# Majority

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A majority, also called a simple majority or absolute majority to distinguish it from related terms, is more than half of the total.[1] A "majority" means, literally, "more than half". Compare this with plurality, which means "the largest number of the group". For example, if a group consists of 9 individuals, a majority would be 5 or more individuals, while having 4 or fewer individuals would not constitute a majority. "Majority" can be used to specify the voting requirement, as in a "majority vote", which means more than half of the votes cast (also sometimes called "50%+1"), since exactly 50% is only a plurality.See plurality for more.

When applied to specific situations, majority can take on different meanings, depending on how you apply it:

• "relative majority" is not a true majority. This is a plurality.
• simple majority of voters means "more than half of the voters" ("50%+1")
• absolute majority means "more than half of eligible voters". This can be used in contexts where less than 100% of eligible voters cast ballots.
• a supermajority is a fraction of the voters between half and all (e.g. 2/3)
• consensus usually means complete agreement or "all voters"

Note that when there are only two candidates, either one of them will have a majority of the voters with any preference preferring them, or there will be a tie. This is because in essence, a two-candidate election is like a two-sided scale where either there is a side with more weight (more voters), or both sides of the scale are evenly balanced (a tie). This is the reason why pairwise counting looks at all possible two-candidate elections when determining the majority winner.

There will not always be a majority winner, depending on the context and definition that's used.

## Majority rule

### Majority rule/Majority winner - Four Criteria

Many methods claim to elect the "majority winner" or work by "majority rule" (See, for example, the CVD's talking points re: IRV: [1]). However, Condorcet's paradox raises an issue: with some groups of voters, no matter which candidate wins, some majority of the voters will prefer a different candidate. Below is a list of criteria, in ascending order of strictness, which could be used to rank the relative strengths of a "majority." (See the following section for criticism of this explanation; this is a more Condorcet-based explanation rather than a neutral, comprehensive look).

• Criterion 1: If a majority of the electorate coordinates their efforts, they can assure that a given candidate is elected, or that another given candidate is not elected. (Weak form of majority criterion).
• Criterion 2: Mutual majority criterion
• Criterion 3: Condorcet criterion
• Criterion 4: Minimal dominant set (Smith, GeTChA) efficiency

• Criterion 1 only: Pseudomajority methods.
• Criteria 1 and 2 only: Weak majority rule methods.
• Criteria 1, 2, and 3: Intermediate majority rule methods.
• Criteria 1, 2, 3, and 4: Strong majority rule methods.

In pseudo-majority methods (like plurality and range voting), a given majority of the electorate can coordinate their intentions and decide the winner, but this merely postpones the question of how they do this. The stronger majority methods not only enable firmly coordinated majorities to assert themselves, but they allow un-coordinated majorities to reveal themselves, without any need for prior coordination. Voting methods that facilitate this process of revelation are considered superior to those that do not by majoritarian advocates.

The remaining three categories allow mutual majorities to reveal themselves (in the absence of a self-defeating strategy by supporters of this majority, or something like a chicken dilemma). Strong majority rule methods not only reveal mutual majorities, but they reveal minimal dominant sets and Condorcet winners (in the absence of a severe burying strategy). This is considered especially valuable because it means revealing possible compromises on divisive issues, thus avoiding a lot of political polarization and strife.

#### Criterion 1 only - Pseudo-Majority Rule Methods

Methods which pass criterion 1 only include Plurality, Approval, Cardinal Ratings, and the Borda count. Although it is always possible in these systems for a coordinated majority to elect their preferred candidate, coordination may be difficult. For example, take an electorate with preferences as follows:

31 A > B > C
29 B > A > C
20 C > B > A
20 C > A > B

In a plurality election, a clear majority (60-40) prefer both A and B to C. But unless A and B voters know whether to vote for A or whether to vote for B, C may win a plurality of votes. In addition, voters for A and B may play a game of "chicken", refusing to vote for the other, because they believe their candidate should win.

