Selected topics in finite mathematics/Majority criterion

The majority fairness criteria is a method used to determine a "fair" voting system. In this criteria, if a person has more than 50% of the first place votes and does not win the election then the majority fairness criteria has been violated.

Objectives
Our learning objectives are:
 * 1) Acknowledge what is Majority criterion
 * 2) Try different voting systems to test if they satisfy the majority criterion (voting systems include plurality, condorcet, sequential pairwise elections, borda count, and sequential runoff)
 * 3) Find out which methods satisfy the majority criterion and which do not

Details
If a candidate wins more than 50% of the first place votes in an election, then to satisfy the majority criterion, that candidate should win the election. If no candidate wins more than 50% of the first place votes, the criterion does not apply to that election.

An election method satisfies the Majority Criterion if a candidate favored by a majority of voters is guaranteed to win.

If every possible election in a voting system satisfies the majority criterion, then the voting system satisfies the majority criterion.

Examples
Plurality does satisy the majority criterion. The winner of the plurality voting method is the person with the most first place votes. If someone has the majority of first place votes, then they will win the election. This satisfies the majority criteria because the person with the majority of first place votes will always win the election. However, a candidate can win the election according to plurality without the majority of the votes. For example if candidate A has 15 votes, candidate B as 13 and candidate D has 7 votes then candidate A wins even though they don't have the majority of the votes. In this case, the majority criteria would not be applicable so plurality still satisfies the majority criteria.

The condorcet method does satisfy the majority criterion because if one candidate receives the majority of the first place votes then no matter how the other votes are distributed the candidate with the majority of votes will win the election. Using the condorcet method, a candidate can win the election without the majority of first place votes, but if no one candidate had the majority of first place votes then the majority criterion would not be applicable. For example:

A is going to beat every candidate in a head to head contest because it has the majority of the votes. No matter how the rest of the votes are spread out, A will always win the election. The condorcet voting method satisfies the majority criteria because if a candidate has the majority of the first place votes, then it will always win the election using this voting method.

Sequential pairwise elections do satisfy the majority criteria because if a candidate has more than 50% of the first place votes, then no matter the agenda, that candidate will be able to beat every other candidate in the head to head battle and win the election. Sequential pairwise elections consist of head to head battles between candidates and if one candidate beats another, then that candidate is out of the election even if they could beat candidates in an election that beat the candidate that beat them. However, the majority criterion only applies if one person has more than 50% of the first place votes, and if a candidate has this, then they will be able to defeat any other candidate and the specific agenda wouldn't make a difference. For example in the table below, candidate B is going to win the election no matter what the agenda is because B has more than 50% of the first place votes so the candidate would defeat every other candidate.

However, the majority criterion only applies when a candidate has more than 50% of the first place votes, so if no candidate has this, then the criteria doesn't apply. The criteria also fails to apply in case of a tie. If the criteria does not apply, the specific agenda chosen can alter the results of the election.

Sequential runoffs does satisfy the majority criteria because if one candidate has the majority of first place votes then that candidate will never be the candidate eliminated from the preference schedule. In sequential runoffs, the least favored candidate is eliminated and as this happens the preference schedule changes to reflect the elimination of a candidate. However, no matter which candidates are eliminated, the candidate with the majority of the first place votes will always win the election because since the candidate has the majority number, he or she will never lose and therefore never be eliminated. However, this criteria does not always apply. If no one candidate has the majority of the first place votes, then the majority criteria does not apply.

Nonexamples
Borda count fails to satisfy the majority criterion. By definition, Borda count declares a winner of an election to be the candidate with the most points. Points are assigned to each place, like so: 3 points for first place, 2 points for second place, 1 point for third place, and 0 points for last place. The following is a nonexample:

In this example, candidate A has the majority of first place votes but does not win the election according to the borda count voting method. While A had the most first place votes, other voters ranked this candidate very low on their preference schedule and therefore candidate A didn't get any more points. According to the borda count, candidate B won the election with 17 votes. Because candidate A had the majority of first place votes and did not win the election, this method fails to satisfy the majority criteria.

Like other voting methods, the majority criteria does not apply when no one candidate has more than 50% of the first place votes.

FAQ
If no one has a majority of first-place votes, is the majority criterion automatically considered satisfied?

Homework
[Create two elections: one which satisfies the criteria and one which does not?]

An election that satisfies the criteria: The below example is an example of the plurality voting system.

Solution [''Person A has 4 votes, Person B has 2 votes, and person C has 0 votes (first place votes). In this example, A has the majority of the first place votes, so majority criteria does apply and using the plurality method, A should win the election because they have the most votes. Since A has the majority of the first place votes and should win the election, the criteria is satisfied.'']

An election that does not satisfy the majority criteria: Borda count method

Solution [''In this example, candidate B has the majority of the first place votes, so according to the majority criteria, B should win the election, however candidate A wins according to the borda count method. Based on how to calculate each candidates points (borda count section) candidate A has 21 points, B has 18 points, C has 8 points, and D has 13 points. Even though candidate B had the majority of the first place votes, candidate A still wins the election according to Borda count because A had the most points.'']