Mutual majority criterion

The mutual majority criterion is a criterion for evaluating electoral system. It requires that whenever a majority of voters prefer a group of candidates (often candidates from the same political party) above all others, someone from that group must win. It is the single-winner case of Droop-Proportionality for Solid Coalitions.

Formal definition
Let L be a subset of candidates. A solid coalition in support of L is a group of voters who strictly prefer all members of L to all candidates outside of L. In other words, each member of the solid coalition ranks their least-favorite member of L higher than their favorite member outside L. Note that the members of the solid coalition may rank the members of L differently.

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.

Relationships to other criteria
This is similar to but stricter than the favorite 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.

The mutual majority criterion is the single-winner case of the Droop proportionality criterion.

All Smith-efficient Condorcet methods pass the mutual majority criterion.

Methods which pass mutual majority but fail the Condorcet criterion can nullify the voting power of voters outside the mutual majority. Instant runoff voting is notable for excluding up to half of voters by this combination.

By method
Anti-plurality voting, range voting, and the Borda count fail the majority-favorite criterion and hence fail the mutual majority criterion.

The Schulze method, ranked pairs, instant-runoff voting, Nanson's method, and Bucklin voting pass this criterion.

Plurality, Black's method, and minimax satisfy the majority-favorite criterion but fail the mutual majority criterion. Methods which pass the majority-favorite criterion but fail mutual majority suffer from vote-splitting effects: a majority party or political coalition can lose simply by running too many candidates. If all but one of the candidates in the mutual majority-preferred set drop out, the remaining mutual majority-preferred candidate will win, which is an improvement from the perspective of all voters in the majority. This effect likely allowed George W. Bush to win the 2000 election in Florida.

Borda count

 * Majority criterion#Borda count

Borda fails the majority-favorite criterion and therefore mutual majority.

Minimax
Assume four candidates A, B, C, and D with 100 voters and the following preferences:

The results would be tabulated as follows:


 * [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
 * [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption

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. Nevertheless, Minimax declares D the winner because its biggest defeat is significantly the smallest compared to the defeats A, B and C caused each other.

Plurality
Assume the Tennessee capital election example.

There are 58% of the voters who prefer Nashville, Chattanooga and Knoxville over Memphis, so the three cities build a set S as described in the definition. But since the supporters of the three cities split their votes, Memphis wins under Plurality.