[EM] STV method based on previously introduced clone-proof Condorcet single winner method

[Ross Hyman]

Here is an alternative that I think satisfies the generalized multiwinner Condorcet criterion.  

Scores are calculated for each candidate set via:
S(Alpha) = Sum_Beta (V(Alpha,B)-V(Beta,A))V(Beta, not Beta) where the sum is over all candidate sets Beta that differ from Alpha by just one candidate.  A is the candidate in Alpha that is not in Beta and B is the candidate in Beta that is not in Alpha. V(Alpha,B) and V(Beta, not Beta) are defined as previously.
The candidate set with the lowest score is eliminated.  Its score is not calculated again and that character set is removed from the sum when recalculating the scores of the other candidate sets.  If a candidate set is eliminated that is the last one to contain a certain candidate, then that candidate is eliminated.  The V(Beta, not Beta)'s are recalculated with that character eliminated from all ballots.  Repeat until only one candidate set remains.


  


--- On Thu, 4/4/13, Ross Hyman <rahyman at sbcglobal.net> wrote:

> From: Ross Hyman <rahyman at sbcglobal.net>
> Subject: Re: STV method based on previously introduced clone-proof Condorcet single winner method
> To: election-methods at electorama.com
> Date: Thursday, April 4, 2013, 8:24 PM
> If there is a candidate set Alpha
> such that V(Alpha,B) > V(Alpha+B-A,A) for all A in Alpha
> and all B not in Alpha, then I would consider the method to
> be a Condorcet method for multiple seats if the candidates
> in Alpha are elected.  The method satisfies this in the
> single winner limit and when the number of candidates is
> just one more than the number of winners.  What about
> the general case of N+K candidates and N winners?
> 
> --- On Thu, 4/4/13, Ross Hyman <rahyman at sbcglobal.net>
> wrote:
> 
> > From: Ross Hyman <rahyman at sbcglobal.net>
> > Subject: STV method based on previously introduced
> clone-proof Condorcet single winner method
> > To: election-methods at electorama.com
> > Date: Thursday, April 4, 2013, 6:48 PM
> > Here is a tentative proposal for an
> > STV method based on an extension to multiple seats of
> the
> > clone-proof Condorcet method I introduced earlier. 
> It
> > is based on several ideas from Schulze STV.  
> > 
> > Start with a variant way of defining the previous
> single
> > winner method that is exactly equivalent if each ballot
> is
> > complete:
> > V(B,A) is the number of ballots that rank B above A.
> > V(B) is the number of ballots that rank B at the top.
> > S(A) = Sum_B V(B,A) V(B) is the score for candidate A.
> > Eliminate the candidate with highest score.
> Recalculate
> > V(B)’s.  Repeat until one candidate remains. 
> > The method is clone proof and for complete ballots it
> will
> > elect a Condorcet winner if there is one.
> > 
> > This method is generalizable to proportional elections
> for N
> > winners by replacing the sum over candidates, B, with a
> sum
> > over all of the N-candidate sets, Beta:
> > First, ballots are proportionally completed like in
> Schulze
> > STV.
> > Scores are calculated for each candidate via:
> > S(A) = Sum_Beta V(Beta,A) V(Beta,not Beta).
> > Eliminate the candidate with highest score. 
> > Recalculate V(Beta, not Beta).  (The V(Beta,A)'s do
> not
> > need to be recalculated).  Repeat until N candidates
> > remain.  
> > 
> > V(Beta,A) is defined as in Schulze STV and is
> determined by
> > a proportional approval voting method where candidates
> in
> > Beta that are ranked above A on a ballot are
> approved. 
> > V(Beta, not Beta) is determined similarly where
> candidates
> > in Beta that are ranked above all candidates not in
> Beta are
> > approved.  
> > 
> > Here is an alternative scaling method for calculating
> > V(Beta,A) (which I introduced in an earlier post for
> > proportional approval voting):
> > 1.    Each ballot approves those candidates
> > from candidate set Beta that are ranked higher than
> > candidate A.
> > 2.    V_i,C is the vote from ballot i for
> > candidate C. Initially the vote value of each ballot
> is
> > distributed equally amongst each of its approved
> candidates.
> > 
> > 3.     The total vote for each
> > candidate C, V_C = sum_i V_i,C is calculated. 
> > V_Smallest is the vote total of the candidate with the
> > smallest vote total.  
> > 4.    V_i,C is reduced to V_i,C
> > V_smallest/V_C.
> > 5.     The reduced total vote on
> > each ballot i, W_i = sum_C V_i,C is calculated.
> > 6.    V_I,C is increased to V_i,C V_i/W_i.
> > 7.    Go to step 3.
> > The count is repeated until V_smallest has
> converged. 
> > V(Beta,A) is the converged value of V_smallest.
> > V(Beta, not Beta) is calculated similarly, except that
> each
> > ballot approves those candidates from candidate set
> Beta
> > that are ranked higher than every candidate not in
> Beta.
> > 
> > 
> > 
> > 
> > 
> > 
> >
>