Difference between revisions of "Definite Majority Choice"

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'''Definite Majority Choice''' (DMC) is a [[voting method]] proposed by several (name suggested by [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015164.html Forest Simmons]) to select a single winner using ballots that express preferences, with an additional indication of Approval Cutoff.
+
'''Definite Majority Choice''' (DMC), also known as '''Ranked Approval Voting''' (RAV) is a single-winner [[voting method]] which uses a hybrid ballot combining both ordinal ranking and approval rating. The method is summarized as
 +
:'''While no undefeated candidates exist, eliminate the least-approved candidate.'''
 +
See also [[Proposed Statutory Rules for DMC]].
  
If there is a candidate who is preferred over the other candidates,
+
== [[Range voting]] implementation ==
when compared in turn with each of the others, DMC guarantees that that candidate will win.
+
From a voter's standpoint, the simplest ballot would use [[Range voting]].  Then the candidates' total score, as a measure of approval, would be used to resolve cycles.
Because of this property, DMC is (by definition) a '''[[Condorcet method]]'''.
+
# Voters cast [[ratings ballot]]s, rating as many candidates as they like.  Equal rating and ranking of candidates is allowed.  Separate ranking of equally-rated candidates is provided.  Write-in candidates are allowed.  Unrated candidates are allowed.
Note that this is different from some other preference voting systems such as [[Borda count|Borda]] and
+
# Ordinal (rank) information is inferred from the candidate rating plus additional ranking.  For example, candidates might be rated from 0 to 99, with 99 most favored.
[[Instant-runoff voting]], which do not make this guarantee.
+
# Each ballot's ordinal ranking is tabulated into a pairwise array containing results for each head-to-head contest (see [[Definite_Majority_Choice#Tallying_Votes|example]] below).  The total rating for each candidate is also tabulated.
 +
# The winner is the candidate who, when compared with every other (un-dropped) candidate, is preferred over the other candidate.
 +
# If no undefeated candidates exist, the candidate with lowest total rating is dropped, and we return to step 4.
  
The main difference between DMC and other 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 is equivalent to each of these other methods [''This needs to be verified! --[[User:Araucaria|Araucaria]] 12:22, 21 Mar 2005 (PST)'']
+
Quick example:  A:99 , B:98, C:50, D:25, E:25 would be counted as
 +
A>B>C>D=E
 +
{| border="1"
 +
! !! A !! B !! C !! D !! E !! F
 +
|-
 +
! A || 99 || 1 || 1 || 1 || 1 || 1
 +
|-
 +
! B || 0 || 98 || 1 || 1 || 1 || 1
 +
|-
 +
! C || 0 || 0 || 50 || 1 || 1 || 1
 +
|-
 +
! D || 0 || 0 || 0 || 25 || 0 || 1
 +
|-
 +
! E || 0 || 0 || 0 || 0 || 25 || 1
 +
|-
 +
! F || 0 || 0 || 0 || 0 || 0 || 00
 +
|}
  
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).
+
== Properties ==
 +
DMC satisfies the following properties:
 +
* DMC satisfies the four [[Majority#Majority_rule.2FMajority_winner_-_Four_Criteria|strong majority rule]] criteria.
 +
* When defeat strength is measured by the pairwise winner's approval rating, DMC is equivalent to [[Ranked Pairs]], [[Schulze method|Schulze]] and [[River]], and is the only strong majority method.
 +
* No candidate can win under DMC if defeated by a higher-approved candidate.
  
== Procedure ==
+
== Background ==
 +
The name "DMC" was first suggested [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015164.html here]. Equivalent methods have been suggested several times on the EM mailing list:
 +
* The [[Pairwise Sorted Approval]] equivalent was first proposed by Forest Simmons in [http://lists.electorama.com/pipermail/election-methods-electorama.com/2001-March/005448.html March 2001].
 +
* The Ranked Approval Voting equivalent was first proposed by Kevin Venzke in [http://lists.electorama.com/pipermail/election-methods-electorama.com/2003-September/010799.html September 2003].  The name was suggested by Russ Paielli in 2005.
  
