Majority criterion
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The majority criterion is a voting system criterion, used to objectively compare voting systems. As applied to ranked ballots, the criterion states that if a majority of voters ranks a given candidate higher than every other candidate, then that candidate should win. For non-ranked ballots (e.g. Approval voting or range voting) it can be expressed as follows: "If more than half of the voters give candidate X a higher rating than any other candidate, the winner should be candidate X."[1]
Condorcet methods (such as the Schulze method and Ranked Pairs), plurality voting, instant-runoff voting, and Bucklin voting comply with the majority criterion, while the Borda count and range voting do not.
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[edit] Difference to 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?'.
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?'.
Condorcet criterion implies majority criterion, but not vice versa. In 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] Examples of failure of the majority criterion
[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 459 + 410 = 869 rating points, versus 510 for candidate A.
Since candidate A is rated higher than candidate B by a majority of the voters, this voting system fails to satisfy the criterion.
[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.