#### Criteria 1 and 2 - Weak Majority Rule Methods

Instant-runoff voting (aka IRV, Single-winner STV) passes the mutual majority criterion. In the example above, IRV enables A and B to coordinate. If all voters voted their sincere preferences, B would be eliminated first, but their votes would transfer to A, resulting in a majority for A.

However, IRV doesn't pass the Condorcet criterion. In an election with preferences as follows:

31 A > B > C
29 B > C > A
40 C > B > A

Looking at this election pairwise, there are three majorities: a majority (69 to 31) prefer B to A, a majority (69-31) prefer C to A, and a majority (60-40) prefer B to C. If you were to award the title "majority winner" to any candidate, B has the fairest claim to that title, as (different) majorities of voters prefer B to each other candidate. However, in IRV, B is eliminated first and does not win.

#### Criteria 1,2, and 3 - Intermediate Majority Rule Methods

Methods that pass the Condorcet criterion would always elect B, the Condorcet winner, in that election.

Minimax (aka Simpson-Kramer, PC, etc.), Nanson, Black, etc.

#### Criteria 1,2,3, and 4 - Strong Majority Rule Methods

Derived from an e-mail by James Green-Armytage

#### Criticism of this scheme

While criteria 2-4 above are popular, only criterion 2 (the Majority criterion for solid coalitions a.k.a. the Mutual majority criterion) deals with "majority" in the sense of "more than half of the voters," and even this criterion applies only in the peculiar special case that more than half of the voters rank the same set of candidates uninterrupted, in some order, in the top positions of the ballot.

Criterion 1 (that a coordinated majority can always elect a specific candidate) is extremely weak, and satisfied by almost any deterministic method.

Criteria 3 and 4 (the Condorcet criterion and Smith criterion) only deal with a "majority" in the sense of "more than half of the voters expressing an opinion between two given candidates." They don't make any assurance that a "majority" in the stronger sense will take precedence over a "majority" in this weaker sense.

### Majority rule criteria based on sincerity

An alternative criterion to these four might guarantee that a majority of the voters (in the sense of "more than half of the voters") with a given preference (such as, "candidate A is preferable to candidate B") can always prevail over the other voters, simply by voting sincerely, without having to use a strategic vote.

For instance, one wording of the Minimal Defense criterion guarantees that if such a majority ranks A sincerely, and simply doesn't rank B above anyone (by leaving B out of the ranking), then B can't win. If we assume that B is a rival frontrunner to A, then very little strategy is demanded of the A voters, since they will likely be inclined to not vote for B, anyway.

This property doesn't imply satisfaction of any of the above criteria except for criterion 1, and none of the above criteria implies this property.

In the following methods, a majority sincerely preferring A to B can ensure that B loses merely by voting for A and not voting for B: Approval voting, Bucklin voting, the River method, the Schulze method and Ranked Pairs (assuming with these that defeat strength is measured as the number of voters favoring the winning side). Most methods with an approval base also guarantee this.

The following methods lack this property: Plurality, Borda count, instant runoff voting, and Minmax. Also, methods which don't allow voters to "not vote for" some candidates would lack this property.

### Majority rule criteria based on beatpaths

If more voters prefer candidate A to candidate B, then A pairwise beats B, and the strength of this pairwise win is equal to the literal number of voters who rank A above B. (It is possible to define strength in other ways, but not for this purpose.)

Candidate A has a beatpath to candidate B if there is some sequence of candidates such that A is the first candidate, B is the last candidate, and for every pair of adjacent candidates in this sequence I followed by J, I pairwise beats J. The strength of this beatpath is equal to the strength of the weakest pairwise win in this sequence (that is, of one candidate over the following candidate).

A pairwise win or a beatpath is of majority strength if its strength is equal to more than half of the voters.

At least two all-in-one majority rule criteria have been proposed which use the concept of beatpaths:

1. If A has a majority-strength pairwise win against B, but B does not have even a majority-strength beatpath to A, then B must not be elected. (Attributed to Stephen Eppley.)
2. If A has a majority-strength beatpath to B, but B does not have a majority-strength beatpath back to A, then B must not be elected. (Attributed to Markus Schulze.)

