[EM] Schulze ("Approval-Domination prioritised Margins")

[Chris Benham]

I have an idea for a new defeat-strength measure for the Schulze algorithm
(and  similar such as Ranked Pairs and River), which I'll call:

"Approval-Domination prioritised Margins":

*Voters rank from the top however many candidates they wish.
Interpreting ranking (in any position, or alternatively above at least one other
candidate) as approval, candidate A is considered as "approval dominating"
candidate B if  A's approval-opposition to B (i.e. A's approval score on ballots
that don't approve B) is greater than B's total approval score.

All pairwise defeats/victories where the victor "approval dominates" the loser
are considered as stronger than all the others.

With that sole modification, we use Margins  as the measure of  defeat strength.*

This aims to meet  SMD  (and so Plurality and Minimal Defense, criteria failed
by regular Margins) and my recently suggested "Smith- Comprehensive 3-slot
Ratings Winner" criterion (failed by Winning Votes).

http://lists.electorama.com/pipermail/election-methods-electorama.com/2008-December/023595.html

Here is an example where the result differs from regular Margins, Winning Votes
and  Schwartz//Approval.

44: A
46: B>C
07: C>A
03: C

A>B  51-46 =  5 * 
B>C  46-10 = 36 
C>A  56-44 = 12

Plain Margins would consider B's defeat to be the weakest and elect B, but that is the only
one of the three pairwise results where the victor "approval-dominates" the loser.  A's approval
opposition to B is 51, higher than B's total approval score of 46.

So instead my suggested alternative considers A's defeat (with the next smallest margin) to be
the weakest  and elects A.  Looking at it from the point of view of the Ranked Pairs algorithm
(MinMax, Schulze, Ranked Pairs, River are all equivalent with three candidates), the A>B result
is considered strongest  and so "locked", followed by the B>C result (with the greatest margin)
to give the final order A>B>C.

Winning Votes  considers C's defeat to be weakest and so elects C.  Schwartz//Approval also
elects C.

Margins election of  B is a failure of  Minimal Defense. Maybe the B supporters are Burying
against A and A is the sincere Condorcet winner.

I have a second suggestion for measuring "defeat strengths" which I think is equivalent to
Schwartz//Approval, and that is simply "Loser's Approval" (interpreting ranking as approval as
above, defeats where the loser's total approval score is higher are considered to be weaker than
those where the loser's total approval score is lower).

Some may see this as more elegant than Schwartz//Approval, and maybe in some more complicated
example it can give a different result.


Chris Benham


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