=== The Ballot ===
+
The [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015144.html philosophical basis] of DMC is to eliminate candidates that the voters strongly agree should ''not'' win, using two strong measures, and choose the undefeated candidate from those remaining.
  
Voters rank their preferred candidates, from favorite to least preferred, and may optionally specify an Approval cutoff.
+
An equivalent, more technical explanation follows.
  
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 optionVoters 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.
+
We call a candidate [[Techniques_of_method_design#Defeats_and_defeat_strength|definitively defeated]] when that candidate is defeated in a head-to-head contest against any other candidate with higher Approval ratingThis kind of defeat is also called an ''Approval-consistent defeat''.
  
 +
To find the DMC winner:
 +
# Eliminate all definitively defeated candidates.  The remaining candidates are called the '''definite majority set'''.  We also call these candidates the '''provisional set''' (or '''P-set'''), since the winner will be found from among that set.
 +
# Among P-set candidates, eliminate any candidate who is defeated by a lower-rated P-set opponent.
 +
# When there are no pairwise ties, there will be one remaining candidate.
 +
Note that the least-approved candidate in the P-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 count|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), [[Schulze method|Schulze]] and [[River]] is the use of the additional Approval rating to break cyclic ambiguities.  If defeat strength is measured by the Total Approval rating of the pairwise winner, all three other methods become equivalent to DMC (See [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015405.html proof]).  Therefore,
 +
* DMC is a strong majority rule method.
 +
* When defeat strength is measured by the approval rating of the defeating candidate, DMC is the only possible immune ([[Condorcet_method#Key_terms_in_ambiguity_resolution|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 Criterion|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 until a single undefeated candidate 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_voting|Approval]] winner will always be a member of the definite majority set, because it cannot be definitively defeated.
 +
 +
== Example ==
 +
Here's a set of preferences taken from Rob LeGrand's [http://cec.wustl.edu/~rhl1/rbvote/calc.html online voting calculator].  We indicate the approval cutoff using '''>>'''.
 +
 +
The ranked ballots:
 
<pre>
 
<pre>
            A    B    C    D    F      + / -
+
  98: Abby > Cora > Erin >> Dave > Brad
           
+
  64: Brad > Abby > Erin >> Cora > Dave
      X1  ( ) ( ) ( ) ( ) ( )    ( ) ( )
+
  12: Brad >  Abby > Erin >> Dave > Cora
           
+
  98: Brad > Erin > Abby >> Cora > Dave
      X2  ( ) ( ) ( ) ( ) ( )    ( ) ( )
+
  13: Brad > Erin > Abby >> Dave > Cora
           
+
125: Brad > Erin >> Dave > Abby > Cora
      X3  ( ) ( ) ( ) ( ) ( )    ( ) ( )
+
124: Cora >  Abby >  Erin >> Dave > Brad
           
+
  76: Cora >  Erin > Abby >> Dave > Brad
      X3  ( ) ( ) ( ) ( ) ( )    ( ) ( )
+
  21: Dave > Abby >> Brad > Erin > Cora
           
+
  30: Dave >> Brad > Abby > Erin > Cora
  Lowest ( ) ( ) ( ) ( ) ( )    ( ) ( )
+
  98: Dave > Brad > Erin >> Cora > Abby
  Passing
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139: Dave >  Cora >  Abby >> Brad > Erin
  Grade
+
23: Dave >  Cora >> Brad >  Abby > Erin
  ("C-" Default)
 
