[EM] Margins Sorted Top-Ratings

[C.Benham]

(This is a re-send with an error corrected)

I think  Margins Sorted Top Ratings would be a good  relatively burial 
resistant public proposal.

* Voters rank from the top however many candidates they wish. 
Equal-ranking allowed.

Give each candidate a score according to the number of ballots on which 
they are ranked below
no others.

Line them up in that order, highest to lowest.

Check to see if all the candidates above bottom in this order pairwise 
beat the candidate immediately
below them.

If they do then elect the candidate highest in the order.

If not begin with the pair that is pairwise out of order by the lowest 
margin and swap them.
(if there is an exact tie in the size of the margin then swap the 
tied-margin pair lowest in the order).

Repeat until no pair of adjacent candidates are pairwise out of order 
and then elect the highest-ordered
candidate. *

This could also use ratings ballots.

This meets Condorcet, but can be at least be explained (if not sold) 
without reference to Condorcet or Smith.

It would be as monotonic as it is possible for a Condorcet method to be.

For the sake of simplicity (and elegance) it has some short-comings.  
When there is a top cycle, voters who
didn't top-rate (rank below no other candidates) any of the candidates 
in the Smith set are disadvantaged by comparison
those that did.  It would also fail Clone-Independence.

A much more complicated method idea I had (that would be the same thing 
with three candidates):

*Voters rank from the top however many candidates they wish. 
Equal-ranking allowed.

(1) Eliminate (drop from the ballots and henceforth ignore) all 
candidates not in the Smith set.

(2) Score the remaining candidates according to their minimum pairwise 
scores, with ballots that rank two candidates
equal-top contributing a whole vote to each of the two candidate's 
scores against each other. Otherwise ballots that
rank two candidates equal below top contribute zero to their pairwise 
scores against each other.

(A possible variation is that they contribute half a vote to each if 
they are ranked below top and above bottom.)

(3) Eliminate all candidates that don't have a "short" (one or two 
steps) beatpath to every candidate with a higher minimum
pairwise score.

(4).  Repeat step 2.  Then margins-sort the resulting scores and elect 
the highest-ordered candidate.*

This is trying to meet Clone Independence, Mono-raise, Chicken Dilemma, 
Non-Drastic Defense.

Chris Benham