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'''Definite Majority Choice''' (DMC) is a [[voting method]] proposed by several (name suggested by [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015164.html Forest Simmons]) to select a single winner using ballots that express both ranked preferences and approval.
'''Definite Majority Choice''' (DMC) is a [[voting method]] proposed by several (name suggested by [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015164.html Forest Simmons]) to select a single winner using ballots that express both ranked preferences and approval.


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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 different strong measures, and choose the winner from among those that remain.
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:The Definite Majority Choice winner is the ''least approved'' candidate who is preferred pairwise over each ''higher-approved'' candidate.


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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 score. This kind of defeat is also called an ''Approval-consistent defeat''.
If there is a candidate who is preferred over the other candidates,

when compared in turn with each of the others, DMC guarantees that that candidate will win.
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To find the DMC winner, the candidates are divided into two groups:
Because of this property, DMC is (by definition) a '''[[Condorcet method]]'''.
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# Definitively defeated candidates.
Note that this is different from some other preference voting systems such as [[Borda count|Borda]] and
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# Candidates that pairwise defeat all higher-approved candidates. We call this group the '''definite majority set'''.
[[Instant-runoff voting]], which do not make this guarantee.
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The least-approved candidate in the definite majority set pairwise defeats ''all'' higher-approved candidates, including all other members of the definite majority set, and is the DMC winner.

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]]:


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: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), [[Cloneproof Schwartz Sequential Dropping]] (Beatpath or Schulze) and [[River]] is the use of the additional Approval score to break ties. If defeat strength is measured by the Total Approval score of the pairwise winner, all three other methods become equivalent to DMC (For proof, see [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015405.html this]).


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The main difference between DMC and Condorcet methods such as [[Ranked Pairs]] (RP), [[Cloneproof Schwartz Sequential Dropping]] (Beatpath or Schulze) and [[River]] is the use of the additional Approval score to break cyclic ambiguities. If defeat strength is measured by the Total Approval score 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]).
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DMC chooses the same winner as [[Ranked Approval Voting]] (RAV) (also known as Approval Ranked Concorcet), and [[Pairwise Sorted Approval]] (PSA).


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DMC is also equivalent to [[Ranked Approval Voting]] (RAV) (also known as Approval Ranked Concorcet), and [[Pairwise Sorted Approval]] (PSA).
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The philosophical basis of DMC (also due to [http://lists.electorama.com/pipermail/election-methods-electorama.com/2005-March/015144.html Forest Simmons]) is to first eliminate candidates that the voters strongly agree should ''not'' win, using two different measures, and choose the winner from among those that remain.


Some people believe that DMC is currently the best candidate for a Condorcet Method that meets the [[Public Acceptability Criterion|Public Acceptability "Criterion"]].
Some people believe that DMC is currently the best candidate for a Condorcet Method that meets the [[Public Acceptability Criterion|Public Acceptability "Criterion"]].
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For example, the X2>X4 ("for X2 over X4") vote is entered in {row 2, column 4}.
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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 score. This kind of defeat is also called an ''Approval-consistent defeat''.


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 {row 2, column 2} (X2>X2 total approval score) exceeds {row 4, column 4} (X4>X4 total approval score).
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To determine the winner, the candidates are divided into two groups:
The winner is then determined as described above.
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# Definitively defeated candidates.
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# Candidates that pairwise defeat all higher-approved candidates. We call this group the '''definite majority set'''.
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The least-approved candidate in the definite majority set pairwise defeats ''all'' higher-approved candidates, including all other members of the definite majority set, and is declared the winner.


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 in the Smith set and then recalculating the new Smith set until a single winner exists. But the definite majority set may also contain higher-approved candidates outside the Smith set. For example, the Approval Winner will always be a member of the definite majority set, because it cannot be definitively defeated.
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 in the Smith set and then recalculating the new Smith set until a single winner exists. But the definite majority set may also contain higher-approved candidates outside the Smith set. For example, the Approval Winner will always be a member of the definite majority set, because it cannot be definitively defeated.
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