[EM] how to turn any single winner method into a proportional multiwinner method

[Ross Hyman]

Any single winner method that produces a societal ranking can be the basis for a proportional multi-winner method with a nice property.  

For now, let's call the method the Societal Ranked Proportional method.  (But maybe that is not a good name since the method does not produce a societal ranking it only uses it).  The method is inspired by DMC.

Here is the nice property
For any n+1 candidate election for n seats that includes any or all of the n-seat SRP winners amongst the n+1 candidates, all SRP winners that are lower ranked than the lowest ranked SRP looser amongst the n+1 candidates, are guaranteed to win.  
Corollary:
If all n SRP winners are 
in any n+1 candidate election with one SRP loser, then each of the n SRP winners are either all 
guaranteed to win or the one SRP winner that is displaced will be of higher societal ranking than the SRP looser that took its place.
Corollary:
If all n SRP winners are 
in any n+1 candidate election with one SRP loser that is higher ranked than all of the SRP winners, then all of the SRP winners will win.

The n seat SRP method:
You need a societal ranking, such as from Schulze beat path, and 
any proportional method, such as QPQ or STV, that can choose n winners from n+1 candidates 
using the same ballots that produced the societal ranking.  I recommend QPQ since you don't need to worry about adjusting 
quotas.

Candidates are either hopeful, elected or excluded.  All candidates are initially hopeful.  Do every possible n+1 candidate election for
n seats that includes all elected candidates (initially none), the lowest
ranked hopeful candidate, and any combination of other hopeful candidates.  If for any of these n+1 candidate elections the lowest ranked hopeful
candidate loses, set that candidate from hopeful to excluded.  Go on to the next lowest ranked hopeful
candidate.  If the lowest ranked hopeful
candidate is a winner of every n+1 candidate election that includes it and all
elected candidates, then set that candidate from hopeful to elected and go on
to the next hopeful candidate.  Continue until n candidates are elected or the total
number of elected plus hopeful candidates equals n.

The method has the virtue of requiring fewer n+1 elections than other methods that are inspired by Condorcet single winner methods such as Schulze STV. It requires only one pass through the societal rankings.  As soon as a candidate is declared excluded one does not need to calculate the results of any other n+1 elections in which it is a candidate.  And as soon as one candidate is declared elected, no further n+1 candidate elections need be performed.
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