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# Voting Methods: Majority Rule, Plurality, Single Run-off

### Learning Outcomes

• Determine the winner of an election using majority rule.
• Determine the winner of an election using the plurality method.
• Construct a preference schedule.
• Determine the winner of an election using a single runoff.

## Majority Rule

In the case that there are only two choices, the simplest voting method is majority rule.  Voters select one choice and the choice that receives more than 50% of the votes wins.  Our U.S. Presidential Election is an example of majority rule, though it is not the majority of the population that is considered.  Rather it is a simple majority of electoral votes that determines the election outcome.

### Majority Rule

Majority rule is a voting method in which the winner must receive more than 50% of the votes.

### Example

A vacation club is trying to decide which destination to visit this year: Hawaii (H) or Anaheim (A). Their votes are shown below:
 Bob Ann Marv Alice Eve Omar Lupe Dave Tish Jim Erika 1st choice A A H H A H H H H A A
These individual ballots can be combined in a table showing the number of voters in the top row that voted for each option:
 Number of votes 5 6 1st choice A H
Notice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: $5+6=11$ total votes.  Since there are only two choices and Hawaii has won the majority (~55%) of the votes, then Hawaii is the winning destination.

## Plurality Method

In this method, the choice with the most first-preference votes is declared the winner. Ties are possible, and would have to be settled through some sort of run-off vote. This method is sometimes assumed to be a "Majority Rules" method, but in fact, it is not necessary for a choice to have gained a majority of votes to win.  A majority is over 50%; in a contest with more than two choices, it is possible for a winner to have a plurality (the most first-preference votes) without having a majority.

### Plurality Method

The plurality method is a voting method in which the choice with the most votes wins.  As noted above, the winning choice need not have captured a majority of votes in a content with more than two choices.

### Example

The following table illustrates the results of a four-way race for Psychology Club President.  Notice that while Araceli received the most votes, no candidate received a majority (more than 50%) of the vote.  Using the plurality method, Araceli would win the election.
 Percentage of the Vote Liam 26% Sophia 19% Araceli 30% Enrique 25%
However, there are some potential problems with the plurality method.  Suppose, for example that all of the voters who chose Sophia and Enrique actually prefer Liam (over Araceli) as their second choice.  Then, while Araceli would still have the most first place votes, Liam would actually be preferred over Araceli by 70% of the voters.  One way to address this issue is to consider all of the voters' preferences.

### Preference Schedules

A traditional ballot usually asks you to pick your favorite from a list of choices and fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.  As we saw in the previous example, in the case of more than two candidates (or choices), this type of ballot can lead to an unfair outcome.  A preference ballot is a ballot that records all of the voter's choices in order of preference.  Tabulating a preference ballot requires a type of table called a preference schedule that details the number of voters who chose each ranking order among the candidates.   The following video gives an excellent summary of how the plurality method can lead to unfair results and how to construct and use a preference schedule in elections. https://www.youtube.com/watch?v=WdtH_8lAqQo

### Example

Consider again our vacation club.  They have decided to consider a third possible destination and so now they are trying to decide between: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below:
 Bob Ann Marv Alice Eve Omar Lupe Dave Tish Jim Erika 1st choice A A H H A O H O H A A 2nd choice O H A A H H A H A H O 3rd choice H O O O O A O A O O H
The individual ballots listed above are typically combined into one preference schedule (table below), which shows the number of voters in the top row that voted for each option.  You might notice that two of our six possible city-orders received no votes: OAH and HOA.  The four remaining orderings are listed in the preference schedule below under the number of votes each received.
 2 3 2 4 1st choice A A O H 2nd choice O H H A 3rd choice H O A O
For the plurality method, we only care about the first choice options. Totaling them up: Anaheim: $2+3=5$ first-choice votes Orlando: 2 first-choice votes Hawaii: 4 first-choice votes Anaheim is the winner using the plurality voting method. Again, we notice that by totaling the vote counts across the top of the preference schedule we obtain the total number of votes cast: $2+3+2+4=11$ total votes.  Thus, we see that Anaheim won with 5 out of 11 votes, ~45% of the votes, which is a plurality of the votes, but not a majority.   Looking again at our preference schedule, we observe that while Anaheim received the most first place votes (five), the other six people prefer Hawaii over Anaheim.  It doesn't quite seem fair that Anaheim is the winning trip despite the fact that 6 out of 11 voters (~55%) would have preferred Hawaii! Hawaii vs Anaheim: 6 to 5 preference for Hawaii over Anaheim
 2 3 2 4 1st choice A A O H 2nd choice O H H A 3rd choice H O A O
Since no option earned a majority (more than 50%) of first place votes, we can use the additional information included in our preference ballot to have a single run-off election.

### Single Run-off Method

In a race with three or more choices in which no choice receives a majority, the single run-off method eliminates all but the top two first place vote-getters and then reevaluates the votes to determine the winner using majority rule.

### Example

In the vacation club example, we would eliminate Orlando, since Anaheim and Hawaii both received more than two votes.
 2 3 2 4 1st choice A A O H 2nd choice O H H A 3rd choice H O A O
Our revised table would look like this:
 2 3 2 4 1st choice A A H H 2nd choice H H A A
Using this method, we see that Hawaii would win with a majority, 6 out of 11 votes (~55%).
Note that the single run-off method is similar to simply holding a second run-off election, but since every voter’s order of preference is recorded on the ballot, the runoff can be computed without requiring a separate (possibly costly) election.