# Definite Majority Choice

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 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 finds the same winner as those other three 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).

The philosophical basis of DMC (also due to Forest Simmons) is to first eliminate candidates that the voters strongly agree should not win, using two different measures, and choose the winner from among those that remain.

DMC is currently the best candidate for a Condorcet Method that meets the 'Public Acceptability Criterion'.

## Procedure

### The Ballot

A voter ranks candidates in order of preference, and may decide to rank some candidates without giving them approval.

#### Using Grades to Rank Candidates

A Graded Ballot ballot implementation would infer the ordinal ranking from the grades given to candidates.

```            A    B    C    D    F       +  /  -

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

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

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

X3   ( )  ( )  ( )  ( )  ( )     ( )   ( )
```

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

A candidate gets one vote in the one-to-one contest with any other candidate with a lower grade (rank).

C is the "Lowest Passing Grade" (LPG): any candidate with a grade of C or higher gets one Approval point. No Approval points are given to candidates graded at C-minus or below or to ungraded candidates.

A candidate's total approval score will be used like the 'seed' rating in sports tournaments, to decide which one-to-one contests have greater weight.

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

In small races it should be sufficient to grade 2 or 3 candidates, but in crowded races, there is the option to add a plus or minus to the grade, allowing a voter to rank candidates at up to 16 levels: 8 approved (A-plus to C) and 8 unapproved (C-minus to unranked).

With the Approval Cutoff / Lowest Passing Grade at C instead of C-minus, an indecisive voter can be hesitant about granting approval by initially filling in a grade of C. If after reconsideration the voter decides to withold approval, the minus can then be checked.

#### Ranking Candidates using a Ranked Choice ballot

If the Graded Ballot is deemed too complex, a ranked ballot could be used instead. Here is one possible format:

```           |<-- Approved -->|
1    2    3    4      5    6    7

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

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

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

X3   ( )  ( )  ( )  ( )    ( )  ( )  ( )
```

Ranks 1 through 4 would be approved, 5 through 7 and ungraded would be unapproved.

The voting method would be unchanged otherwise:

1. Candidates receiving rank 1 through 4 would get 1 approval point.
2. A higher-ranked candidate gets one vote in each one-to-one contest with lower-ranked candidates.

#### Discussion

What is a voter saying by giving a candidate a non-approved grade or rank?

One could consider the Approval Cutoff / Lowest Passing Grade (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 want X to win, but of all the alternatives, X would make fewest changes in the wrong direction. I also won't give 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.

The rankings on a single ballot are added into a round-robin table using the standard Condorcet pairwise matrix method: the higher ranked candidates are given one vote in each of their pairwise contests with lower ranked candidates.

The Condorcet pairwise matrix can also be used to store each candidate's Approval point score, in the otherwise-unused diagonal cell.

We call a candidate Strongly defeated when that candidate is pairwise defeated by any other candidate with higher Approval rating. This kind of defeat is also called an Approval-consistent or definitive defeat.

To determine the winner:

1. Eliminate all strongly defeated candidates.
2. The winner is the candidate that pairwise defeats all other remaining candidates.

DMC always selects the Condorcet Winner, if one exists, and otherwise selects a member of the Smith Set. Step 1 has the effect of successively eliminating the least approved candidate in the Smith set, but allows higher-approved candidates outside the Smith set (such as the Approval Winner) to remain in the set of non-strongly-defeated candidates.

### Handling Ties and Near Ties

#### Approval Ties

During the initial ranking of candidates, two candidates may have the same approval score.

If equal Approval scores affect the outcome, there are several alternatives for Approval-tie-breaking. The procedure that would be most in keeping with the spirit of DMC, however would be to initially rank candidates

1. In descending order of Approval
2. If equal, in descending order of "Grade Point Average" (i.e., total Cardinal Rating)
3. If equal, in descending order of total first- and second-place votes
4. If equal, in descending order of total first-, second- and third-place votes.

#### Pairwise Ties

When there are no ties, the winner is the candidate in Forest Simmon's P set, the set of candidates which are not approval-consistently defeated.

In the event of a pairwise tie or near tie (say, margin within 0.01%), it is sometimes possible to proceed anyway, since another member of P may defeat the tied pair. But if there is no clear winner, ties should be handled using the same Random Ballot procedure as in Maximize Affirmed Majorities.