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

[Ross Hyman]

oops.  Messed up the nice property.  Here is the corrected version.
An n-seat SRP winner that is a candidate in an n+1 candidate election for n seats will be one of the winners provided that every lower ranked n-seat SRP winner is also a candidate in the election.

The corollaries remain unchanged.
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.


One way of thinking about the procedure is the following.  Imagine the set of all n+1 candidate elections for n seats.  For the lowest ranked candidate in the societal ranking, if it is the looser 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.

As mentioned in the previous post, in practice one does not really need to calculate the results of every election since the procedure removes many of them before one would need to calculate their results.
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