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Mutual majority criterion

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Title: Mutual majority criterion  
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Subject: Nanson's method, Instant-runoff voting, Voting system, Majority loser criterion, Independence of Smith-dominated alternatives
Collection: Voting System Criteria
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Mutual majority criterion

The mutual majority criterion is a criterion used to compare voting systems. It is also known as the majority criterion for solid coalitions and the generalized majority criterion. The criterion states that if there is a subset S of the candidates, such that more than half of the voters strictly prefer every member of S to every candidate outside of S, this majority voting sincerely, the winner must come from S. This is similar to but stricter than the majority criterion, where the requirement applies only to the case that S contains a single candidate.

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

The plurality vote, approval voting, range voting, the Borda count, and minimax fail this criterion.


  • Examples 1
    • Borda count 1.1
    • Minimax 1.2
    • Plurality 1.3
    • Range voting 1.4
  • See also 2


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.


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.

Range voting

Majority criterion#Range voting

Range voting does not satisfy the Majority criterion. The set solely containing candidate A is a set S as described in the definition, but B is the winner. Thus, range voting does not satisfy the mutual majority criterion.

See also

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