Satisfaction of either of these criteria also implies satisfaction of the Minimal Defense criterion and the Majority criterion for solid coalitions.

When criterion #2 is used, this is equivalent to requiring that the winner always comes from the Condorcet (doubly-augmented gross) top tier or CDTT.

The most popular method which satisfies these properties is the Schulze method.

## Comparison to Utilitarian systems

Mathematically, majoritarianism is a necessary approximation of utilitarianism under the assumption that satisfying any of a voter's preferences has the same utility as any other voter. Any voter's utility towards A>B exactly cancels any other voter's utility towards B>A, so the favoring the majority maximizes net utility. It is, effectively, a rule of thumb under the assumption of strict utilitarian equality of all voters.

Utilitarian systems are based on not making such an assumption, by instead letting voters give their own estimates of utility for different scenarios. Since cardinal ballots are capped (there are no "utility monsters"), voters can effectively only express degrees of indifference between options, reducing the intensity of their preferences. Critics of utilitarianism claim this inherently penalizes voters who choose to do so, and therefore the assumption of equal intensity of preferences of majority rule is more "fair to all".

In contrast, under the philosophy of utilitarianism choosing a majoritarian winner when a utilitarian winner is available would be considered bad (according to the voters themselves), as in such scenarios the majoritarian rule-of-thumb has likely failed. For utilitarians, taking the indifferences into account allows voters to concede towards a greater good, if they choose to do so. Under any utilitarian system, a majority can always bullet vote and get its way if they choose to do so. The point, according to utilitarians, is to grant them the choice of not doing so.

All utilitarian systems are cardinal voting methods because other ballot structures do not contain enough information to distinguish the utility of voters. All such systems fail the majority criterion by design. Common examples of such are Approval, Score, and STAR voting. Following from Utilitarian theory it is argued that the Utilitarian winner is preferred in situations where they are well-liked by all voters rather than a candidate who is narrowly preferred by a majority but loathed by the minority.

Note that a utilitarian winner need not have significantly more utility than the majoritarian winner to win, nor need satisfy significantly more people. See Smith//Score#Notes for some ideas on mixing the two philosophies.

If voters are considered imprecise judges of their own utilities (i.e. comparisons are not infinitely precise, so indifferences do exist whether voters express them in their ballots or not), then comparisons between candidates are based on slightly noisy utilities. This means even rankings may be regarded as probabilistically accurate. In this case, both utilitarianism and majoritarianism represent different philosophies of addressing this inherent noise. Ranked systems under majoritarianism assume all preferences are 100% accurate and there are no degrees of indifference, regardless of how closely the candidates may be perceived to the ranked voter. Cardinal systems would consider the ballot as a sample of the underlying distribution of utilities, such that their aggregation would reproduce the statistical likelihood of satisfying the individual distributions on an average election.

### Utilitarian critique of majoritarianism

(This critique can be found at https://forum.electionscience.org/t/utilitarian-vs-majoritarian-in-single-winner/602)

An illustrative score votingexample for 100 voters with candidates A, B, C ,D is

20 = A:5, B:2, C:2, D:0

20 = A:2, B:5, C:2, D:0

20 = A:2, B:2, C:5, D:0

40 = A:0, B:0, C:0, D:5

D is the score winner with 200 and all others have 180. D is the Utilitarian winner because they yield the most Utility. If only A and D were running the ballots would likely be.

