Plurality Method

Learning Outcomes

Determine the winner of an election using preference ballots
Evaluate the fairness of an election using preference ballots

Preference Schedules

To begin, we’re going to want more information than a traditional ballot normally provides. A traditional ballot usually asks you to pick your favorite from a list of choices. This ballot fails to provide any information on how a voter would rank the alternatives if their first choice was unsuccessful.

Preference ballot

A preference ballot is a ballot in which the voter ranks the choices in order of preference.

Data presentation

In the tables below, note the two variables present: city-order and number of votes. The first table lists the results of the vote. After tallying up the order preferences though, the voter name no longer matters. The second table lists how many votes each possible city-order received.

We can see that, given a list of three cities A, O, and H, there are 6 possible orderings that can be made.

The calculation to find the number of possible orderings from [latex]n[/latex] given choices is [latex]n![/latex]. The [latex]![/latex] symbol is called factorial . It is the product of the given integer [latex]n[/latex] with each of the integers below it, up to and including [latex]1[/latex]. In this case, [latex]3[/latex] choices provide [latex]3 \cdot 2 \cdot 1 = 6[/latex] choices. Here they are.

AOH, OAH,OHA, AHO, HAO, HOA.

There were two possible city-orders that received no votes: OAH and HOA. The four remaining orderings are listed in the second table under the number of votes each received.

Example

A vacation club is trying to decide which destination to visit this year: Hawaii (H), Orlando (O), or Anaheim (A). Their votes are shown below:

	Bob	Ann	Marv	Alice	Eve	Omar	Lupe	Dave	Tish	Jim
1st choice	A	A	O	H	A	O	H	O	H	A
2nd choice	O	H	H	A	H	H	A	H	A	H
3rd choice	H	O	A	O	O	A	O	A	O	O

These individual ballots are typically combined into one preference schedule, which shows the number of voters in the top row that voted for each option:

	1	3	3	3
1st choice	A	A	O	H
2nd choice	O	H	H	A
3rd choice	H	O	A	O

Notice that by totaling the vote counts across the top of the preference schedule we can recover the total number of votes cast: [latex]1+3+3+3=10[/latex] total votes.

The following video will give you a summary of what issues can arise from elections, as well as how a preference table is used in elections.

Plurality

The voting method we’re most familiar with in the United States is the 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

In our election from above, 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

For the plurality method, we only care about the first choice options. Totaling them up:

Anaheim: [latex]1+3=4[/latex] 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

Three candidates are running in an election for County Executive: Goings (G), McCarthy (M), and Bunney (B)^[1] The voting schedule is shown below. Which candidate wins under the plurality method?

	44	14	20	70	22	80	44
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.

Solution

What’s Wrong with Plurality?

The election from the above example 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.

Fairness Criteria

The fairness criteria are statements that seem like they should be true in a fair election.

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.

Study strategy

Work through the EXAMPLES and TRY IT problems below carefully, using pencil and paper, perhaps more than once.

Remember, math must be performed to be understood well.

Example

In the election, 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
1st choice	A	A	O	H
2nd choice	O	H	H	A
3rd 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

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 [latex]342+214+298=854[/latex] 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.

If you prefer to watch a video of the previous example being worked out, here it is.

Example

The following preference schedule came from an election whose candidates are Eric (E), Bridgette (B), Kelsie (K), and Daniel (D). Determine if there is a Condorcet winner in this election.

	29	21	18	10	1
1st choice	D	E	B	K	K
2nd choice	K	K	E	B	B
3rd choice	E	B	K	E	D
4th choice	B	D	D	D	E

Remember that in using the Condorcet method, you want to examine head-to-head comparisons:

Daniel vs Bridgette: 29 prefer Daniel; 50 (21+18+10+1) prefer Bridgette: Bridgette is preferred
Daniel vs Kelsie: 29 prefer Daniel; 50 (21+18+10+1) prefer Kelsie: Kelsie is preferred
Daniel vs Eric: 30 (29+1) prefer Daniel; 49 (21+18+10) prefer Eric: Eric is preferred
Bridgette vs Kelsie: 18 prefer Bridgette; 61 (29+21+10+1) prefer Kelsie: Kelsie is preferred
Bridgette vs Eric: 29 (18+10+1) prefer Bridgette; 50 (29+21) prefer Eric: Eric is preferred
Kelsie vs Eric: 40 (29+10+1) prefer Kelsie; 39 (21+18) prefer Eric: Kelsie is preferred

In this case, Kelsie is the Condorcet winner, because she is preferred in the majority of the head-to-head comparisons with the other candidates.

Try It

Consider the election from the previous Try It. Is there a Condorcet winner in this election?

	44	14	20	70	22	80	44
1st choice	G	G	G	M	M	B	B
2nd choice	M	B		G	B	M
3rd choice	B	M		B	G	G

Solution

Insincere Voting

Situations 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.

The following video gives another mini lesson that covers the plurality method of voting as well as the idea of a Condorcet Winner.

This data is loosely based on the 2008 County Executive election in Pierce County, Washington. See http://www.co.pierce.wa.us/xml/abtus/ourorg/aud/Elections/RCV/ranked/exec/summary.pdf ↵

Module 7: Voting Theory