Majority criterion
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The majority criterion is a voting system criterion, used to objectively compare voting systems. The criterion states that if a majority of voters strictly prefers a given candidate to every other candidate (i.e. the given candidate is the first preference of more than half the voters) and they vote sincerely, then that candidate should win.
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 does not. Additionally, methods that do not use a ranked ballot, such as Approval voting, cannot satisfy it.
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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] Approval voting
For example 55 voters prefer 3 candidates in the order A>B>C, 35 voters prefer the candidates in the order B>C>A and 10 voters prefer the candidates in the order C>B>A. The 35 B>C>A voters approve only candidate B, the 10 C>B>A voters approve candidates C and B, 35 of the A>B>C voters approve only candidate A and 20 of the A>B>C voters approve both candidates A and B.
| A 55 |
| B 65 |
| C 10 |
A majority of the voters prefer candidate A to all other candidates but candidate B wins.
[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.