 
</pre>
 
</pre>
  
A voter may give the same grade (rank) to more than one candidate.  Ungraded candidates are graded (ranked) below all graded candidates.
+
The pairwise matrix, with the victorious and approval scores highlighted:
 +
<table border cellpadding=3>
 +
<tr align="center"><td colspan=2 rowspan=2></td><th colspan=5>against</th></tr>
 +
<tr align="center"><td class="against"><span class="cand">Abby</span></td><td class="against"><span class="cand">Brad</span></td><td class="against"><span class="cand">Cora</span></td><td class="against"><span class="cand">Dave</span></td><td class="against"><span class="cand">Erin</span></td></tr>
 +
<tr align="center">
 +
<th rowspan=5>for</th>
 +
<td class="for"><span class="cand">Abby</span></td>
 +
<td bgcolor="yellow">645</td>
 +
<td class="loss">458</td>
 +
<td bgcolor="yellow">461</td>
 +
<td bgcolor="yellow">485</td>
 +
<td bgcolor="yellow">511</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Brad</span></td>
 +
<td bgcolor="yellow">463</td>
 +
<td bgcolor="yellow">410</td>
 +
<td bgcolor="yellow">461</td>
 +
<td class="loss">312</td>
 +
<td bgcolor="yellow">623</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Cora</span></td>
 +
<td class="loss">460</td>
 +
<td class="loss">460</td>
 +
<td bgcolor="yellow">460</td>
 +
<td class="loss">460</td>
 +
<td class="loss">460</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Dave</span></td>
 +
<td class="loss">436</td>
 +
<td bgcolor="yellow">609</td>
 +
<td bgcolor="yellow">461</td>
 +
<td bgcolor="yellow">311</td>
 +
<td class="loss">311</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Erin</span></td>
 +
<td class="loss">410</td>
 +
<td class="loss">298</td>
 +
<td bgcolor="yellow">461</td>
 +
<td bgcolor="yellow">610</td>
 +
<td bgcolor="yellow">708</td>
 +
</tr>
 +
</table>
  
Any candidate at the Lowest Passing Grade or higher is given one Approval vote.  Unless changed, the Lowest Passing Grade is C-minus by default.
+
The candidates in descending order of approval are Erin, Abby, Cora, Brad, Dave.
  
No Approval votes are given to ungraded candidates or candidates graded below the Lowest Passing Grade.
+
After reordering the pairwise matrix, it looks like this:
  
Grades assigned to non-passing (disapproved) candidates help determine which of them will win if the voter's approved candidates do not win.
+
<table border cellpadding=3>
 +
<tr align="center"><td colspan=2 rowspan=2></td><th colspan=5>against</th></tr>
 +
<tr align="center">
 +
<td class="against"><span class="cand">Erin</span></td>
 +
<td class="against"><span class="cand">Abby</span></td>
 +
<td class="against"><span class="cand">Cora</span></td>
 +
<td class="against"><span class="cand">Brad</span></td>
 +
<td class="against"><span class="cand">Dave</span></td>
 +
</tr>
 +
<tr align="center">
 +
<th rowspan=5>for</th>
 +
<td class="for"><span class="cand">Erin</span></td>
 +
<td bgcolor="yellow">708</td>
 +
<td class="loss">410</td>
 +
<td bgcolor="yellow">461</td>
 +
<td class="loss">298</td>
 +
<td bgcolor="yellow">610</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Abby</span></td>
 +
<td bgcolor="yellow">511</td>
 +
<td bgcolor="yellow">645</td>
 +
<td bgcolor="yellow">461</td>
 +
<td class="loss">458</td>
 +
<td bgcolor="yellow">485</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Cora</span></td>
 +
<td class="loss">460</td>
 +
<td class="loss">460</td>
 +
<td bgcolor="yellow">460</td>
 +
<td class="loss">460</td>
 +
<td class="loss">460</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Brad</span></td>
 +
<td bgcolor="yellow">623</td>
 +
<td bgcolor="yellow">463</td>
 +
<td bgcolor="yellow">461</td>
 +
<td bgcolor="yellow">410</td>
 +
<td class="loss">312</td>
 +
</tr>
 +
<tr align="center">
 +
<td class="for"><span class="cand">Dave</span></td>
 +
<td class="loss">311</td>
 +
<td class="loss">436</td>
 +
<td bgcolor="yellow">461</td>
 +
<td bgcolor="yellow">609</td>
 +
<td bgcolor="yellow">311</td>
 +
</tr>
 +
</table>
  
