# 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).

## 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
("C" by default)
```

Any candidate at the Lowest Passing Grade or higher is given one Approval vote. Unless explicitly changed, the Lowest Passing Grade is assumed to be C.

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 enables 16 rank levels (including no rank given).

99% of the time, there should be no need to change the LPG -- with 8 default approved grade levels from A+ to C, there is plenty of room to express relative preferences.

Setting the default LPG at C instead of C-minus allows an indecisive voter to be hesitant about granting approval by initially filling in a grade of C. If after reconsideration the voter decides to disapprove the candidate, the minus can then be checked.

Having an LPG option may be useful as a last resort --- it allows a voter to move the cutoff higher or lower after entering grades if the voter makes a mistake. But it is not necessary to have an LPG option in a first public proposal.

#### 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 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.

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.

We call a candidate Strongly defeated when that candidate is pairwise defeated by any other higher-ranked candidate. 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 it 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

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.