Definite Majority Choice: Difference between revisions

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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.
# 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.
# 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.
# 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.
# 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.
# 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
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
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{| border="1"
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== Properties ==
== Properties ==
DMC satisfies the following properties:
DMC satisfies the following properties:
* DMC satisfies the four [[Majority#Majority_rule.2FMajority_winner_-_Four_Criteria|strong majority rule]] criteria.
* 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.
* 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.
* No candidate can win under DMC if defeated by a higher-approved candidate.
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An equivalent, more technical explanation follows.
An equivalent, more technical explanation follows.


We call a candidate [[Techniques_of_method_design#Defeats_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''.
We call a candidate [[Techniques of method design#Defeats 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:
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.
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 satisfies 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 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,
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.


DMC is also equivalent to [[Ranked Approval Voting]] (RAV) (also known as
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_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.


== Example ==
== Example ==
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=== Tallying Votes ===
=== Tallying Votes ===
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.
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.
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For example, the X2>X4 ("for X2 over X4") vote is entered in {row 2, column 4}.
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_method#Counting_with_Matrices|elsewhere]]. But in DMC, X2 ''definitively defeats'' X4 if
When pairwise totals are completed, we determine the outcome of a particular pairwise contest as described [[Condorcet method#Counting 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 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 {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 ====
==== Pairwise Ties ====
When there are no ties, the winner is the least approved member of the definite majority ([[Techniques_of_method_design#Special_sets|P]]) set.
When there are no ties, the winner is the least approved member of the definite majority ([[Techniques of method design#Special 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_Majorities#Compute_Tiebreak|Random Ballot]] procedure as in [[Maximize Affirmed Majorities]].
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 Majorities#Compute 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_Choice|Democratic Fair Choice]].
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 Choice|Democratic Fair Choice]].


== 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.
*[[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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