Majority criterion
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The majority criterion is a single-winner voting system criterion, used to objectively compare such systems. The criterion states that if a majority of voters prefer a given candidate over every other candidate, then that candidate should win. [1]
Some methods that comply with this criterion include any Condorcet method, Plurality voting, instant-runoff voting, and Bucklin voting.
Generally, methods which give weight to preference strength may fail the majority criterion, thus the Borda count and Range voting fail the majority criterion. The application of the majority criterion to methods which cannot provide a full ranking, such as Approval voting, 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, this additional vote is may aid their less-preferred candidate to defeat their more-preferred candidate. Whether or not voters will actually do this may depend on the voter's knowledge of who the front-runners are, and the strength of their preferences between them. As an example, if a majority of voters prefer one frontrunner to another, for there to be a failure to elect that candidate would require voters to, in Approval voting, vote for two frontrunners in spite of having a preference between them. This may happen in nonpartisan elections, but is unlikely to be of consequence in a two-party system.
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[edit] 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 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.
[edit] Application of the Majority Criterion: Controversy
There is controversy over how the Majority Criterion is to be applied, based on the wording of the Criterion and the intention behind it. 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 world "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 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.
[edit] Examples
[edit] Approval voting
Approval voting does not satisfy the majority criterion. Approval voting is not a preferential voting 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 frontrunners, they would be more likely to vote strategically, and it would be unusual for a majority of voters to approve both frontrunners, as is the case in this example.
[edit] 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.
[edit] Range voting
For example 100 voters cast the following votes:
| # Voters | Ballot | ||
| 51 | A:10 | B:9 | C:0 |
| 49 | A:0 | B:10 | C:0 |
Candidate B would win with a total of 51*9 + 49*10 = 459 + 490 = 949 rating points, versus 510 for candidate A.
Because candidate A is rated higher than candidate B by a majority of the voters, this voting system fails to satisfy the criterion.
