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'''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]].
 
It can be extended to use [[Range voting]] instead of [[Approval voting]] as its base: in that case, the method eliminates the least-rated candidate.
''Page refactoring in progress. Comments welcome.--[[User:Araucaria|Araucaria]] 11:41, 30 Sep 2005 (PDT)''
 
Its elimination logic is the same as [[Benham's method]], and the method can thus be thought of as a rated version of it.
== Procedure ==
 
# Voters cast [[Preferential voting|ranked ballots]], ranking as many candidates as they like. Equal ranking of candidates is allowed.
== [[Range voting]] implementation ==
# By default, all ranked candidates are considered [[Approval voting|approved]]. If the voter doesn't approve of some of the ranked candidates, an [[Approval Cutoff|approval cutoff]] can be placed at some rank, indicating that all candidates at that rank and below are disapproved.
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.
# Ballots are tabulated into a pairwise array containing results for each head-to-head contest (see [[Definite_Majority_Choice#Tallying_Votes|example]] below). Total approval rating for each candidate is also tabulated.
# 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 least-approved 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
{| class="wikitable" 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
|}
 
== Alternative implementation ==
 
This implementation is called '''Pairwise Sorted Approval'''. It is the simplest of a class of [[Pairwise Sorted Methods]].
 
A voter ranks candidates, and specifies approval, either by using an [[Approval Cutoff]] or by ranking above and below a fixed approval cutoff rank.
 
To determine the winner,
# sort candidates in descending order of approval.
# For each candidate, move it higher in the list as long as it pairwise beats the next-higher candidate, and only after all candidates above it have moved upward as far as they can.
 
This procedure can be used to produce a social ordering. It finds the same winner as the Benham-form implementation.
 
== Properties ==
DMC satisfies the following properties:
* DMC satisfies the four [[Majority#Majority_ruleMajority rule.2FMajority_winner_2FMajority winner -_Four_Criteria 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|Cloneproof Schwarz Sequential DroppingSchulze]] 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]. EquivlalentEquivalent methods have been suggested several times on the EM mailing list:
* The [[Pairwise Sorted Approval]] equivalentmethod/implementation 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 equivalentmethod/implementation 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.
Line 26 ⟶ 67:
An equivalent, more technical explanation follows.
 
We call a candidate [[Techniques_of_method_designTechniques of method design#Defeats_and_defeat_strengthDefeats and defeat strength|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:
Line 36 ⟶ 77:
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 satisifiessatisfies 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|Cloneproof Schwartz Sequential DroppingSchulze]] (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,
* 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_methodCondorcet method#Key_terms_in_ambiguity_resolutionKey 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_votingApproval 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 [https://www.cs.angelo.edu/~rlegrand/rbvote/calc.html online voting calculator]. We indicate the approval cutoff using '''>>'''.
Here is a simple example of how ranking and approval information is used to determine the DMC winner.
 
The ranked ballots:
Suppose that the candidates (in order of approval) are
<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>
 
The pairwise matrix, with the victorious and approval scores highlighted:
John
{| class="wikitable" cellpadding="3" border=""
|- align="center"
| colspan=2 rowspan=2 |
! colspan=5 | against
|- align="center"
! class="against" | Abby
! class="against" | Brad
! class="against" | Cora
! class="against" | Dave
! class="against" | Erin
|- align="center"
! rowspan=5 | for
! class="for" | Abby
| bgcolor="yellow" | 645
| class="loss" | 458
| bgcolor="yellow" | 461
| bgcolor="yellow" | 485
| bgcolor="yellow" | 511
|- align="center"
! class="for" | Brad
| bgcolor="yellow" | 463
| bgcolor="yellow" | 410
| bgcolor="yellow" | 461
| class="loss" | 312
| bgcolor="yellow" | 623
|- align="center"
! class="for" | Cora
| class="loss" | 460
| class="loss" | 460
| bgcolor="yellow" | 460
| class="loss" | 460
| class="loss" | 460
|- align="center"
! class="for" | Dave
| class="loss" | 436
| bgcolor="yellow" | 609
| bgcolor="yellow" | 461
| bgcolor="yellow" | 311
| class="loss" | 311
|- align="center"
! class="for" | Erin
| class="loss" | 410
| class="loss" | 298
| bgcolor="yellow" | 461
| bgcolor="yellow" | 610
| bgcolor="yellow" | 708
|}
 
The candidates in descending order of approval are Erin, Abby, Cora, Brad, Dave.
Jane
 
After reordering the pairwise matrix, it looks like this:
Jill
 
{| class="wikitable" cellpadding="3" border=""
Jack
|- align="center"
 
| colspan=2 rowspan=2 |
Jean
! colspan=5 | against
 
|- align="center"
and that the only two "downward" majority preferences are Jill to Jack and Jane to Jean.
! class="against" | Erin
 
