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*'''consensus''' usually means complete agreement or "all voters"
== Majority rule/Majority winner - Four
Many methods claim to elect the "majority winner" or work by "majority rule" (See, for example, the [[Center for Voting and Democracy|CVD]]'s talking points re: IRV: [http://www.fairvote.org/irv/talking.htm]). 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 criterion, in ascending order of strictness, which could be used to rank the relative strengths of a "majority."
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''Derived from an e-mail by James Green-Armytage''
== An alternate perspective ==
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 set|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.
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.
== Majority rule definitions 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:
#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.)
#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 [[CDTT|Condorcet (doubly-augmented gross) top tier]] or [[CDTT]].
[[Category:Voting theory]]
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