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((Short description|Property of electoral systems))
((More references|article|date=September 2010))
The '''mutual majority criterion''', also known as '''majority for solid coalitions''' or the '''generalized [[majority criterion]]''', is a [[voting system criterion]] that says that if a [[political faction]] or [[Political party|party]] wins a majority of the vote, they should win the election.

== Formal definition ==
A ''solid coalition'' in support of L is a group of voters who strictly prefer any member of L (their least-favorite member of L) to every candidate outside of L (their favorite member of not-L). The solid coalition is the standard definition of a political party in (as it can be defined without )

Majority for solid coalitions says that if there is a [[solid coalition]] of voters in support of an [[electoral list]] of candidates, called L, where this solid coalition consists of more than half of all voters.

=== 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 a 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>

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

All [[Smith criterion|Smith-efficient]] [[Condorcet methods]] pass the mutual majority criterion.<ref>((Cite periodical |author=James 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]] 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 [[Condorcet criterion#Instant-runoff voting|combination]].((Clarify|date=March 2024))

== By method ==
[[Anti-plurality voting]], [[range voting]], and the [[Borda count]] fail the [[majority 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. 

[[First-past-the-post|Plurality]], [[Black's method]], and [[Minimax Condorcet|minimax]] satisfy the [[majority criterion]] but fail the mutual majority criterion.<ref>((Cite journal |last1=Kondratev |first1=Aleksei Y. |last2=Nesterov |first2=Alexander S. |year=2020 |title=Measuring Majority Power and Veto Power of Voting Rules |journal=Public Choice |volume=183 |issue=1–2 |pages=187–210 |arxiv=1811.06739 |doi=10.1007/s11127-019-00697-1 |s2cid=53670198))</ref> Methods which pass the majority criterion but fail mutual majority suffer from [[Vote splitting|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 [[2000 United States presidential election in Florida|George W. Bush in Florida]].

[[Rated voting]] methods such as [[Score voting|score]] typically fail the mutual majority criterion; however, the applicability of mutual majority criteria to cardinal methods is contested, as it is possible for one 

=== Borda count ===
:''[[Majority criterion#Borda count]]''
Borda fails the majority criterion and therefore mutual majority.

=== 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 <br/>[Y] 67
| bgcolor="#e0e0ff" | [X] 69 <br />[Y] 31
| bgcolor="#ffe0e0" | [X] 48 <br />[Y] 52
|-
| bgcolor="#ffc0c0" | B
| bgcolor="#e0e0ff" | [X] 67 <br />[Y] 33
|
| bgcolor="#ffe0e0" | [X] 36 <br />[Y] 64
| bgcolor="#ffe0e0" | [X] 48 <br />[Y] 52
|-
| bgcolor="#ffc0c0" | C
| bgcolor="#ffe0e0" | [X] 31 <br />[Y] 69
| bgcolor="#e0e0ff" | [X] 64 <br />[Y] 36
| 
| bgcolor="#ffe0e0" | [X] 48 <br />[Y] 52
|-
| bgcolor="#ffc0c0" | D
| bgcolor="#e0e0ff" | [X] 52 <br />[Y] 48
| bgcolor="#e0e0ff" | [X] 52 <br />[Y] 48
| bgcolor="#e0e0ff" | [X] 52 <br />[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 ===
Assume the [[Plurality voting system#A hypothetical example|Tennessee capital election example]].

{| class="wikitable"
! width=25% style="background-color: #ffdddd" | 42% of voters<br /><small>(close to Memphis)</small>
! width=25% style="background-color: #ccffcc" | 26% of voters<br /><small>(close to Nashville)</small>
! width=25% style="background-color: #ddddff" | 15% of voters<br /><small>(close to Chattanooga)</small>
! width=25% style="background-color: #ffeedd" | 17% of voters<br /><small>(close to Knoxville)</small>
|-
|
# '''Memphis'''
# Nashville
# Chattanooga
# Knoxville
|
# '''Nashville'''
# Chattanooga
# Knoxville
# Memphis
|
# '''Chattanooga'''
# Knoxville
# Nashville
# Memphis
|
# '''Knoxville'''
# Chattanooga
# Nashville
# Memphis
|}

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.

=== Score voting ===
Score voting does not satisfy the majority criterion, and so fails the MMC. However, the applicability of majoritarian criteria such as mutual majority or the Smith criterion to cardinal systems, and especially score voting, is contentious. 

=== STAR voting ===
((Empty section|date=June 2021))

==See also==
* [[Majority criterion]]
* [[Majority loser criterion]]
* [[Voting system]]
* [[Voting system criterion]]

==References==
((Reflist))

((voting systems))

[[Category:Electoral system criteria]]

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