! class="against" | Abby
We are assuming that all other majority preferences are directed upward:
! class="against" | Cora
 
! class="against" | Brad
Jane defeats John,
! class="against" | Dave
 
|- align="center"
Jill defeats both Jane and John,
! rowspan=5 | for
 
! class="for" | Erin
Jack defeats both Jane and John, and
| bgcolor="yellow" | 708
 
| class="loss" | 410
Jean defeats John, Jill, and Jack.
| bgcolor="yellow" | 461
 
| class="loss" | 298
The downward or "approval consistent" preferences are enforced by eliminating Jack and Jean.
| bgcolor="yellow" | 610
 
|- align="center"
Jill (pairwise) defeats both of the remaining candidates, so Jill is the DMC winner.
! class="for" | Abby
 
| bgcolor="yellow" | 511
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.
| bgcolor="yellow" | 645
 
| bgcolor="yellow" | 461
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.
| class="loss" | 458
 
| bgcolor="yellow" | 485
The lower the candidate with the "defeat all above" property, the greater the solid list of highly seeded candidates that it defeats.
|- align="center"
! class="for" | Cora
| class="loss" | 460
| class="loss" | 460
| bgcolor="yellow" | 460
| class="loss" | 460
| class="loss" | 460
|- align="center"
! class="for" | Brad
| bgcolor="yellow" | 623
| bgcolor="yellow" | 463
| bgcolor="yellow" | 461
| bgcolor="yellow" | 410
| class="loss" | 312
|- align="center"
! class="for" | Dave
| class="loss" | 311
| class="loss" | 436
| bgcolor="yellow" | 461
| bgcolor="yellow" | 609
| bgcolor="yellow" | 311
|}
 
To find the winner,
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.
* 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.
Line 91 ⟶ 219:
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>
 
As an example, say a voter ranked candidates as follows:
<pre>
┌───────────────────────────────────────┐
+-----------------------+---------------+
|│ RANKING |│
├───────┬───────┬───────┬───────┬───────┤
+-------+-------+-------+-------+-------+
|│ 1 |│ 2 |│ 3 |│ 4 |│ 5 |│
────────────┼───────┼───────┼───────┼───────┼───────┤
------------+-------+-------+-------+-------+-------+
X1 |│ ( ) |│ ( ) |│ ( ) |│ (X●) |│ ( ) |│
|│ |│ |│ |│ |│ |│
X2 |│ (X●) |│ ( ) |│ ( ) |│ ( ) |│ ( ) |│
|│ |│ |│ |│ |│ |│
X3 |│ ( ) |│ ( ) |│ ( ) |│ ( ) |│ (X●) |│
|│ |│ |│ |│ |│ |│
X4 |│ ( ) |│ (X●) |│ ( ) |│ ( ) |│ ( ) |│
|│ |│ |│ |│ |│ |│
DISAPPROVED |│ ( ) |│ ( ) |│ (X●) |│ ( ) |│ ( ) |│
────────────┴───────┴───────┴───────┴───────┴───────┘
------------+-------+-------+-------+-------+-------+
</pre>
 
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
Line 142 ⟶ 270:
 
=== Tallying Votes ===
As in other [[Condorcet method]]s, the rankings on a single ballot are added into a round-robin tablegrid using the standard [[Condorcet_methodCondorcet method#Counting_with_matricesCounting 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.
Line 148 ⟶ 276:
For example, the single example ballot above,
X2 > X4 >> X1 > X3
, the following votes would be added into the pairwise array:
{| class="wikitable" border="1"
|
! !! X1 !! X2 !! X3 !! X4
! X1 !! X2 !! X3 !! X4
|-
! X1 || 0 || 0 || 1 || 0
| 0 || 0 || 1 || 0
|-
! X2 || 1 || 1 || 1 || 1
| 1 || 1 || 1 || 1
|-
! X3 || 0 || 0 || 0 || 0
| 0 || 0 || 0 || 0
|-
! X4 || 1 || 0 || 1 || 1
| 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 [[Condorcet_methodCondorcet method#Counting_with_MatricesCounting 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.
 
==== 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 ====
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.
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 ===
Line 254 ⟶ 314:
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 Bucklintotal countfirst- and second-place vote
# If equal, in descending order of total first-, second- and third-place votes
# If equal, in descending order of totalranks first-above andlast second-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 ====
When there are no ties, the winner is the least approved member of the definite majority ([[Techniques_of_method_designTechniques of method design#Special_setsSpecial 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_MajoritiesMaximize Affirmed Majorities#Compute_TiebreakCompute Tiebreak|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 [[Imagine_Democratic_Fair_ChoiceImagine Democratic Fair Choice|Democratic Fair Choice]].
 
== See Alsoalso ==
 
*[[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:Condorcet-cardinal methodhybrid methods]]
[[Category:Smith-efficient Condorcet methods]]
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