Definite Majority Choice

From Electowiki
Revision as of 15:39, 13 April 2005 by Araucaria (talk | contribs) (Moved ''least-approved'' definition to top of article)

Jump to: navigation, search


The Definite Majority Choice winner is the least approved candidate who is preferred pairwise over each higher-approved 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 ties. If defeat strength is measured by the Total Approval score of the pairwise winner, all three other methods become equivalent to DMC (For proof, see this).

DMC chooses the same winner as Ranked Approval Voting (RAV) (also known as Approval Ranked Concorcet), and Pairwise Sorted Approval (PSA).


Some people 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


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

On this ballot,


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

Tallying Votes



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


To determine the winner,

  1. Eliminate all definitively defeated candidates. We call the remaining candidates the definite majority set.
  2. The winner is the single candidate who pairwise defeats (wins head-to-head contests with) all other candidates in the definite majority set.

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 and then recalculating the new Smith set until a single winner exists. 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.

DMC has some interesting properties:

  • The DMC winner has the lowest total approval score of any candidate in the Definite Majority set.
  • When defeat strength is measured by the approval of the defeating candidate, there is only one possible immune method, namely DMC.
  • DMC is a strong majority rule method.

A more intuitive ballot --- Ranking Candidates using Grades




            A    B    C    D    F       +  /  -







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.





An even simpler ballot --- Voting by slate


                     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)   |



Say the Libertarian slate for this race is








Handling Ties and Near Ties

Approval Ties



  1. In descending order of Approval

Pairwise Ties



See Also