Majority criterion

# Majority criterion

Discussion

Encyclopedia
The majority criterion is a single-winner voting system criterion, used to compare such system
Voting system
A voting system or electoral system is a method by which voters make a choice between options, often in an election or on a policy referendum....

s. The criterion states that "if one candidate is preferred by a majority (more than 50%) of voters, then that candidate must win".

Some methods that comply with this criterion include any Condorcet method
Condorcet method
A Condorcet method is any single-winner election method that meets the Condorcet criterion, which means the method always selects the Condorcet winner if such a candidate exists. The Condorcet winner is the candidate who would beat each of the other candidates in a run-off election.In modern...

, instant-runoff voting
Instant-runoff voting
Instant-runoff voting , also known as preferential voting, the alternative vote and ranked choice voting, is a voting system used to elect one winner. Voters rank candidates in order of preference, and their ballots are counted as one vote for their first choice candidate. If a candidate secures a...

, and Bucklin voting
Bucklin voting
Bucklin voting is a class of voting systems that can be used for single-member and multi-member districts. It is named after its original promoter, James W. Bucklin of Grand Junction, Colorado, and is also known as the Grand Junction system...

.

Some methods which give weight to preference strength fail the majority criterion, while others pass it. Thus the Borda count
Borda count
The Borda count is a single-winner election method in which voters rank candidates in order of preference. The Borda count determines the winner of an election by giving each candidate a certain number of points corresponding to the position in which he or she is ranked by each voter. Once all...

and range voting
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

fail the majority criterion, while the Majority judgment
Majority Judgment
Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner. If more than one candidate has the...

passes it. The application of the majority criterion to methods which cannot provide a full ranking, such as approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

, is disputed.

These methods that fail the majority criterion may offer a strategic incentive to voters to bullet vote, i.e., vote for one candidate only, not providing any information about their possible support for other candidates, since, with such methods, these additional votes may aid their less-preferred.

## Comparison with the Condorcet criterion

By the majority criterion, a candidate X should win if a majority of voters answers affirmatively to the question 'Do you prefer X to every other candidate?'.

The Condorcet criterion
Condorcet criterion
The Condorcet candidate or Condorcet winner of an election is the candidate who, when compared with every other candidate, is preferred by more voters. Informally, the Condorcet winner is the person who would win a two-candidate election against each of the other candidates...

is stronger. According to it, a candidate X should win if for every other candidate Y there is a majority of voters that answers affirmatively to the question 'Do you prefer X to Y?'.

Satisfaction of the Condorcet criterion implies that of the majority criterion, but not vice versa. With the Condorcet criterion the individuals comprising the majorities of voters answering affirmatively may vary according to Y, but the majority criterion requires a single majority which has X as their first choice, preferred to every other candidate.

In the statement that Condorcet criterion is stronger than the majority criterion, the word criterion must be understood as a criterion that a voting system may or may not satisfy, not as a criterion that a candidate must satisfy in order to win the election.

## Application of the majority criterion: Controversy

The majority criterion was initially defined with respect to voting systems based only on preference order. It is ambiguous how to apply it to systems with absolute rating categories such as Approval
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

, Range
Range voting
Range voting is a voting system for one-seat elections under which voters score each candidate, the scores are added up, and the candidate with the highest score wins.A form of range voting was apparently used in...

, and Majority Judgment
Majority Judgment
Majority Judgment is a single-winner voting system proposed by Michel Balinski and Rida Laraki. Voters freely grade each candidate in one of several named ranks, for instance from "excellent" to "bad", and the candidate with the highest median grade is the winner. If more than one candidate has the...

.

