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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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 ==
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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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== 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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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
|}
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* [[Marginal Ranked Approval Voting]]: chooses the winner from a subset of the definite majority set.
[[Category:
[[Category:Smith-efficient Condorcet methods]]
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