The mutual majority criterion is a criterion for evaluating

electoral systems An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, nonprofit organizations and inf ...

. It is also known as the majority criterion for solid coalitions and the generalized majority criterion. This criterion requires that whenever a

majority A majority is more than half of a total; however, the term is commonly used with other meanings, as explained in the "#Related terms, Related terms" section below. It is a subset of a Set (mathematics), set consisting of more than half of the se ...

of voters prefer a group of candidates above all others, then the winner must be a candidate from that group. The mutual majority criterion may also be thought of as the single-winner case of Droop- Proportionality for Solid Coalitions.

Formal definition

Let L be a subset of candidates. A

solid coalition In social choice theory, a solid coalition or voting bloc is a group of voters who support a given group of candidates over any opponent outside the group. Solid coalitions formalize the idea of a political faction or voting bloc, allowing soci ...

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

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 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. All Smith-efficient Condorcet methods pass the mutual majority criterion. Methods which pass mutual majority but fail the

Condorcet criterion A Condorcet winner (, ) is a candidate who would receive the support of more than half of the electorate in a one-on-one race against any one of their opponents. Voting systems where a majority winner will always win are said to satisfy the Condo ...

may nullify the voting power of voters outside the mutual majority whenever they fail to elect the Condorcet winner.

By method

Anti-plurality voting Anti-plurality voting describes an electoral system in which each voter votes ''against'' a single candidate, and the candidate with the fewest votes against wins. Anti-plurality voting is an example of a positional voting method. Example I ...

range voting Score voting, sometimes called range voting, is an electoral system for single-seat elections. Voters give each candidate a numerical score, and the candidate with the highest average score is elected. Score voting includes the well-known approva ...

, and the

Borda count The Borda method or order of merit is a positional voting rule that gives each candidate a number of points equal to the number of candidates ranked below them: the lowest-ranked candidate gets 0 points, the second-lowest gets 1 point, and so on ...

fail the majority-favorite criterion and hence fail the mutual majority criterion. In addition,

minimax Minimax (sometimes Minmax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, combinatorial game theory, statistics, and philosophy for ''minimizing'' the possible loss function, loss for a Worst-case scenari ...

, the

contingent vote The contingent vote is a two-stage electoral system that elects a single representative, in which the winner receives a majority of votes. It uses ranked voting. The voter ranks the candidates in order of preference, and when the votes are f ...

, Young's method,

first past the post First-past-the-post (FPTP)—also called choose-one, first-preference plurality (FPP), or simply plurality—is a single-winner voting rule. Voters mark one candidate as their favorite, or First-preference votes, first-preference, and the cand ...

, and Black fail, even though they pass the majority-favorite criterion. The Schulze method, ranked pairs,

instant-runoff voting Instant-runoff voting (IRV; ranked-choice voting (RCV), preferential voting, alternative vote) is a single-winner ranked voting election system where Sequential loser method, one or more eliminations are used to simulate Runoff (election), ...

, Nanson's method, and

Bucklin voting Bucklin voting is a class of voting methods that can be used for single-member and multi-member districts. As in highest median rules like the majority judgment, the Bucklin winner will be one of the candidates with the highest median ranking ...

pass this criterion.

Borda count

:'' Majority criterion#Borda count'' The mutual majority criterion implies the majority criterion so the Borda count's failure of the latter is also a failure of the mutual majority criterion. The set solely containing candidate A is a set S as described in the definition.

Minimax

Assume four candidates A, B, C, and D with 100 voters and the following preferences: The results would be tabulated as follows: * indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption * 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 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

58% of the voters prefer Nashville, Chattanooga and Knoxville to Memphis. Therefore, the three eastern 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 voting.

References

{{voting systems Electoral system criteria