Section 11.2 The Plurality Method
Objectives
Find the winner of an election using the plurality method
Determine if an election has a Condorcet candidate
Understand whether an election has violated the Condorcet criterion
Subsection 11.2.1 Plurality
Definition 11.2.1. 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.
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 11.2.2.
In our election from previous pages, we had the preference table:
| 1 | 3 | 3 | 3 | |
|---|---|---|---|---|
| 1st choice | A | A | O | H |
| 2nd choice | O | H | H | A |
| 3rd choice | H | O | A | O |
Who wins the election with the plurality method?
Solution.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.
Problem 11.2.4. Try It Now.
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?
| 14 | 44 | 20 | 70 | 22 | 80 | 39 | |
|---|---|---|---|---|---|---|---|
| 1st choice | G | G | G | M | M | B | B |
| 2nd choice | M | B | G | B | M | ||
| 3rd 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.
Answer.Using plurality method:
G gets \(44+14+20=78\) first-choice votes
M gets \(70+22=92\) first-choice votes
B gets \(80+39=119\) first-choice votes
Bunney (B) wins under plurality method.
Subsection 11.2.2 What's Wrong with Plurality?
The election from Example 11.2.2 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 | |
|---|---|---|---|---|
| 1st choice | A | A | O | H |
| 2nd choice | O | H | H | A |
| 3rd choice | H | O | A | O |
Anaheim vs Hawaii: 6 out of 10 would prefer Hawaii over Anaheim
| 1 | 3 | 3 | 3 | |
|---|---|---|---|---|
| 1st choice | A | A | O | H |
| 2nd choice | O | H | H | A |
| 3rd 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 11.2.8. Fairness Criteria.
The fairness criteria are statements that seem like they should be true in a fair election.
Definition 11.2.9. 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 11.2.10.
In the election in Example 11.2.2 is there a Condorcet candidate? If so, who?
Solution.We see above that Hawaii is preferred over Anaheim (6 out of 10 people prefer Hawaii to Anaheim). Comparing Hawaii to Orlando, we can that 6 out of 10 people ranked Hawaii abover Orlando.
Since Hawaii is preferred in a one-to-one comparison to both other choices, Hawaii is the Condorcet Winner.
Example 11.2.11.
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 | |
|---|---|---|---|
| 1st choice | Elle | Don | Key |
| 2nd choice | Don | Key | Don |
| 3rd 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: \(214 \div 854=25.1\%\)
Key: \(298 \div 854=34.9\%\)
Elle: \(342 \div 854=40.0\%\)
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.
Problem 11.2.13. Try It Now.
Consider the election in the preference schedule below. Is there a Condorcet winner in this election?
| 14 | 44 | 20 | 70 | 22 | 80 | 39 | |
|---|---|---|---|---|---|---|---|
| 1st choice | G | G | G | M | M | B | B |
| 2nd choice | M | B | G | B | M | ||
| 3rd choice | B | M | B | G | G |
Determining the Condorcet Winner:
G vs M: \(44+14+20=78\) prefer G, \(70+22+80=172\) prefer M: M preferred
G vs B: \(44+14+20+70=148\) prefer G, \(22+80+39=141\) prefer B: G preferred
M vs B: \(44+70+22=136\) prefer M, \(14+80+39=133\) prefer B: M preferred
M is preferred over both B and G, so M is the Condorcet winner, based on the information we have.
Subsection 11.2.3 Insincere Voting
Situations like the one in Example 11.2.11 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.