Difference between revisions of "Definite Majority Choice"

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'''Definite Majority Choice''' (DMC) is a single-winner [[voting method]] that
+
'''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
uses ballots expressing both ordinal rank and approval rating. The name "DMC" was first suggested [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015164.html here]. Equivlalent methods have been suggested several times on the EM mailing list:
+
:'''While no undefeated candidates exist, eliminate the least-approved candidate.'''
* 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].
+
See also [[Proposed Statutory Rules for DMC]].
* 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 [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 different strong measures, and choose the undefeated candidate from those remaining.
+
== [[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.
 +
# 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.
 +
# 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.
 +
# 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 one-line ([[Ranked Approval Voting]]) heuristic for determining the DMC winner is:
+
Quick example:  A:99 , B:98, C:50, D:25, E:25 would be counted as
* While no undefeated candidates exist, eliminate the least-approved candidate.
+
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
 +
|}
 +
 
 +
== 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.
 +
 
 +
== 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 [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.
  
 
An equivalent, more technical explanation follows.
 
An equivalent, more technical explanation follows.
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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.
 
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.
+
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 DMC winner satisifies this variant of the [[Condorcet Criterion]]:
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: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 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|Cloneproof Schwartz Sequential Dropping]] (Beatpath or 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,
+
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.
 
* 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.
 
* 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.
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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.
 
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.
  
Some believe that DMC is currently the best candidate for a Condorcet Method that meets the [[Public Acceptability Criterion|Public Acceptability "Criterion"]].
+
== 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 '''>>'''.
  
== Procedure ==
+
The ranked ballots:
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.
+
<pre>
 +
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
 +
</pre>
  
Suppose that the candidates (in order of approval) are
+
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>
  
John
+
The candidates in descending order of approval are Erin, Abby, Cora, Brad, Dave.
  
Jane
+
After reordering the pairwise matrix, it looks like this:
  
Jill
+
<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>
  
Jack
+
To find the winner,
 
+
* We start at the lower right diagonal entry, and start moving upward and leftward along the diagonal.
Jean
+
* 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.
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 Criterion|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.
 
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 ===
 
=== 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:
+
A voter ranks candidates in order of preference, additionally indicating approval cutoff, using a ballot like the following:
 
<pre>
 
<pre>
        +-----------------------+---------------+
+
            +-----------------------+---------------+
        |<-- Favorite        Least Preferred -->|
+
            |               RANKING                |
        +-----------------------+---------------+
+
            +-------+-------+-------+-------+-------+
        |<--    Approved    -->|  Not Approved |
+
            |  1  |  2  |  3  |  4  |  5  |
        +-------+-------+-------+-------+-------+
+
------------+-------+-------+-------+-------+-------+
        |  1  |  2  |  3  |  4  |  5  |
+
          X1 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
  -------+-------+-------+-------+-------+-------+
+
            |      |      |      |      |      |
      X1 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
+
          X2 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
        |      |      |      |      |      |
+
            |      |      |      |      |      |
      X2 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
+
          X3 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
        |      |      |      |      |      |
+
            |      |      |      |      |      |
      X3 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
+
          X4 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
        |      |      |      |      |      |
+
            |      |      |      |      |       |
       X4 |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
+
DISAPPROVED |  ( )  |  ( )  |  ( )  |  ( )  |  ( )  |
  -------+-------+-------+-------+-------+-------+
+
------------+-------+-------+-------+-------+-------+
 
</pre>
 
</pre>
 
On this ballot,
 
# Candidates ranked at 1st through 3rd choice get 1 approval point each.
 
# Candidates ranked fourth, fifth and unranked receive no approval points.
 
# 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:
 
As an example, say a voter ranked candidates as follows:
 
