Mutual majority criterion

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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 majority of voters ranks a certain group of candidates at the top of their ballot, then one of these candidates should win the election.

Formal definition[edit]

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[edit]

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.[1]

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

All Smith-efficient Condorcet methods pass the mutual majority criterion.[2]

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.[clarification needed]

By method[edit]

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.

Plurality, Black's method, and minimax satisfy the majority criterion but fail the mutual majority criterion.[3] Methods which pass the majority 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 in Florida.

Rated voting methods such as 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[edit]

Majority criterion#Borda count

Borda fails the majority criterion and therefore mutual majority.


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

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:

Pairwise election results
Y A [X] 33
[Y] 67
[X] 69
[Y] 31
[X] 48
[Y] 52
B [X] 67
[Y] 33
[X] 36
[Y] 64
[X] 48
[Y] 52
C [X] 31
[Y] 69
[X] 64
[Y] 36
[X] 48
[Y] 52
D [X] 52
[Y] 48
[X] 52
[Y] 48
[X] 52
[Y] 48
Pairwise election results (won-tied-lost): 2-0-1 2-0-1 2-0-1 0-0-3
worst pairwise defeat (winning votes): 69 67 64 52
worst pairwise defeat (margins): 38 34 28 4
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.


Assume the Tennessee capital election example.

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. 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[edit]

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.

See also[edit]


  1. ^ Tideman, Nicolaus (2006). Collective Decisions and Voting: The Potential for Public Choice. Ashgate Publishing. ISBN 978-0-7546-4717-1. Note that mutual majority consistency implies majority consistency.
  2. ^ James Green-Armytage (October 2011). "Four Condorcet-Hare Hybrid Methods for Single-Winner Elections" (PDF). Voting Matters. No. 29. pp. 1–14. S2CID 15220771. Meanwhile, they possess Smith consistency [efficiency], along with properties that are implied by this, such as [...] mutual majority.
  3. ^ Kondratev, Aleksei Y.; Nesterov, Alexander S. (2020). "Measuring Majority Power and Veto Power of Voting Rules". Public Choice. 183 (1–2): 187–210. arXiv:1811.06739. doi:10.1007/s11127-019-00697-1. S2CID 53670198.