• Determine the winner and assess the fairness of an election using preference ballots
• Determine the winner and assess the fairness of an election using the Instant Runoff method
• Determine the winner and assess the fairness of an election using a Borda count
• Determine the winner and assess the fairness of an election using Copeland’s method
• Determine the winner and assess the fairness of an election using the Approval Voting method
• Apply Arrow’s Impossibility Theorem

In many decision making situations, it is necessary to gather the group consensus. This happens when a group of friends decides which movie to watch, when a company decides which product design to manufacture, and when a democratic country elects its leaders.

While the basic idea of voting is fairly universal, the method by which those votes are used to determine a winner can vary. Amongst a group of friends, you may decide upon a movie by voting for all the movies you’re willing to watch, with the winner being the one with the greatest approval. A company might eliminate unpopular designs then revote on the remaining. A country might look for the candidate with the most votes.

In deciding upon a winner, there is always one main goal: to reflect the preferences of the people in the most fair way possible.

Preference Ballot

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. A preference ballot is a voting system where voters rank the candidates or options in order of preference. It allows voters to indicate their preferences for multiple choices rather than selecting just one option.

preference ballot

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

Let’s look at an 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

Once you have the preference votes of all individuals it is now time to count the ballots. It’s not merely a matter of tallying the number of supporters for each candidate. Consideration must be given to how many voters ranked each candidate as their first, second, and third choice. A more systematic approach to presenting these results is required. This process is commonly referred to as creating a preference schedule or a preference table.

preference schedule

A preference schedule is a table used to organize how people rank different options or choices in an election. It helps us see and understand the preferences of different individuals or groups.

Let’s return to our example and create a preference schedule from the results.

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

 

For these results, we create the following preference schedule. Notice the top row shows the number of voters that voted for each ranking:

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
1st choice	A	A	O	H
2nd choice	O	H	H	A
3rd choice	H	O	A	O

 

Reading this table we can gather one person voted for the ranking A, O, H, three people voted for the ranking A, H, O, three people voted for the ranking O, H, A, and three people voted for the ranking H, A, O.

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.

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.

plurality method

In the plurality method of voting, voters choose their preferred candidate or option, and the candidate with the most votes, even if it’s not an absolute majority, wins the election.

In our vacation club election example from the previous page, we had the preference table:

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
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: [latex]3[/latex] first-choice votes

Hawaii: [latex]3[/latex] first-choice votes

Anaheim is the winner using the plurality voting method.

Notice that Anaheim won with [latex]4[/latex] out of [latex]10[/latex] votes, [latex]40\%[/latex] of the votes, which is a plurality of the votes, but not a majority.

What’s Wrong with Plurality?

The election from the vacation club example may seem totally clear, 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: [latex]7[/latex] out of the [latex]10[/latex] would prefer Anaheim over Orlando

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
1st choice	A	A	O	H
2nd choice	O	H	H	A
3rd choice	H	O	A	O

 

Anaheim vs Hawaii: [latex]6[/latex] out of [latex]10[/latex] would prefer Hawaii over Anaheim

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
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 criterion

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

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.

Condorcet Winner

The Condorcet Winner refers to a candidate who would win in a one-to-one comparison against every other candidate in a multi-candidate election.

Returning to our vacation club election example, what choice is the Condorcet Winner?

We see above that Hawaii is preferred over Anaheim. Comparing Hawaii to Orlando, we can see [latex]6[/latex] out of [latex]10[/latex] would prefer Hawaii to Orlando.

	[latex]1[/latex]	[latex]3[/latex]	[latex]3[/latex]	[latex]3[/latex]
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.

Consider a city council election in a district that is historically [latex]60\%[/latex] Democratic voters and [latex]40\%[/latex] 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:

	[latex]342[/latex]	[latex]214[/latex]	[latex]298[/latex]
1st choice	Elle	Don	Key
2nd choice	Don	Key	Don
3rd choice	Key	Elle	Elle

Show Solution

Using plurality method:

We can see a total of [latex]342+214+298=854[/latex] voters participated in this election. Computing percentage of first place votes:

Don: [latex]\frac{214}{854} = 25.1\%[/latex]

Key: [latex]\frac{298}{854} = 34.9\%[/latex]

Elle: [latex]\frac{342}{854} = 40.0\%[/latex]

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 [latex]40\%[/latex] of the vote.

Analyzing this election closer, we see that it violates the Condorcet Criterion. Analyzing the one-to-one comparisons:

Elle vs Don: [latex]342[/latex] prefer Elle; [latex]512[/latex] prefer Don: Don is preferred

Elle vs Key: [latex]342[/latex] prefer Elle; [latex]512[/latex] prefer Key: Key is preferred

Don vs Key: [latex]556[/latex] prefer Don; [latex]298[/latex] 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.

Here is a video showing the worked solution to this example:

You can view the transcript for “Condorcet winner and insincere voting with plurality method” here (opens in new window).

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.