Mutual majority criterion

The Mutual majority criterion is a criterion for evaluating voting systems. It applies to ranked ballot elections. It can be stated as follows:

If there is a majority of voters for which it is true that they all rank a set of candidates above all others, then one of these candidates must win.

It is an extension of the majority criterion for sets of candidates.

[Merge: The mutual majority criterion says that if a majority of voters unanimously vote a given set of candidates above a given rating or ranking, and all other candidates below that rating or ranking, then the winner must be from that set.]

This is often called Majority criterion for solid coalitions or simply (and confusingly) Majority criterion.

Systems which pass

Borda-Elimination, Bucklin, Coombs, IRV, Kemeny-Young, Nanson (original), Pairwise-Elimination, Ranked Pairs, Schulze, Smith//Minmax, Descending Solid Coalitions, Majority Choice Approval, any Smith-efficient Condorcet method, Condorcet-IRV hybrid methods

Systems which fail

Black, Borda, Dodgson, Minmax, Sum of Defeats

Notes

Voting methods which pass the majority criterion but not the mutual majority criterion (some ranked methods) possess a spoiler effect, since if all but one candidate in the mutual majority drops out, the remaining candidate in the mutual majority is guaranteed to win, whereas if nobody had dropped out, a candidate not in the mutual majority might have won.

The mutual majority criterion doesn't apply to situations where there are two largest "sides" if enough voters are indifferent to the two sides.

51 A>C

49 B

10 C

Even though A and B are the two candidates with the most dedicated support, with A pairwise beating B and C, A technically is not preferred by a majority, and C would beat A in some mutual majority-passing methods, such as Bucklin. Smith-efficiency implies both the mutual majority criterion and the election of one of the largest side's candidates (A) in these types of scenarios. Likewise, if there are some losing candidates ranked above the mutual majority set of candidates by some voters in the majority, this voids the criterion guarantee.

26 A>B

25 B

49 C

It has been argued that to avoid the Chicken dilemma, C must win here. But methods that do so create a spoiler effect, since if A drops out, B must win by the mutual majority criterion.