Definite Majority Choice

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. The method is summarized as

While no undefeated candidates exist, eliminate the least-approved candidate.

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 ballots, 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 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 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

A>B>C>D=E

	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 strong majority rule criteria.
When defeat strength is measured by the pairwise winner's approval rating, DMC is equivalent to Ranked Pairs, Schulze and River, and is the only strong majority method.
No candidate can win under DMC if defeated by a higher-approved candidate.

Background

The name "DMC" was first suggested here. Equivalent methods have been suggested several times on the EM mailing list:

The Pairwise Sorted Approval method/implementation was first proposed by Forest Simmons in March 2001.
The Ranked Approval Voting method/implementation was first proposed by Kevin Venzke in September 2003. The name was suggested by Russ Paielli in 2005.

The 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.

An equivalent, more technical explanation follows.

We call a candidate 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:

Eliminate all definitively defeated candidates. The remaining candidates are called the definite majority set. We also call these candidates the provisional set (or P-set), since the winner will be found from among that set.
Among P-set candidates, eliminate any candidate who is defeated by a lower-rated P-set opponent.
When there are no pairwise ties, there will be one remaining candidate.

Note that the least-approved candidate in the P-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 and Instant-runoff voting, which do not make this guarantee.

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 main difference between DMC and Condorcet methods such as Ranked Pairs (RP), 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 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 (cloneproof) method.

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 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 winner will always be a member of the definite majority set, because it cannot be definitively defeated.

Example

Here's a set of preferences taken from Rob LeGrand's online voting calculator. We indicate the approval cutoff using >>.

The ranked ballots:

 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

The pairwise matrix, with the victorious and approval scores highlighted:

		Abby	Brad	Cora	Dave	Erin
		against
for	Abby	645	458	461	485	511
	Brad	463	410	461	312	623
	Cora	460	460	460	460	460
	Dave	436	609	461	311	311
	Erin	410	298	461	610	708

The candidates in descending order of approval are Erin, Abby, Cora, Brad, Dave.

After reordering the pairwise matrix, it looks like this:

		Erin	Abby	Cora	Brad	Dave
		against
for	Erin	708	410	461	298	610
	Abby	511	645	461	458	485
	Cora	460	460	460	460	460
	Brad	623	463	461	410	312
	Dave	311	436	461	609	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.

Simple ballot example

A voter ranks candidates in order of preference, additionally indicating approval cutoff, using a ballot like the following:

             ┌───────────────────────────────────────┐
             │                RANKING                │
             ├───────┬───────┬───────┬───────┬───────┤
             │   1   │   2   │   3   │   4   │   5   │
 ────────────┼───────┼───────┼───────┼───────┼───────┤
          X1 │  ( )  │  ( )  │  ( )  │  ( )  │  ( )  │
             │       │       │       │       │       │
          X2 │  ( )  │  ( )  │  ( )  │  ( )  │  ( )  │
             │       │       │       │       │       │
          X3 │  ( )  │  ( )  │  ( )  │  ( )  │  ( )  │
             │       │       │       │       │       │
          X4 │  ( )  │  ( )  │  ( )  │  ( )  │  ( )  │
             │       │       │       │       │       │
 DISAPPROVED │  ( )  │  ( )  │  ( )  │  ( )  │  ( )  │
 ────────────┴───────┴───────┴───────┴───────┴───────┘

As an example, say a voter ranked candidates as follows:

             ┌───────────────────────────────────────┐
             │                RANKING                │
             ├───────┬───────┬───────┬───────┬───────┤
             │   1   │   2   │   3   │   4   │   5   │
 ────────────┼───────┼───────┼───────┼───────┼───────┤
          X1 │  ( )  │  ( )  │  ( )  │  (●)  │  ( )  │
             │       │       │       │       │       │
          X2 │  (●)  │  ( )  │  ( )  │  ( )  │  ( )  │
             │       │       │       │       │       │
          X3 │  ( )  │  ( )  │  ( )  │  ( )  │  (●)  │
             │       │       │       │       │       │
          X4 │  ( )  │  (●)  │  ( )  │  ( )  │  ( )  │
             │       │       │       │       │       │
 DISAPPROVED │  ( )  │  ( )  │  (●)  │  ( )  │  ( )  │
 ────────────┴───────┴───────┴───────┴───────┴───────┘

We summarize this ballot as

 X2 > X4 >> X1 > X3

where the ">>" indicates the approval cutoff — candidates to the right of that sign receive no approval votes. This ballot is counted as

 X2 > X2  (approval point)
 X2 > X4
 X2 > X1
 X2 > X3
 X4 > X4  (approval point)
 X4 > X1
 X4 > X3
 X1 > X3

Alternatively, we treat Disapproved (D) as another candidate, and treat votes against D as approval points.

Tallying Votes

As in other Condorcet methods, the rankings on a single ballot are added into a round-robin grid using the standard 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.

For example, the single example ballot above,

X2 > X4 >> X1 > X3

the following votes would be added into the pairwise array:

	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 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 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

Approval Ties

During the initial ranking of candidates, two candidates may have the same approval score.

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 total first- and second-place vote
If equal, in descending order of total first-, second- and third-place votes
If equal, in descending order of ranks above last place
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 (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 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 Democratic Fair Choice.