Adding a plus or minus to a candidate's grade is optional, but allows up to 15 rankings.
+
To find the winner,
 +
* We start at the lower right diagonal entry, and start moving upward and leftward along the diagonal.
 +
* We eliminate the least approved candidate until one of the higher-approved remaining candidates has a solid row of victories in non-eliminated columns.
 +
* Dave is eliminated first, and Brad pairwise defeats all remaining candidates.  So Brad is the DMC winner.
  
99% of the time, there should be no need to change the LPG -- with 9 grade levels from A+ to C-, there is plenty of room to express relative preferences.
+
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.
  
But the LPG option allows a voter to move the cutoff higher or lower if sentiment changes before finalizing the ballot, without having to get a new ballot.
+
=== Simple ballot example ===
 +
A voter ranks candidates in order of preference, additionally indicating approval cutoff, using a ballot like the following:
 +
<pre>
 +
            +-----------------------+---------------+
 +
            |                RANKING                |
 +
            +-------+-------+-------+-------+-------+
 +
            |  1  |  2  |  3  |  4  |  5  |
 +
------------+-------+-------+-------+-------+-------+
 +
          X1 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
 +
            |      |      |      |      |      |
 +
          X2 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
 +
            |      |      |      |      |      |
 +
          X3 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
 +
            |      |      |      |      |      |
 +
          X4 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
 +
            |      |      |      |      |      |
 +
DISAPPROVED |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
 +
------------+-------+-------+-------+-------+-------+
 +
</pre>
  
==== Discussion ====
+
As an example, say a voter ranked candidates as follows:
What is a voter saying by giving a candidate a grade below the Approval Cutoff?
+
<pre>
 +
            +-----------------------+---------------+
 +
            |                RANKING                |
 +
            +-------+-------+-------+-------+-------+
 +
            |  1  |  2  |  3  |  4  |  5  |
 +
------------+-------+-------+-------+-------+-------+
 +
          X1 |  ( )  |  ( )  |  ( )  |  (X)  |  ( )  |
 +
            |      |      |      |      |      |
 +
          X2 |  (X)  |  ( )  |  ( )  |  ( )  |  ( )  |
 +
            |      |      |      |      |      |
 +
          X3 |  ( )  |  ( )  |  ( )  |  ( )  |  (X)  |
 +
            |      |      |      |      |      |
 +
          X4 |  ( )  |  (X)  |  ( )  |  ( )  |  ( )  |
 +
            |      |      |      |      |      |
 +
DISAPPROVED |  ( )  |  ( )  |  (X)  |  ( )  |  ( )  |
 +
------------+-------+-------+-------+-------+-------+
 +
</pre>
  
One could consider the LPG to be like Gerald FordAnybody better would make a good president, and anybody worse would be bad.
+
We summarize this ballot as
 +
  X2 > X4 >> X1 > X3
 +
where the ">>" indicates the approval cutoff --- candidates to the right of that sign receive no approval votesThis ballot is counted as
 +
  X2 > X2  (approval point)
 +
  X2 > X4
 +
  X2 > X1
 +
  X2 > X3
 +
  X4 > X4  (approval point)
 +
  X4 > X1
 +
  X4 > X3
 +
  X1 > X3
  
Grading candidate X below the LPG gives the voter a chance to say "I don't like X and don't want him to win, but of all the alternatives, he would make the least change in the wrong direction.  I won't give him 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.
+
Alternatively, we treat '''Disapproved''' (D) as another candidate, and treat votes against D as approval points.
  