<pre>
 
<pre>
        +-----------------------+---------------+
+
            +-----------------------+---------------+
        |<-- Favorite        Least Preferred -->|
+
            |               RANKING                |
        +-----------------------+---------------+
+
            +-------+-------+-------+-------+-------+
        |<--    Approved    -->|  Not Approved |
+
            |  1  |  2  |  3  |  4  |  5  |
        +-------+-------+-------+-------+-------+
+
------------+-------+-------+-------+-------+-------+
        |  1  |  2  |  3  |  4  |  5  |
+
          X1 |  ( )  |  ( )  |  ( )  |  (X)  |  ( )  |
  -------+-------+-------+-------+-------+-------+
+
            |      |      |      |      |      |
      X1 |  ( )  |  ( )  |  ( )  |  (X)  |  ( )  |
+
          X2 |  (X)  |  ( )  |  ( )  |  ( )  |  ( )  |
        |      |      |      |      |      |
+
            |      |      |      |      |      |
      X2 |  (X)  |  ( )  |  ( )  |  ( )  |  ( )  |
+
          X3 |  ( )  |  ( )  |  ( )  |  ( )  |  (X)  |
        |      |      |      |      |      |
+
            |      |      |      |      |      |
      X3 |  ( )  |  ( )  |  ( )  |  ( )  |  (X)  |
+
          X4 |  ( )  |  (X)  |  ( )  |  ( )  |  ( )  |
        |      |      |      |      |      |
+
            |      |      |      |      |      |
      X4 |  ( )  |  (X)  |  ( )  |  ( )  |  ( )  |
+
DISAPPROVED |  ( )  |  ( )  |  (X)  |  ( )  |  ( )  |
  -------+-------+-------+-------+-------+-------+
+
------------+-------+-------+-------+-------+-------+
 
</pre>
 
</pre>
  
This ballot could be summarized briefly using the notation
+
We summarize this ballot as
 
   X2 > X4 >> X1 > X3
 
   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
 
where the ">>" indicates the approval cutoff --- candidates to the right of that sign receive no approval votes.  This ballot is counted as
Line 130: Line 245:
 
   X4 > X3
 
   X4 > X3
 
   X1 > X3
 
   X1 > X3
 +
 +
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 table 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.
+
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.
  
 
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.
 
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.
Line 157: Line 274:
 
* the {row 2, column 2} (X2>X2) total approval score exceeds the {row 4, column 4} (X4>X4) total approval score.
 
* 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 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.
 
<pre>
 
            A    B    C    D    F      +  /  -
 
 
      X1  ( )  ( )  ( )  ( )  ( )    ( )  ( )
 
 
      X2  ( )  ( )  ( )  ( )  ( )    ( )  ( )
 
 
      X3  ( )  ( )  ( )  ( )  ( )    ( )  ( )
 
 
      X3  ( )  ( )  ( )  ( )  ( )    ( )  ( )
 
</pre>
 
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 [[Imagine Democratic Fair Choice|Democratic Fair Choice]] method:
 
<pre>
 
                    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)  |
 
</pre>
 
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
 
<pre>
 
    Deirdre (Lib.) >> Cecil (Reb.) > Ellen (Dem.) > Bob (Ind.) > Anna (Green)
 
</pre>
 
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
 
<pre>
 
    Deirdre (Lib.) > Bob (Ind.) > Anna (Green) >> Cecil (Reb.) > Ellen (Dem.)
 
</pre>
 
  
 
==== Discussion ====
 
==== Discussion ====
 
What is a voter saying by giving a candidate a non-approved grade or rank?
 
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.
+
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.
 
 
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 ===
 
=== Handling Ties and Near Ties ===
Line 244: Line 288:
 
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
 
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
 
# In descending order of approval score
# If equal, in descending order of Bucklin count
+
# 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 total first-, second- and third-place votes
# If equal, in descending order of total first- and second-place votes
+
# If equal, in descending order of ranks above last place
 
# If equal, in descending order of total first-place votes
 
# If equal, in descending order of total first-place votes
# If equal, in descending order of "Grade Point Average" (i.e., total Cardinal Rating)
 
With ranked choice ballots, the Bucklin count is determined by first counting all first place votes, then successively adding in lower preference votes until one candidate has more than 50%.  This is a graduated form of approval.  When an approval cutoff is added to the ballot, however, we make this additional change -- the lower preference votes are not added into the Bucklin scores if they are below the cutoff.
 
  
 
==== Pairwise Ties ====
 
==== Pairwise Ties ====
Line 260: Line 302:
 
== See Also ==
 
== 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.
 
* [[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]]
 
* [[Pairwise Sorted Methods]]
 
* [[Marginal Ranked Approval Voting]]:  chooses the winner from a subset of the definite majority set.
 
* [[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]]
 
<!--
 
<!--

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