Definite Majority Choice: Difference between revisions
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:'''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 ==
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# 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 [[
# 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.
Quick example: A:99
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|Schulze]] and [[River]], and is the only strong majority method.
* No candidate can win under DMC if defeated by a higher-approved candidate.
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== 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
* 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.
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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 ([[
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 [
The ranked ballots:
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The pairwise matrix, with the victorious and approval scores highlighted:
|- align="center"
| colspan=2 rowspan=2 |
! colspan=5 | against
|- align="center"
! class="against" | Brad
! class="against" | Dave
! class="against" | Erin
|- align="center"
! rowspan=5 | for
! class="for" | Abby
| bgcolor="yellow" | 645
| class="loss" | 458
| bgcolor="yellow" | 511
|- align="center"
! class="for" | Brad
| bgcolor="yellow" | 463
| bgcolor="yellow" | 410
| bgcolor="yellow" | 461
|- align="center"
| class="loss" | 460
| class="loss" | 460
| bgcolor="yellow" | 460
| class="loss" | 460
|- align="center"
! class="for" | Dave
| bgcolor="yellow" | 609
| bgcolor="yellow" | 461
| bgcolor="yellow" | 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.
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After reordering the pairwise matrix, it looks like this:
|- align="center"
| colspan=2 rowspan=2 |
! colspan=5 | against
|- align="center"
! 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" | Cora
| class="loss" | 460
|- align="center"
! class="for" | Brad
| bgcolor="yellow" | 623
| bgcolor="yellow" | 463
| class="loss" | 312
|- align="center"
| class="loss" | 311
| class="loss" | 436
| bgcolor="yellow" | 461
| bgcolor="yellow" | 609
| bgcolor="yellow" | 311
|}
To find the winner,
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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 grid using the standard [[
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.
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==== 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.
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* [[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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