# Maximum Majority Voting

Maximum majority voting (MMV) is a single-winner, ranked ballot voting method. MMV can also be used to create a sorted list of winners. MMV and maximize affirmed majorities (MAM) are variations of the ranked pairs (RP) Condorcet method.

## Procedure

The MMV procedure is as follows:

1. Voting
2. Preferences
3. Counting
4. Sorting
5. Candidate ordering
6. Winner selection

MMV can also be used to create a sorted list of preferred candidates. To create a sorted list, repeatedly use MMV to select a winner, remove that winner from the list of candidates, and repeat (to find the next runner up, and so forth).

### Voting, preferences, counting

In the votes, each voter's preferences are considered. For example, if a voter states "A > B > C" (A is better than B, and B is better than C), then the count should add one for A in A vs. B, one for A in A vs. C, and one for B in B vs. C. Voters may also express indifference (e.g., A = B), and unstated candidates are assumed to be equally worse than the stated candidates.

After counting the majorities can be determined. If "Vxy" is the number of Votes that rank x over y, then "x" wins if Vxy > Vyx, and "y" wins if Vyx > Vxy.

### Sorting

The pairs of winners, called the "majorities", are then sorted from the largest majority to the smallest majority. If two matchups have the same number of winning votes, the one with the largest margin (weakest loser) is listed first.

### Candidate ordering, winner selection

Using the sorted list, the candidates are ordered from greatest majority to least, unless the pair will create a circularity (e.g., where A is more than B, B is more than C, but C is more than A), then the smallest majority in the circularity is eliminated. The candidate with the most paiwise wins (i.e. the Maximum Majority) wins the election.

## An example

### The situation (voting)

Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. Let's say the candidates for the capital are Memphis (on the far west end), Nashville (in the center), Chattanooga (129 miles southeast of Nashville), and Knoxville (on the far east side, 114 northeast of Chattanooga). Here's the population breakdown by metro area (surrounding county):

• Memphis (Shelby County): 826,330
• Nashville (Davidson County): 510,784
• Chattanooga (Hamilton County): 285,536
• Knoxville (Knox County): 335,749

Let's say that in the vote, the voters vote based on geographic proximity. Assuming that the population distribution of the rest of Tennessee follows from those population centers, one could easily envision an election where the percentages of votes would be as follows:

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

The results would be tabulated as follows:

Pairwise Election Results
A
Memphis Nashville Chattanooga Knoxville
BMemphis[A] 58%
[B] 42%
[A] 58%
[B] 42%
[A] 58%
[B] 42%
Nashville[A] 42%
[B] 58%
[A] 32%
[B] 68%
[A] 32%
[B] 68%
Chattanooga[A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 17%
[B] 83%
Knoxville[A] 42%
[B] 58%
[A] 68%
[B] 32%
[A] 83%
[B] 17%
Pairwise election results (won-lost-tied): 0-3-0 3-0-0 2-1-0 1-2-0
Votes against in worst pairwise defeat: 58%N/A68%83%
• [A] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption
• [B] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
• [NP] indicates voters who expressed no preference between either candidate

### Counting

First, list every pair, and determine the winner:

Pair Winner
Memphis (42%) vs. Nashville (58%) Nashville 58%
Memphis (42%) vs. Chattanooga (58%) Chattanooga 58%
Memphis (42%) vs. Knoxville (58%) Knoxville 58%
Nashville (68%) vs. Chattanooga (32%) Nashville 68%
Nashville (68%) vs. Knoxville (32%) Nashville 68%
Chattanooga (83%) vs. Knoxville (17%) Chattanooga: 83%

Note that absolute counts of votes can be used, or percentages of the total number of votes; it makes no difference.

### Sorting

The votes are then sorted. The largest majority is "Chattanooga over Knoxville"; 83% of the voters prefer Chattanooga. Nashville (68%) beats both Chattanooga and Knoxville by a score of 68% over 32% (an exact tie, which is unlikely in real life for this many voters). Since Chattanooga > Knoxville, and they're the losers, Nashville vs. Knoxville will be added first, followed by Nashville vs. Chattanooga.

Thus, the pairs from above would be sorted this way:

Pair Winner
Chattanooga (83%) vs. Knoxville (17%) Chattanooga 83%
Nashville (68%) vs. Knoxville (32%) Nashville 68%
Nashville (68%) vs. Chattanooga (32%) Nashville 68%
Memphis (42%) vs. Nashville (58%) Nashville 58%
Memphis (42%) vs. Chattanooga (58%) Chattanooga 58%
Memphis (42%) vs. Knoxville (58%) Knoxville 58%

### Candidate ordering, winner selection

The pairs are then sorted in order, skipping any pairs that would create a cycle:

• Chattanooga over Knoxville.
• Nashville over Knoxville.
• Nashville over Chattanooga.
• Nashville over Memphis.
• Chattanooga over Memphis.
• Knoxville over Memphis.

In this case, no cycles are created by any of the pairs, so none are eliminated.

In this example, Nashville is the winner using MMV - and would be in any Condorcet method.

### Ambiguity resolution example

Let's say there was an ambiguity. For a simple situation involving candidates A, B, and C.

• A > B 72%
• B > C 68%
• C > A 52%

In this situation we sort the majorities starting with the greatest one first.

• Lock A > B
• Lock B > C
• We eliminate the final C > A as it creates an ambiguity or cycle.

Therefore, A is the winner.

### Summary

In the example election, the winner is Nashville. This would be true for any Condorcet method. Using the first-past-the-post system and some other systems, Memphis would have won the election by having the most people, even though Nashville won every simulated pairwise election outright. Using Instant-runoff voting in this example would result in Knoxville winning, even though more people preferred Nashville over Knoxville.

## Notes

MMV has similarities to Copeland's method.

MMV is Smith-efficient. This is because all candidates in the Smith set pairwise beat all candidates not in the Smith set, meaning that there can't be any Condorcet cycles involving candidate(s) in the Smith set and candidates not in the Smith set. Therefore, at worst, all candidates in the Smith set will have at least (number of candidates not in the Smith set) pairwise victories, whereas all candidates not in the Smith set will have at most ((number of candidates not in the Smith set) - 1) victories, since they can't beat anyone in the Smith set, and can't beat themselves. Therefore, Smith set members will always have more pairwise victories, and thus beat, all candidates not in the Smith set.

MMV passes ISDA. This is because adding or removing candidates not in the Smith set can only increase or decrease every candidate in the Smith set's number of pairwise victories correspondingly, and since Smith set members can't be in cycles with non-Smith set members, there is no way for one Smith member's number of victories to increase or decrease any more than another's when running MMV.