[EM] Does it satisfy Droop proportionality, monotonicity, and clone independence?

[Ross Hyman]

In December I proposed a proportional
multi-winner method  that satisfies a Condorcet property.  The method is constructed from a single winner societal ranking method, such asSchulze beat path or Tideman ranked pairs, and a conventional proportional multi-winner method used to choose n winners from n+1 candidates, such as Woodhall QPQ or STV.
 
 
However I have not proved that the method shares
the desirable properties of the societal ranking and proportional methods that
make it up.  
 
If the proportional method that makes it up satisfies Droop proportionality,
which is the case for QPQ and STV, is that also the case for the combined
method?
 
If the societal ranking that makes it up is monotonic and clone independent,
which is the case for Schulze beat path and Tideman ranked pairs, then does the
combined method satisfy multi-winner generalizations of those single winner
properties?
 If it can be shown that the method satisfies Droop proportionality, monotonicity, and clone independence, then I think it would be preferable to conventional QPQ or STV.  Otherwise not.  

The Method 
Consider the set of all n+1 candidate elections for n seats.   For
the lowest ranked candidate in the societal ranking, if it is the loser of any
of the elections in which it is a candidate, then remove all elections in which
it is a candidate.  If instead, the
lowest ranked candidate wins every election in which it is a candidate, then
remove all election in which it is not a candidate.  Continue this process for the next lowest
ranked candidate in the societal ranking, etc, until all of the remaining
elections have the same set of n winners.  These are the n winners of the method.



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