Preference Ballots and the Plurality Method: Videos & Practice Problems

Determine the winner of the election using the plurality method and determine whether they received the majority of the votes.
In the election for debate team president, Nancy got 26 votes, Frank got 12 votes, and George got 7 votes.

• A. Nancy, Without Majority votes
• B. Nancy, With Majority votes
• C. Frank, Without Majority votes
• D. Frank, With Majority votes

Answer: (B)

Determine the winner of the election using the plurality method and determine whether they received the majority of the votes.
In a vote to decide the school mascot, a tiger got 23 votes, a wildcat got 18 votes, a lion got 13 votes, and a panther got 7 votes.

• A. Tiger, Without majority votes
• B. Tiger, With majority votes
• C. Wildcat, Without majority votes
• D. Wildcat, With majority votes

Answer: (A)

Here's what students ask on this topic:

What is the plurality method in voting and how does it determine the winner?

The plurality method is a voting system where the winner is the candidate or option with the most first-place votes. Unlike other methods that consider full rankings or preferences, the plurality method only counts the number of first-place votes each candidate receives. The candidate with the highest number of these votes wins, regardless of whether they have a majority. This method is straightforward and commonly used in elections where voters select only their top choice. However, it can result in a winner who does not have more than half of the total votes, which means the winner may not have majority support but simply the largest share of votes.

How do you distinguish between a plurality winner and a majority winner?

A plurality winner is the candidate who receives the most first-place votes, but not necessarily more than half of all votes. A majority winner, on the other hand, is a candidate who receives more than 50% of all votes cast. In mathematical terms, a majority means the candidate's vote count $V$ satisfies $V>0.5\times T$, where $T$ is the total number of votes. This distinction is important because a plurality winner might win with less than half the votes if the vote is split among many candidates, while a majority winner has clear support from more than half the voters.

Is it necessary to have a full preference table to determine the winner using the plurality method?

No, a full preference table is not necessary when using the plurality method. Since the plurality method only considers first-place votes, knowing the total number of first-place votes for each candidate is sufficient to determine the winner. The rankings beyond the first place do not affect the outcome. This simplifies the counting process and makes the plurality method easier to implement compared to methods that require full preference rankings.

Why might the plurality method not always reflect the majority preference in an election?

The plurality method might not reflect the majority preference because it only counts first-place votes and ignores the preferences of voters who ranked other candidates first. If there are multiple candidates, the vote can be split so that the winner has the most first-place votes but less than 50% of the total votes. This means the winner may not have the support of the majority of voters, just the largest single group. This can lead to outcomes where a candidate wins despite most voters preferring other candidates, which is a limitation of the plurality method.

How can you calculate if a candidate has a majority using the plurality method?

To calculate if a candidate has a majority using the plurality method, you compare the number of first-place votes the candidate received to the total number of votes cast. The candidate has a majority if their first-place votes $V$ satisfy $V>0.5\times T$, where $T$ is the total votes. If this inequality holds, the candidate has more than half the votes and is a majority winner. If not, the candidate is only a plurality winner, having the most votes but less than a majority.