 
=== Tallying Votes ===
 
=== Tallying Votes ===
 +
As in other [[Condorcet method]]s, the rankings on a single ballot are added into a round-robin grid using the standard [[Condorcet_method#Counting_with_matrices|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.
  
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.
+
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.
  
To determine the winner:
+
For example, the single example ballot above,
# Eliminate any candidate that is defeated in a one-to-one match with any other higher-approved candidate. So by 2 different measures, a definite majority agrees that candidate should be eliminated.
+
  X2 > X4 >> X1 > X3
#If more than one candidate remains, the winner is the single candidate that defeats all others in one-to-one (pairwise)contests.
+
, the following votes would be added into the pairwise array:
 +
{| border="1"
 +
! !! X1 !! X2 !! X3 !! X4
 +
|-
 +
! X1 || 0 || 0 || 1 || 0
 +
|-
 +
! X2 || 1 || 1 || 1 || 1
 +
|-
 +
! X3 || 0 || 0 || 0 || 0
 +
|-
 +
! X4 || 1 || 0 || 1 || 1
 +
|}
  
A simple way to implement this is to permute the pairwise array so that candidates are listed in order of approval.
+
For example, the X2>X4 ("for X2 over X4") vote is entered in {row 2, column 4}.
  
Then, starting with the diagonal cell corresponding to the least-approved candidate, examine all the cells in the column above the diagonal.  If the candidate loses to any higher-approved candidates, proceed up the diagonal to the approval rating of the next-higher-approved candidate.
+
When pairwise totals are completed, we determine the outcome of a particular pairwise contest as described [[Condorcet_method#Counting_with_Matrices|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.
  
The winner is the first candidate found with no losses to higher-approved candidates.
+
==== Discussion ====
 +
What is a voter saying by giving a candidate a non-approved grade or rank?
  
''Can this be made clearer? --[[User:Araucaria|Araucaria]] 15:05, 21 Mar 2005 (PST)''
+
Disapproving a ranked candidate X 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 won't approve X 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 in the hands of the strongest group of core supportors within the majority faction.
  
 
=== Handling Ties and Near Ties ===
 
=== Handling Ties and Near Ties ===
  
In ordinary DMC, the winner is the candidate in Forest Simmon's '''[http://wiki.electorama.com/wiki/Techniques_of_method_design#Special_sets P]''' set, the ''set of candidates which are not approval-consistently defeated''.
+
==== Approval Ties ====
 +
 
 +
During the initial ranking of candidates, two candidates may have the same approval score.
 +
 
 +
If equal approval scores affect the outcome (which only occurs when there is no candidate who defeats all others), 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
 +
# In descending order of approval score
 +
# If equal, in descending order of total first- and second-place vote
 +
# If equal, in descending order of total first-, second- and third-place votes
 +
# If equal, in descending order of ranks above last place
 +
# If equal, in descending order of total first-place votes
 +
 
 +
==== Pairwise Ties ====
 +
When there are no ties, the winner is the least approved member of the definite majority ([[Techniques_of_method_design#Special_sets|P]]) set.
 +
 
 +
When the least-approved member of the definite majority set has a pairwise tie or disputed contest (say, margin within 0.01%) with another member of the definite majority set, there is no clear winner.  In that case, pairwise ties could be handled using the same [[Maximize_Affirmed_Majorities#Compute_Tiebreak|Random Ballot]] procedure as in [[Maximize Affirmed Majorities]].
  
But in the event of a tie or near tie (say, margin within 0.01%), there may be no clear winner.
+
Alternatively, the winner could be decided by using a random ballot to choose the winner from among the definite majority set, as in [[Imagine_Democratic_Fair_Choice|Democratic Fair Choice]].
  
