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.

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.
  • Let Z be the least-approved candidate who strongly defeats Y. Note that if Y has a clear upward defeat over X, Z must have greater approval than X.
  • Approval(X) - Approval(Y) < Approval(Z) - Approval(X)
• 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 "buoyancy" is smaller than the "ballast" 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 buoyancy from defeating a candidate Z who defeats Y isn't large enough to overcome the ballast 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.

If desired, the secondary defeat strength used to measure buoyancy,

Approval(X) - Approval(Y) < Approval(Z) - Approval(X),

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 scores highlighted:

		against
		Abby	Brad	Cora	Dave	Erin
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:

		against
		Erin	Abby	Cora	Brad	Dave
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'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 {4,4} 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 entry {4,4}, 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 candidate encountered.
• Looking down the 2nd column from Abby's approval score in {2,2}, the only loss is to Brad, an already eliminated marginal loser.
• Abby is the least-approved member of the strong set and wins the election.

What if the secondary defeat strength were calculated by a different measure?

If winning votes are used, wv(Dave>Brad)=609 > wv(Brad>Abby)=463, so Brad is still marginally defeated by Dave and Abby still wins.

Strong preference votes are those that span the approval cutoff. So sp(Dave>Brad)=213 > sp(Brad>Abby)=98. Brad is still marginally defeated by Dave and Abby still wins.