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 both ranked preferences and approval.

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, additionally giving approval points to some or all of those ranked.

Using Grades to Rank Candidates

Many people are familiar with the standard method of giving grades A-plus through F-minus. Most are also familiar with the Pass/Fail form of grading. A student receives grades from many instructors and on finishing school has a total grade point average or pass/fail total.

A similar idea could be used to rank candidates -- a voter could grade candidates as if the voter were the instructor and the candidates were the students. Determining the winner of the election would be similar to finding the student with the best set of grades.

            A    B    C    D    F       +  /  -
				       	    	  
      X1   ( )  ( )  ( )  ( )  ( )     ( )   ( )
				       	    	  
      X2   ( )  ( )  ( )  ( )  ( )     ( )   ( )
				       	    	  
      X3   ( )  ( )  ( )  ( )  ( )     ( )   ( )
				       	    	  
      X3   ( )  ( )  ( )  ( )  ( )     ( )   ( )

Like an instructor grading students, a voter may give the same grade (rank) to more than one candidate. But here, there is one additional grade -- no grade at all. Ungraded candidates are ranked lower than all graded candidates. By giving one candidate a higher grade than another, the voter gives the higher-graded candidate one vote in its one-to-one contest with the lower-graded candidate.

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 (that includes ungraded candidates).

A candidate's total approval score will be used like the 'seed' rating in sports tournaments, to decide which one-to-one victories are worth more than others.

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 elections it should be adequate for a voter to grade only 2 or 3 candidates, but in crowded races, the voter could also add a plus or minus on the grade. That allows a voter to specify up to 16 different rank 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 (rank 8) would be unapproved.

The voting method would be unchanged otherwise:

1. Candidates ranked at 1st through 4th choice get 1 approval point each.
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.

Tallying Votes

The rankings on a single ballot are added into a round-robin table using the standard Condorcet pairwise matrix method: When a ballot ranks / grades one candidate higher than another, it is saying that the first candidate receives one vote in the one-to-one contest against the other.

Since the diagonal cells in the Condorcet pairwise matrix are usually left blank, those locations can be used to store each candidate's Approval point score.

We call a candidate Definitively defeated when that candidate is defeated in a one-to-one contest against any other candidate with higher Approval score. This kind of defeat is also called an Approval-consistent defeat.

To determine the winner:

1. Eliminate all definitively 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 definitively 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.