Definite Majority Choice

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Definite Majority Choice (DMC) is a voting method to select a single winner using ballots that express both ranked preferences and approval. [Name suggested here.]

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

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

To find the DMC winner, the candidates are divided into two groups:

  1. Definitively defeated candidates.
  2. Candidates that pairwise defeat all higher-approved candidates. We call this group the definite majority set.

The least-approved candidate in the definite majority set pairwise defeats all higher-approved candidates, including all other members of the definite majority set, and is the DMC winner.

If there is a candidate who, when compared in turn with each of the others, is preferred over the other candidate, DMC guarantees 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 DMC winner satisifies this variant of the Condorcet Criterion:

The Definite Majority Choice winner is the least-approved candidate who, when compared in turn with each of the other higher-approved candidates, is preferred over the other candidate.

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 cyclic ambiguities. If defeat strength is measured by the Total Approval score of the pairwise winner, all three other methods become equivalent to DMC (See proof). Therefore,

  • DMC is a strong majority rule method.
  • When defeat strength is measured by the approval of the defeating candidate, DMC is the only possible immune (cloneproof) method.

DMC is also equivalent to Ranked Approval Voting (RAV) (also known as Approval Ranked Concorcet), and Pairwise Sorted Approval (PSA): DMC always selects the Condorcet Winner, if one exists, and otherwise selects a member of the Smith set. Eliminating the definitively defeated candidates from consideration has the effect of successively eliminating the least approved candidate in the Smith set and then recalculating the new Smith set until a single winner exists, which is why DMC is equivalent to RAV. But the definite majority set may also contain higher-approved candidates outside the Smith set. For example, the Approval winner will always be a member of the definite majority set, because it cannot be definitively defeated.

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


Before explaining how the ballots elicit the approval and pairwise preference information from the voters, let's consider a simple example of how that information is used to determine the DMC winner.

Suppose that the candidates (in order of approval) are






and that the only two "downward" majority preferences are Jill to Jack and Jane to Jean.

We are assuming that all other majority preferences are directed upward:

Jane defeats John,

Jill defeats both Jane and John,

Jack defeats both Jane and John, and

Jean defeats John, Jill, and Jack.

The downward or "approval consistent" preferences are enforced by eliminating Jack and Jean.

Jill (pairwise) defeats both of the remaining candidates, so Jill is the DMC winner.

Note that Jill is the lowest approval candidate that pairwise defeats each of the higher approved candidates. This property is obviously true of the Condorcet Winner when there is one, and completely determines the DMC winner, as well.

At first blush "least approved" may sound bad, but if we did not use the least approved candidate with the "defeat all above" property, then there would be another candidate that defeated everybody "seeded" above our candidate while defeating our candidate, too.

The lower the candidate with the "defeat all above" property, the greater the solid list of highly seeded candidates that it defeats.

The ideal state of affairs is that the highest approval candidate pairwise defeats all candidates below it, in which case it is simultaneously the Approval Winner, the Condorcet Winner, and the DMC winner. This is the expected state of affairs when there is no ambiguity in the will of the voters.

Now for the nitty gritty of collecting approval and pairwise preference information from the voters. First we'll illustrate how the method works with a deliberately crude ballot and then explore other ballot formats.

Simple ballot example

A voter ranks candidates in order of preference, additionally giving approval points to some or all of those ranked, using a ballot like the following:

         |<-- Favorite        Least Preferred -->|
         |<--    Approved     -->|  Not Approved |
         |   1   |   2   |   3   |   4   |   5   |
      X1 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
         |       |       |       |       |       |
      X2 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
         |       |       |       |       |       |
      X3 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
         |       |       |       |       |       |
      X4 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |

On this ballot,

  1. Candidates ranked at 1st through 3rd choice get 1 approval point each.
  2. Candidates ranked fourth, fifth and unranked receive no approval points.
  3. A higher-ranked candidate is given one vote in each of its head-to-head contests with lower-ranked candidates. In particular, all explicitly ranked candidates are given 1 vote in each of their contests with unranked candidates.