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 [http://wiki.electorama.com/wiki/Techniques_of_method_design#Orderings Random Ballot Order].
+
== See Also ==
  
 +
*[[Proposed Statutory Rules for DMC]]: The rules for DMC in a form that would be suitable for adoption by a state legislature.
 +
* [[Imagine Democratic Fair Choice]]:  a method that picks its winner from the same P set as DMC.  It currently uses a 'slate' ballot similar to the one suggested above.
 +
* [[Pairwise Sorted Methods]]
 +
* [[Marginal Ranked Approval Voting]]:  chooses the winner from a subset of the definite majority set.
  
 +
[[Category:Single-winner voting methods]]
 
[[Category:Condorcet method]]
 
[[Category:Condorcet method]]
 +
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Latest revision as of 08:23, 11 May 2017

Definite Majority Choice (DMC), also known as Ranked Approval Voting (RAV) is a single-winner voting method which uses a hybrid ballot combining both ordinal ranking and approval rating. The method is summarized as

While no undefeated candidates exist, eliminate the least-approved candidate.

See also Proposed Statutory Rules for DMC.

Range voting implementation

From a voter's standpoint, the simplest ballot would use Range voting. Then the candidates' total score, as a measure of approval, would be used to resolve cycles.

  1. Voters cast ratings ballots, rating as many candidates as they like. Equal rating and ranking of candidates is allowed. Separate ranking of equally-rated candidates is provided. Write-in candidates are allowed. Unrated candidates are allowed.
  2. Ordinal (rank) information is inferred from the candidate rating plus additional ranking. For example, candidates might be rated from 0 to 99, with 99 most favored.
  3. Each ballot's ordinal ranking is tabulated into a pairwise array containing results for each head-to-head contest (see example below). The total rating for each candidate is also tabulated.
  4. The winner is the candidate who, when compared with every other (un-dropped) candidate, is preferred over the other candidate.
  5. If no undefeated candidates exist, the candidate with lowest total rating is dropped, and we return to step 4.

Quick example: A:99 , B:98, C:50, D:25, E:25 would be counted as

A>B>C>D=E
A B C D E F
A 99 1 1 1 1 1
B 0 98 1 1 1 1
C 0 0 50 1 1 1
D 0 0 0 25 0 1
E 0 0 0 0 25 1
F 0 0 0 0 0 00

Properties

DMC satisfies the following properties:

  • DMC satisfies the four strong majority rule criteria.
  • When defeat strength is measured by the pairwise winner's approval rating, DMC is equivalent to Ranked Pairs, Schulze and River, and is the only strong majority method.
  • No candidate can win under DMC if defeated by a higher-approved candidate.

Background

The name "DMC" was first suggested here. Equivalent methods have been suggested several times on the EM mailing list:

The philosophical basis of DMC is to eliminate candidates that the voters strongly agree should not win, using two strong measures, and choose the undefeated candidate from those remaining.

An equivalent, more technical explanation follows.

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

To find the DMC winner:

  1. Eliminate all definitively defeated candidates. The remaining candidates are called the definite majority set. We also call these candidates the provisional set (or P-set), since the winner will be found from among that set.
  2. Among P-set candidates, eliminate any candidate who is defeated by a lower-rated P-set opponent.
  3. When there are no pairwise ties, there will be one remaining candidate.

Note that the least-approved candidate in the P-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), Schulze and River is the use of the additional Approval rating to break cyclic ambiguities. If defeat strength is measured by the Total Approval rating 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 rating 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 until a single undefeated candidate 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.

Example

Here's a set of preferences taken from Rob LeGrand's online voting calculator. We indicate the approval cutoff using >>.

