mutual majority criterion

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.

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.{{Cite book|url=https://books.google.com/books?id=RN5q_LuByUoC|title=Collective Decisions and Voting: The Potential for Public Choice|isbn=978-0-7546-4717-1|quote=Note that mutual majority consistency implies majority consistency.|last1=Tideman|first1=Nicolaus|year=2006|publisher=Ashgate Publishing }}

All Smith-efficient Condorcet methods pass the mutual majority criterion.{{Cite periodical |first=James|last=Green-Armytage |date=October 2011 |title=Four Condorcet-Hare Hybrid Methods for Single-Winner Elections |url=https://www.votingmatters.org.uk/ISSUE29/ISSUE29.pdf |pages=1–14 |issue=29 |quote=Meanwhile, they possess Smith consistency [efficiency], along with properties that are implied by this, such as [...] mutual majority. |periodical=Voting Matters |s2cid=15220771}}

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.

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

In addition, minimax, the contingent vote, Young's method, first past the post, and Black fail, even though they pass the majority-favorite criterion.{{cite journal | last=Kondratev | first=Aleksei Yu. | last2=Nesterov | first2=Alexander | title=Measuring Majority Tyranny: Axiomatic Approach | journal=SSRN Electronic Journal | publisher=Elsevier BV | year=2018 | issn=1556-5068 | doi=10.2139/ssrn.3208580 | page=3}}

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

: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.

{{Main article|Minimax Condorcet}}

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.

{{Tenn voting example}}

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.

• Majority criterion
• Majority loser criterion
• Voting system
• Voting system criterion

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Category:Electoral system criteria