For approval voting, the difficulty is that the criterion refers to an exclusive preference, and it is unstated whether this preference is actually indicated on the ballot or not. The common simple statement of the criterion, as given in the introduction to this article, does not resolve this, for the word "prefer" can refer to a mental state or to an action; a complete statement of the criterion would either refer to actual marks on the ballot showing the required preference, or it could refer to the mental state of the voters. Since an approval voting
Approval voting
Approval voting is a single-winner voting system used for elections. Each voter may vote for as many of the candidates as the voter wishes. The winner is the candidate receiving the most votes. Each voter may vote for any combination of candidates and may give each candidate at most one vote.The...

ballot, for example, allows a voter to conceal the necessary exclusive preference without voting insincerely, the method cannot determine a majority preference based on what is not shown on the ballot. Thus, if "prefer" means an action, approval voting passes this criterion; if it means a mental state, approval voting does not pass.

For Majority Judgment, the difficulty is different. There are presumed to be enough rating categories to express any salient mental preference. But the word "prefer" could be interpreted in a relative sense, as rating the preferred candidate above any other candidate (in which case the method does not pass); or in an absolute sense, as rating the preferred candidate with the highest available rating (in which case it does).

Although the criterion's exact definition with respect to Range voting is unclear, the result is not: unstrategic Range voting does not pass this criterion under any definition.

### Approval voting

Approval voting does not satisfy the majority criterion. Approval voting is not a preferential voting
Preferential voting
Preferential voting is a type of ballot structure used in several electoral systems in which voters rank candidates in order of relative preference. For example, the voter may select their first choice as '1', their second preference a '2', and so on...

method, and therefore its advocates argue that the majority criterion should not be applied to it; this contention is controversial among election methods experts.

An example of approval voting's majority criterion failure:

Suppose 100 voters have the following preferences:
 55: A>B>C 35: B>C>A 10: C>B>A

Next, suppose they cast the following votes:
 55: A+B 35: B+C 10: C+B

B wins with 100 votes to A's 55 and C's 45. Note, however, that 55% of the voters indicated they approved of both B and A, and approval ballots have no way to indicate preferences between two different 'approved' candidates. If those voters had realized they were the majority, they would have voted for A alone, and won with 55 votes to B's 45 and C's 45. Note, however, that if the voters are aware that A and B are the front-runners, they would be more likely to vote strategically, and it would be unusual for a majority of voters to approve both front-runners, as is the case in this example.

### Borda count

For example 100 voters cast the following votes:
 55: A>B>C 35: B>C>A 10: C>B>A

A has 110 Borda points (55 x 2 + 35 x 0 + 10 x 0). B has 135 Borda points (55 x 1 + 35 x 2 + 10 x 1). C has 55 Borda points (55 x 0 + 35 x1 + 10 x 2).
 A 110 B 135 C 55

Candidate A is the first choice of a majority of voters but candidate B wins the election.

### Range voting

For example 100 voters cast the following votes:

 # Voters Ballot 80 A:10 B:9 C:0 20 A:0 B:10 C:0

Candidate B would win with a total of 80*9 + 20*10 = 720 + 200 = 920 rating points, versus 800 for candidate A.

Because candidate A is rated higher than candidate B by a (substantial) majority of the voters, this voting system fails to satisfy the criterion.

### Majority Judgment

If "prefer" means to rank a candidate at the top rating, this method passes.

For example 100 voters cast the following votes:

 # Voters Ballot 51 A:Excellent B:Good C:Poor 49 A:Poor B:Excellent C:Poor

Candidate A would win with a median rating of Excellent, versus Good for candidate B. Thus, this voting system satisfies the criterion.

Note that if the majority criterion is interpreted as in Arrow's theorem, to refer strictly to preference order and not to absolute ratings, this system can fail:

 # Voters Ballot 49 A:Excellent B:Good C:Poor 2 A:Fair B:Poor C:Poor 49 A:Poor B:Excellent C:Poor

B's median is Good. A is preferred by a majority, but some of that majority rated A as only Fair (leaving no candidates rated Excellent or Good), so A's median is Fair. B would win.

With only 3 ranking categories (Excellent, Good, and Poor) A's lowest possible median would be Good, and so a nonmajority B's median could at most tie that of a majority A. But B would win the majority judgment tiebreaker after 2 ratings were removed from each candidate.