60 = A:5, D:0 40 = A:0, D:5

Under Majoritarianism A would win so it looks as if B and C are spoilers for A. Systems which pass the Majority criteria, like IRV, would yield either A, B or C as the winner in the original case. Utilitarian philosophy would not view that B and C are spoilers for A. Score relies on there being enough candidates to properly get utilities. In the absence of a representative sample of candidates score would yield the majoritarian winner A. This does not mean that score gives the correct answer in the case of less candidates but that it does not work as intended because the voters do not have enough choice. It is important to then note that partisanship is fundamentally incompatible with score. If there is no party to put up candidates for a significant portion of the voters then score will not yield accurate utilities. In party based systems there are never parties for all groups. Majoritarian systems are in this way more compatible with partisanship which is why collectivists have always favoured majoritarian solutions and individualists favour utilitarian solutions. Of course one could strategically vote under score and give

20 = A:5, B:5, C:5, D:0

20 = A:5, B:5, C:5, D:0

20 = A:5, B:5, C:5, D:0

40 = A:0, B:0, C:0, D:5

But this would not really happen if you look at it from a game theory perspective. The voters who favour A,B and C are in competition and are actually not so ideologically aligned. In the end some might give a little more and this might be enough to win in this scenario. For this reason some advocate that Approval Voting is the appropriate system because it forces voters into the Nash Equilibrium.

Another game theory perspective to consider is that elections do not exist in a vacuum. Utilitarianism leads to an equilibrium where more candidates run who are centrist to try to find the right balance to please the most. Majoritarianism leads to tyranny of the majority. Polling data does not show that voters are divided ideologically into incompatible factions. Ideologically people are Gaussian distributed around the center.

## Notes

The most common, simple alternative to majority rule is utilitarianism i.e. in the two-candidate case, rather than electing the majority's preference, elect the candidate that makes voters "happier". Note that while in the two-candidate case, a majority can force its preference with no coordination (i.e. each voter can vote strategically according to their own preference) in utilitarian methods by saying that they get maximal utility from their preferred candidate and no utility for their less-preferred candidate (i.e. normalization), this doesn't hold when there are more candidates. However, a "utilitarian" rated method like Majority Judgement passes the majority criterion for rated ballots, ensuring that a normalizing majority can get their preference.

The most basic criterion for majority rule is that a voting method must pass the majority criterion in the two-candidate case. However, this means that all majoritarian methods must fail Independence of irrelevant alternatives, because when there is a Condorcet cycle of more than two candidates, no matter who the voting method elects, all candidates except one that pairwise beats the winner can be eliminated to change the winner i.e. the pairwise-beating candidate who was a loser now wins as a majority's 1st choice. Note that this means that while certain voting methods may nominally pass IIA (i.e. Approval voting, Score voting) because they fail the majority criterion, they will still fail it if the majority of voters would strategically vote in a two-candidate election to elect their preferred candidate.

Almost all extensions of majority rule involve the winner of the election pairwise beating at least one other candidate. This can easily be seen by how many majoritarian methods involve runoffs, and explains why most such methods pass the Condorcet loser criterion (with the notable exception of some methods like Minimax).

Sometimes excluding voters with no preference among any of the named candidates can make a plurality become a majority. Example:

49 A>B

48 B>A

3 A=B

If ignoring the A=B voters (who have no preferences between any of the candidates marked on at least one ballot), then A is a majority's 1st choice, but otherwise, is simply a plurality's 1st choice.

ISDA implies several of the criteria mentioned above. When there is a mutual majority and a minority with a preference among the mutual majority's preferred candidates, the ISDA-based reasoning for deciding who to elect can be thought of as eliminating everyone not in the mutual majority, checking if there is a new mutual majority set, and then repeating. Taking the above example:

31 A > B > C
29 B > C > A
40 C > B > A

There is a mutual majority of 69% of voters for B and C, so by ISDA, A can be eliminated. Then, the example becomes:

60 B > C 40 C > B

The 31 A>B>C and 29 B>C>A voters fuse into one coalition with A gone, and so there is now a majority who put B as their 1st choice, and because ISDA implies the majority criterion, B wins.

A majority is a Droop quota in the single-winner case.

Note that no voting method can guarantee a candidate has "majority support" from an absolute majority of voters, since some voters may not have preferences between certain candidates. Thus, at most voting methods can elect candidates with support from a majority of voters with preferences.