[1] The mutual majority criterion may also be thought of as the single-winner case of Droop-Proportionality for Solid Coalitions.
The mutual majority criterion says that if there is a solid coalition of voters in support of L, and this solid coalition consists of more than half of all voters, then the winner of the election must belong to L. This is similar to but stricter than the majority criterion, where the requirement applies only to the case that L is only one single candidate.
It is also stricter than the majority loser criterion, which only applies when L consists of all candidates except one.
[3] Methods which pass mutual majority but fail the Condorcet criterion may nullify the voting power of voters outside the mutual majority whenever they fail to elect the Condorcet winner.
Assume four candidates A, B, C, and D with 100 voters and the following preferences: The results would be tabulated as follows: Result: Candidates A, B and C each are strictly preferred by more than the half of the voters (52%) over D, so {A, B, C} is a set S as described in the definition and D is a Condorcet loser.