The ranked ballots:

 98: Abby >  Cora >  Erin >> Dave > Brad
 64: Brad >  Abby >  Erin >> Cora > Dave
 12: Brad >  Abby >  Erin >> Dave > Cora
 98: Brad >  Erin >  Abby >> Cora > Dave
 13: Brad >  Erin >  Abby >> Dave > Cora
125: Brad >  Erin >> Dave >  Abby > Cora
124: Cora >  Abby >  Erin >> Dave > Brad
 76: Cora >  Erin >  Abby >> Dave > Brad
 21: Dave >  Abby >> Brad >  Erin > Cora
 30: Dave >> Brad >  Abby >  Erin > Cora
 98: Dave >  Brad >  Erin >> Cora > Abby
139: Dave >  Cora >  Abby >> Brad > Erin
 23: Dave >  Cora >> Brad >  Abby > Erin

The pairwise matrix, with the victorious and approval scores highlighted:

against
AbbyBradCoraDaveErin
for Abby 645 458 461 485 511
Brad 463 410 461 312 623
Cora 460 460 460 460 460
Dave 436 609 461 311 311
Erin 410 298 461 610 708

The candidates in descending order of approval are Erin, Abby, Cora, Brad, Dave.

After reordering the pairwise matrix, it looks like this:

against
Erin Abby Cora Brad Dave
for Erin 708 410 461 298 610
Abby 511 645 461 458 485
Cora 460 460 460 460 460
Brad 623 463 461 410 312
Dave 311 436 461 609 311

To find the winner,

  • We start at the lower right diagonal entry, and start moving upward and leftward along the diagonal.
  • We eliminate the least approved candidate until one of the higher-approved remaining candidates has a solid row of victories in non-eliminated columns.
  • Dave is eliminated first, and Brad pairwise defeats all remaining candidates. So Brad is the DMC winner.

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 indicating approval cutoff, using a ballot like the following:

             +-----------------------+---------------+
             |                RANKING                |
             +-------+-------+-------+-------+-------+
             |   1   |   2   |   3   |   4   |   5   |
 ------------+-------+-------+-------+-------+-------+
          X1 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
             |       |       |       |       |       |
          X2 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
             |       |       |       |       |       |
          X3 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
             |       |       |       |       |       |
          X4 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
             |       |       |       |       |       |
 DISAPPROVED |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
 ------------+-------+-------+-------+-------+-------+

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

             +-----------------------+---------------+
             |                RANKING                |
             +-------+-------+-------+-------+-------+
             |   1   |   2   |   3   |   4   |   5   |
 ------------+-------+-------+-------+-------+-------+
          X1 |  ( )  |  ( )  |  ( )  |  (X)  |  ( )  |
             |       |       |       |       |       |
          X2 |  (X)  |  ( )  |  ( )  |  ( )  |  ( )  |
             |       |       |       |       |       |
          X3 |  ( )  |  ( )  |  ( )  |  ( )  |  (X)  |
             |       |       |       |       |       |
          X4 |  ( )  |  (X)  |  ( )  |  ( )  |  ( )  |
             |       |       |       |       |       |
 DISAPPROVED |  ( )  |  ( )  |  (X)  |  ( )  |  ( )  |
 ------------+-------+-------+-------+-------+-------+

We summarize this ballot as

 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

Alternatively, we treat Disapproved (D) as another candidate, and treat votes against D as approval points.

Tallying Votes

As in other Condorcet methods, the rankings on a single ballot are added into a round-robin grid 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.

Discussion

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

Disapproving a ranked candidate X 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 won't approve X 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 in the hands of the strongest group of core supportors 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 (which only occurs when there is no candidate who defeats all others), 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 score
  2. If equal, in descending order of total first- and second-place vote
  3. If equal, in descending order of total first-, second- and third-place votes
  4. If equal, in descending order of ranks above last place
  5. If equal, in descending order of total first-place votes

Pairwise Ties

When there are no ties, the winner is the least approved member of the definite majority (P) set.

When the least-approved member of the definite majority set has a pairwise tie or disputed contest (say, margin within 0.01%) with another member of the definite majority set, there is no clear winner. In that case, pairwise ties could be handled using the same Random Ballot procedure as in Maximize Affirmed Majorities.

Alternatively, the winner could be decided by using a random ballot to choose the winner from among the definite majority set, as in Democratic Fair Choice.

See Also