is given an independent rating. Candidates are not ranked against each other.

See link for more information.

Ranked Choice Voting

Types of voting methods where voters rank candidates. Three prominent types are instant runoff or Hare's Method, Borda Count, and Condorcet Method.