As an example, say a voter ranked candidates as follows:

         |<-- Favorite        Least Preferred -->|
         |<--    Approved     -->|  Not Approved |
         |   1   |   2   |   3   |   4   |   5   |
      X1 |  ( )  |  ( )  |  ( )  |  (X)  |  ( )  |
         |       |       |       |       |       |
      X2 |  (X)  |  ( )  |  ( )  |  ( )  |  ( )  |
         |       |       |       |       |       |
      X3 |  ( )  |  ( )  |  ( )  |  ( )  |  (X)  |
         |       |       |       |       |       |
      X4 |  ( )  |  (X)  |  ( )  |  ( )  |  ( )  |

This ballot could be summarized briefly using the notation

 X2 > X4 >> X1 > X3

where the ">>" indicates the approval cutoff --- candidates to the right of that sign receive no approval votes. This ballot is counted as

 X2 > X2  (approval point)
 X2 > X4
 X2 > X1
 X2 > X3
 X4 > X4  (approval point)
 X4 > X1
 X4 > X3
 X1 > X3

Tallying Votes

As in other Condorcet methods, the rankings on a single ballot are added into a round-robin table using the standard Condorcet pairwise matrix: when a ballot ranks / grades one candidate higher than another, it means the higher-ranked candidate receives one vote in the head-to-head 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.

For example, the single example ballot above,

X2 > X4 >> X1 > X3

, the following votes would be added into the pairwise array:

X1 X2 X3 X4
X1 0 0 1 0
X2 1 1 1 1
X3 0 0 0 0
X4 1 0 1 1

For example, the X2>X4 ("for X2 over X4") vote is entered in {row 2, column 4}.

When pairwise totals are completed, we determine the outcome of a particular pairwise contest as described elsewhere. But in DMC, X2 definitively defeats X4 if

  • the {row 2, column 4} (X2>X4) total votes exceed the {row 4, column 2} (X4>X2) total votes, and
  • the {row 2, column 2} (X2>X2) total approval score exceeds the {row 4, column 4} (X4>X4) total approval score.

The winner is then determined as described above.

A more intuitive ballot --- Ranking Candidates using Grades

One barrier to public acceptance of DMC is the ballot design. So how could the process be more intuitive, without sacrificing flexibility and expression?

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 head-to-head 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' ranking in sports tournaments, to decide in which order head-to-head contests are to be scheduled.

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 fill in the plus or minus option to fine-tune the grade. Plus/minus options allow a voter to distinguish up to 16 different rank levels: 8 approved (A-plus to C) and 8 unapproved (C-minus to unranked).

Because we have fixed the Approval Cutoff / Lowest Passing Grade at C instead of C-minus, an indecisive voter has the opportunity to 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.

To avoid spoiled ballots, we count a grade with both plus and minus cells filled as no plus or minus at all. So a truly indecisive voter could change a C grade to C-minus and back to C.

An even simpler ballot --- Voting by slate

In our modern world, there are sometimes too many choices available. A voter who is confused by too many choices or hasn't had time to study issues carefully might benefit by using a published preference slate, as has been suggested by the Democratic Fair Choice method:

                     I      |  I also
                  support   |  approve
                 directly:  |    of:
  Anna              (X)     |    ( )
  Bob               ( )     |    ( )
  Cecil             ( )     |    (X)
  Deirdre           ( )     |    (X)
  Ellen             ( )     |    ( )
  Democrat          ( )     |    ---
  Republican        ( )     |    ---
  Libertarian       ( )     |    ---
  Green             ( )     |    ---
  Labor             ( )     |    ---
  Progressive       ( )     |    ---
  <local newspaper> ( )     |    ---
                    (vote   | (vote for as
                     for    |  many candidates
                   exactly  |  as you want)
                     one)   |

Each candidate, political organization or local newspaper could publish a preference and approval ranking, its "slate" for that particular race.

By selecting a slate, the voter is saying that they want to simply copy the ranking, but if they also approve other candidates, they have the opportunity to move those candidates up in the ranking in the order they appear in the slate.

Say the Libertarian slate for this race is

    Deirdre (Lib.) >> Cecil (Reb.) > Ellen (Dem.) > Bob (Ind.) > Anna (Green)

where we denote the approval cutoff using ">>". Say the voter selects the libertarian slate but also approves Bob and Anna. Then the ballot would be counted as

    Deirdre (Lib.) > Bob (Ind.) > Anna (Green) >> Cecil (Reb.) > Ellen (Dem.)


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.

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.

See Also