Mutual majority criterion
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The mutual majority criterion, also known as majority for solid coalitions or the generalized
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 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
All
Methods which pass mutual majority but fail the
By method
The Schulze method, ranked pairs, instant-runoff voting, Nanson's method, and Bucklin voting pass this criterion.
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
Borda fails the majority criterion and therefore mutual majority.
Minimax
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:
X | |||||
A | B | C | D | ||
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.
Plurality
Assume the
42% of voters (close to Memphis) |
26% of voters (close to Nashville) |
15% of voters (close to Chattanooga) |
17% of voters (close to Knoxville) |
---|---|---|---|
|
|
|
|
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.
See also
- Majority criterion
- Majority loser criterion
- Voting system
- Voting system criterion
References
- ISBN 978-0-7546-4717-1.
Note that mutual majority consistency implies majority consistency.
- S2CID 15220771.
Meanwhile, they possess Smith consistency [efficiency], along with properties that are implied by this, such as [...] mutual majority.
- S2CID 53670198.