Definite Majority Choice: Difference between revisions
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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.
:'''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.
Its elimination logic is the same as [[Benham's method]], and the method can thus be thought of as a rated version of it.
== [[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
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#
* When defeat strength is measured by the pairwise winner's approval rating, DMC is equivalent to [[Ranked Pairs]], [[Schulze 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].
* The
* The Ranked Approval Voting
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.
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An equivalent, more technical explanation follows.
We call a candidate [[
To find the DMC winner:
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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
: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|
* 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 ([[
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 [[
== 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 '''>>'''.
The ranked ballots:
<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:
{| 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.
After reordering the pairwise matrix, it looks like this:
{| class="wikitable" cellpadding="3" border=""
|- align="center"
| colspan=2 rowspan=2 |
! colspan=5 | against
|- align="center"
! class="against" | Erin
! class="against" | Abby
! class="against" | Cora
! class="against" | Brad
! class="against" | Dave
|- align="center"
! rowspan=5 | for
! class="for" | Erin
| bgcolor="yellow" | 708
| class="loss" | 410
| bgcolor="yellow" | 461
| class="loss" | 298
| bgcolor="yellow" | 610
|- align="center"
! class="for" | Abby
| bgcolor="yellow" | 511
| bgcolor="yellow" | 645
| bgcolor="yellow" | 461
| class="loss" | 458
| bgcolor="yellow" | 485
|- 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,
* 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.
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A voter ranks candidates in order of preference, additionally indicating approval cutoff, using a ballot like the following:
<pre>
┌───────────────────────────────────────┐
├───────┬───────┬───────┬───────┬───────┤
────────────┼───────┼───────┼───────┼───────┼───────┤
X1
X2
X3
X4
DISAPPROVED
────────────┴───────┴───────┴───────┴───────┴───────┘
</pre>
As an example, say a voter ranked candidates as follows:
<pre>
┌───────────────────────────────────────┐
├───────┬───────┬───────┬───────┬───────┤
────────────┼───────┼───────┼───────┼───────┼───────┤
X1
X2
X3
X4
DISAPPROVED
────────────┴───────┴───────┴───────┴───────┴───────┘
</pre>
We summarize this ballot as
X2 > X4 >> X1 > X3
where the ">>" indicates the approval cutoff
X2 > X2 (approval point)
X2 > X4
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=== Tallying Votes ===
As in other [[Condorcet method]]s, the rankings on a single ballot are added into a round-robin
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.
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For example, the single example ballot above,
X2 > X4 >> X1 > X3
{| class="wikitable" 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
|}
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 [[
* 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 ===
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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
# If equal, in descending order of total first-, second- and third-place votes
# If equal, in descending order of
# 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 ([[
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 [[
Alternatively, the winner could be decided by using a random ballot to choose the winner from among the definite majority set, as in [[
== See
*[[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
[[Category:Smith-efficient Condorcet methods]]
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