Marginal Ranked Approval Voting

Marginal Ranked Approval Voting (MRAV) is a refinement of Definite Majority Choice. It will choose the same winner most of the time, but will eliminate the DMC winner under certain circumstances. It satisfies all the other criteria satisfied by DMC and is implemented in exactly the same manner. The only difference is how the final vote tally is interpreted. MRAV finds the same winner and is equivalent to Approval Sorted Margins, and hence is a symmetric method.

Definitions

 * Strong defeat: Pairwise defeat by higher-approved candidate
 * Strong losers: Set of all strongly defeated candidates
 * Provisional set: Set of non-strongly-defeated candidates
 * Each provisional winner defeats all higher-approved members of the set. This is Forest's "P" set.  Convenient that Provisional starts with P, isn't it? ;-)
 * Clear upward defeat: Y has a clear upward defeat over X when lower-approved candidate Y pairwise defeats higher-approved candidate X and also pairwise defeats every other candidate with lower approval than X and higher approval than Y.
 * Marginal defeat: Pairwise defeat of provisional candidate X by strong loser Y under these conditions:
 * Y has a clear upward defeat over X.
 * If Y is strongly defeated by a candidate with lower approval than X, X is not marginally defeated by Y.
 * There is a candidate Z with greater approval than X who strongly defeats Y.
 * If no such Z exists, X is not marginally defeated by Y.
 * Approval(Y) - Approval(X) > Approval(X) - Approval(Z)
 * This means that Y's approval-margin defeat strength over X is more positive than X's over Z.
 * Note that both sides of the inequality are negative.
 * The inequality can be arranged to Approval(X) < (Approval(Z) + Approval(Y))/2; that is, X's approval is less than the average approval of Y and Z.
 * Another rearrangement of the inequality is:
 * Approval(X) - Approval(Y) < Approval(Z) - Approval(X)
 * Which means that X's approval is closer to the approval of the lower-approved candidate Y (who has a clear upward defeat of X) than to the higher approved candidate Z (who defeats Y).
 * Marginal losers: Set of all marginally defeated candidates
 * Strong set: set of candidates neither strongly nor marginally defeated

Procedure
The least-approved member of the strong set defeats all higher-approved candidates (whether in the strong set or not) and wins the election.

The philosophical motivation for removing marginally defeated candidates from consideration is that their approval "lift" is smaller than the "drag" of lower-ranked candidates who defeat them, and so they are dragged down.

The MRAV winner will differ from the DMC winner only when the DMC winner is marginally defeated. This can occur only when
 * There is a cyclic ambiguity in the pairwise preferences
 * The DMC winner is defeated by a strongly defeated candidate Y
 * The DMC's lift from defeating a candidate Z who defeats Y isn't large enough to overcome the drag of Y's clear upward defeat of X.

The Approval winner and the highest-approved member of the Smith set are always members of the strong set.

The secondary defeat strength used to measure lift is the approval margin:
 * Approval(Y) - Approval(X) > Approval(X) - Approval(Z)

Approval margin could be replaced by other metrics. For example, winning votes,
 * wv(X>Y) > wv(Z>X),

or Approval-Weighted Pairwise's "strong preference":
 * sp(X>Y) > sp(Z>X)

Example
Here's a set of preferences taken from Rob LeGrand's online voting calculator:

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:

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

After reordering the pairwise matrix, it looks like this:

To find the winner,
 * We start at the lower right diagonal entry, and start moving upward and leftward along the diagonal.
 * We're looking for a candidate who has a solid row of victories to the left of the diagonal.
 * Brad is the first such candidate encountered. Under DMC, Brad would be the winner.
 * Starting at Brad's approval score in the Brad>Brad cell, we start looking down the 4th column to see if any lower-approved candidates defeat Brad.
 * We see a defeating score of 609 in the Dave>Brad cell.
 * Dave has a clear upward defeat over Brad, since there are no other candidates with approval scores between Brad and Dave.
 * From the Dave>Brad cell, we move left along Dave's row until we find a losing score against a candidate with higher approval than Brad.
 * The least-approved candidate who defeats Dave is Abby.
 * Approval(Brad) - Approval(Dave) = 99. Approval(Abby) - Approval(Brad) = 235.
 * Brad is therefore marginally defeated by Dave and is a marginal loser.
 * Starting at Brad's diagonal Brad>Brad approval score, we continue up the diagonal, looking for another candidate with a solid row of victories to the left of the diagonal.
 * Abby is the next such candidate encountered.
 * Looking down the 2nd column from Abby's approval score in Abby>Abby, the only loss is to Brad, an already-eliminated marginal loser. Note that Abby's Abby>Dave>Brad beatpath to Brad is stronger than Brad's Brad>Abby beatpath, using approval margins as the defeat strength.
 * By quick inspection, we see that the only remaining strong candidate is Erin.
 * The strong set contains candidates Erin and Abby.
 * Abby is the least-approved strong candidate and defeats Erin directly. So Abby is the MRAV winner.

Marginal defeat alternatives
What if the secondary defeat strength used in the marginal defeat calculation were calculated by a different measure?

In the example above, if winning votes are used, wv(Dave>Brad)=609 > wv(Brad>Abby)=463. Brad is still marginally defeated by Dave and Abby still wins.

We find strong preference votes by totaling only those winning votes that cross the approval cutoff. So sp(Dave>Brad)=213 > sp(Brad>Abby)=98. Brad is still marginally defeated by Dave and Abby still wins.