# Definite Majority Choice

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Definite Majority Choice (DMC) is a voting method proposed by several (name suggested by Forest Simmons) to select a single winner using ballots that express preferences, with an additional indication of Approval Cutoff.

If there is a candidate who is preferred over the other candidates, when compared in turn with each of the others, DMC guarantees that that candidate will win. Because of this property, DMC is (by definition) a Condorcet method. Note that this is different from some other preference voting systems such as Borda and Instant-runoff voting, which do not make this guarantee.

The main difference between DMC and other Condorcet methods such as Ranked Pairs (RP), Cloneproof Schwartz Sequential Dropping (Beatpath or Schulze) and River is the use of the additional Approval score to break ties. If defeat strength is measured by the Total Approval score of the pairwise winner, DMC is equivalent to each of these other methods [This needs to be verified! --Araucaria 12:22, 21 Mar 2005 (PST)]

DMC chooses the same winner as (and could be considered equivalent in most respects to) Ranked Approval Voting (RAV) (also known as Approval Ranked Concorcet), and Pairwise Sorted Approval (PSA).

## Procedure

### The Ballot

Voters rank their preferred candidates, from favorite to least preferred, and may optionally specify an Approval cutoff.

A Graded Ballot ballot implementation would infer the ordinal ranking from the 'grades' given to candidates, and the Approval Cutoff would be determined with a Lowest Passing Grade option. Voters could grade their choices from favorite (A+) to least preferred (ungraded), and give some or all of their graded choices a "passing grade" to signify approval.

```            A    B    C    D    F       + /  -

X1   ( )  ( )  ( )  ( )  ( )     ( )  ( )

X2   ( )  ( )  ( )  ( )  ( )     ( )  ( )

X3   ( )  ( )  ( )  ( )  ( )     ( )  ( )

X3   ( )  ( )  ( )  ( )  ( )     ( )  ( )

Lowest  ( )  ( )  ( )  ( )  ( )     ( )  ( )
Passing
Grade
("C-" Default)
```

A voter may give the same grade (rank) to more than one candidate. Ungraded candidates are graded (ranked) below all graded candidates.

Any candidate at the Lowest Passing Grade or higher is given one Approval vote. Unless changed, the Lowest Passing Grade is C-minus by default.

No Approval votes are given to ungraded candidates or candidates graded below the Lowest Passing Grade.

Grades assigned to non-passing (disapproved) candidates help determine which of them will win if the voter's approved candidates do not win.

Adding a plus or minus to a candidate's grade is optional, but allows up to 15 rankings.

99% of the time, there should be no need to change the LPG -- with 9 grade levels from A+ to C-, there is plenty of room to express relative preferences.

But the LPG option allows a voter to move the cutoff higher or lower if sentiment changes before finalizing the ballot, without having to get a new ballot.

#### Discussion

What is a voter saying by giving a candidate a grade below the Approval Cutoff?

One could consider the LPG to be like Gerald Ford. Anybody better would make a good president, and anybody worse would be bad.

Grading candidate X below the LPG gives the voter a chance to say "I don't like X and don't want him to win, but of all the alternatives, he would make the least change in the wrong direction. I won't give him X a passing grade because I want X to have as small a mandate as possible." This allows the losing minority to have some say in the outcome of the election, instead of leaving the choice to the strongest core support within the majority faction.

### Tallying Votes

Rankings are added into a pairwise array. The approval scores of each candidate can be stored the diagonal cells, which are unused in other non-Approval-Condorcet hybrids.

To determine the winner:

1. Eliminate any candidate that is defeated in a one-to-one match with any other higher-approved candidate. So by 2 different measures, a definite majority agrees that candidate should be eliminated.
2. If more than one candidate remains, the winner is the single candidate that defeats all others in one-to-one (pairwise)contests.

A simple way to implement this is to permute the pairwise array so that candidates are listed in order of approval.

Then, starting with the diagonal cell corresponding to the least-approved candidate, examine all the cells in the column above the diagonal. If the candidate loses to any higher-approved candidates, proceed up the diagonal to the approval rating of the next-higher-approved candidate.

The winner is the first candidate found with no losses to higher-approved candidates.

Can this be made clearer? --Araucaria 15:05, 21 Mar 2005 (PST)

### Handling Ties and Near Ties

In ordinary DMC, the winner is the candidate in Forest Simmon's P set, the set of candidates which are not approval-consistently defeated.

But in the event of a tie or near tie (say, margin within 0.01%), there may be no clear winner.

In that case, form the superset P*, the union of all sets P that result from all possible combinations of reversed ties or near-ties. Then choose the winner from P* using Random Ballot Order.