Mutual majority criterion

{{Short description|Property of electoral systems}} {{More references|article|date=September 2010}} The '''mutual majority criterion''' is a criterion for evaluating electoral systems. It is also known as the '''majority criterion for solid coalitions''' and the '''generalized majority criterion'''. This criterion requires that whenever a majority of voters prefer a group of candidates above all others, then the winner must be a candidate from that group.<ref>{{Cite journal |journal=Voting matters |title=Cardinal-weighted pairwise comparison |url=https://www.votingmatters.org.uk/ISSUE19/i19p2.pdf |year=2004 |access-date=2024-10-19|first=James|last=Green-Armytage}}</ref> 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 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 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.<ref>{{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 }}</ref>

All Smith-efficient Condorcet methods pass the mutual majority criterion.<ref>{{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}}</ref>

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.

== By method == 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.<ref name="x494">{{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}}</ref>

The Schulze method, ranked pairs, instant-runoff voting, Nanson's method, and Bucklin voting 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 === {{Main article|Minimax Condorcet}} Assume four candidates A, B, C, and D with 100 voters and the following preferences:

{| class="wikitable" |- ! 19 voters !! 17 voters !! 17 voters !! 16 voters || 16 voters || 15 voters |- | 1. C || 1. D || 1. B || 1. D || 1. A || 1. D |- | 2. A || 2. C || 2. C || 2. B || 2. B || 2. A |- | 3. B || 3. A || 3. A || 3. C || 3. C || 3. B |- | 4. D || 4. B || 4. D || 4. A || 4. D || 4. C |}

The results would be tabulated as follows: {| class=wikitable border=1 |+ Pairwise election results |- | colspan=2 rowspan=2 | | colspan=4 bgcolor="#c0c0ff" align=center | X |- | bgcolor="#c0c0ff" | A | bgcolor="#c0c0ff" | B | bgcolor="#c0c0ff" | C | bgcolor="#c0c0ff" | D |- | bgcolor="#ffc0c0" rowspan=4 | Y | bgcolor="#ffc0c0" | A | | bgcolor="#ffe0e0" | [X] 33 [Y] 67 | bgcolor="#e0e0ff" | [X] 69 [Y] 31 | bgcolor="#ffe0e0" | [X] 48 [Y] 52 |- | bgcolor="#ffc0c0" | B | bgcolor="#e0e0ff" | [X] 67 [Y] 33 | | bgcolor="#ffe0e0" | [X] 36 [Y] 64 | bgcolor="#ffe0e0" | [X] 48 [Y] 52 |- | bgcolor="#ffc0c0" | C | bgcolor="#ffe0e0" | [X] 31 [Y] 69 | bgcolor="#e0e0ff" | [X] 64 [Y] 36 | | bgcolor="#ffe0e0" | [X] 48 [Y] 52 |- | bgcolor="#ffc0c0" | D | bgcolor="#e0e0ff" | [X] 52 [Y] 48 | bgcolor="#e0e0ff" | [X] 52 [Y] 48 | bgcolor="#e0e0ff" | [X] 52 [Y] 48 | |- | colspan=2 bgcolor="#c0c0ff" | Pairwise election results (won-tied-lost): | 2-0-1 | 2-0-1 | 2-0-1 | 0-0-3 |- | colspan=2 bgcolor="#c0c0ff" | worst pairwise defeat (winning votes): | 69 | 67 | 64 | 52 |- | colspan=2 bgcolor="#c0c0ff" | worst pairwise defeat (margins): | 38 | 34 | 28 | 4 |- | colspan=2 bgcolor="#c0c0ff" | worst pairwise opposition: | 69 | 67 | 64 | 52 |}

* [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 === {{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.

==References== {{Reflist}}

Category:Electoral system criteria