Definition 1.2.1 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.
Section 1.2 Plurality
¶The voting method we’re most familiar with in the United States is the plurality method .
This method is sometimes mistakenly called the majority method, or majority rules , but it is not necessary for a choice to have gained a majority of votes to win. A majority is over 50 % ; it is possible for a winner to have a plurality without having a majority.
Example 1.2.2
In our election from above, we had the preference table:
| 1 | 3 | 3 | 3 | |
| 1 st choice | A | A | O | H |
| 2 nd choice | O | H | H | A |
| 3 rd choice | H | O | A | O |
For the plurality method, we only care about the first choice options. Totaling them up:
Anaheim: 1+3 = 4 first-choice votes
Orlando: 3 first-choice votes
Hawaii: 3 first-choice votes
Anaheim is the winner using the plurality voting method.
Notice that Anaheim won with 4 out of 10 votes, 40 % of the votes, which is a plurality of the votes, but not a majority.
Try it Now 1.2.4
Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B). The voting schedule is shown below. Which candidate wins under the plurality method?
| 44 | 14 | 20 | 70 | 22 | 80 | 39 | |
| 1 st choice | G | G | G | M | M | B | B |
| 2 nd choice | M | B | G | B | M | ||
| 3 rd choice | B | M | B | G | G |
Note: In the third column and last column, those voters only recorded a first-place vote, so we don’t know who their second and third choices would have been.
Subsection 1.2.1 What’s Wrong with Plurality?
¶The election from Example 2 may seem totally clean, but there is a problem lurking that arises whenever there are three or more choices. Looking back at our preference table, how would our members vote if they only had two choices?
Anaheim vs Orlando: 7 out of the 10 would prefer Anaheim over Orlando
| 1 | 3 | 3 | 3 | |
| 1 st choice | A | A | O | H |
| 2 nd choice | O | H | H | A |
| 3 rd choice | H | O | A | O |
Anaheim vs Hawaii: 6 out of 10 would prefer Hawaii over Anaheim
| 1 | 3 | 3 | 3 | |
| 1 st choice | A | A | O | H |
| 2 nd choice | O | H | H | A |
| 3 rd choice | H | O | A | O |
This doesn’t seem right, does it? Anaheim just won the election, yet 6 out of 10 voters, 60 % of them, would have preferred Hawaii! That hardly seems fair. Marquis de Condorcet, a French philosopher, mathematician, and political scientist wrote about how this could happen in 1785, and for him we name our first fairness criterion .
Definition 1.2.5 Fairness Criteria
The fairness criteria are statements that seem like they should be true in a fair election.
Definition 1.2.6 Condorcet Criterion
If there is a choice that is preferred in every one-to-one comparison with the other choices, that choice should be the winner. We call this winner the Condorcet Winner , or Condorcet Candidate.
Example 1.2.7
In the election from Example 2 , what choice is the Condorcet Winner?
We see above that Hawaii is preferred over Anaheim. Comparing Hawaii to Orlando, we can see 6 out of 10 would prefer Hawaii to Orlando.
| 1 | 3 | 3 | 3 | |
| 1 st choice | A | A | O | H |
| 2 nd choice | O | H | H | A |
| 3 rd choice | H | O | A | O |
Since Hawaii is preferred in a one-to-one comparison to both other choices, Hawaii is the Condorcet Winner.
Example 1.2.8
Consider a city council election in a district that is historically 60 % Democratic voters and 40 % Republican voters. Even though city council is technically a nonpartisan office, people generally know the affiliations of the candidates. In this election there are three candidates: Don and Key, both Democrats, and Elle, a Republican. A preference schedule for the votes looks as follows:
| 342 | 214 | 298 | |
| 1 st choice | Elle | Don | Key |
| 2 nd choice | Don | Key | Don |
| 3 rd choice | Key | Elle | Elle |
We can see a total of 342 + 214 + 298 = 854 voters participated in this election. Computing percentage of first place votes:
Don: 214/854 = 25.1 %
Key: 298/854 = 34.9 %
Elle: 342/854 = 40.0 %
So in this election, the Democratic voters split their vote over the two Democratic candidates, allowing the Republican candidate Elle to win under the plurality method with 40 % of the vote.
Analyzing this election closer, we see that it violates the Condorcet Criterion. Analyzing the one-to-one comparisons:
Elle vs Don: 342 prefer Elle; 512 prefer Don: Don is preferred
Elle vs Key: 342 prefer Elle; 512 prefer Key: Key is preferred
Don vs Key: 556 prefer Don; 298 prefer Key: Don is preferred
So even though Don had the smallest number of first-place votes in the election, he is the Condorcet winner, being preferred in every one-to-one comparison with the other candidates.
Try it Now 1.2.10
Consider the election from Try it Now 1. Is there a Condorcet winner in this election?
| 44 | 14 | 20 | 70 | 22 | 80 | 39 | |
| 1 st choice | G | G | G | M | M | B | B |
| 2 nd choice | M | B | G | B | M | ||
| 3 rd choice | B | M | B | G | G |
Subsection 1.2.2 Insincere Voting
¶Situations like the one in Example 8 above, when there are more than one candidate that share somewhat similar points of view, can lead to insincere voting . Insincere voting is when a person casts a ballot counter to their actual preference for strategic purposes. In the case above, the democratic leadership might realize that Don and Key will split the vote, and encourage voters to vote for Key by officially endorsing him. Not wanting to see their party lose the election, as happened in the scenario above, Don’s supporters might insincerely vote for Key, effectively voting against Elle.