Dear Markus Schulze, I think got the idea of the Schulze proportional method after your definition and Raph Frank's explanation and example.
I am however not sure that the Schulze proportional method "satisfies the proportionality criterion for the top-down approach to create party lists". You wrote (6.5.2010): a. Suppose x and y are the only hopeful candidates. Suppose N is the number of voters. Suppose Droop proportionality for n seats requires that x must be elected and that y must not be elected, then we get H[A(1),...,A(n-1),x,y] > N/(n+1) and H[A(1),...,A(n-1),y,x] < N/(n+1), and, therefore, H[A(1),...,A(n-1),x,y] > H[A(1),...,A(n-1),y,x]. This guarantees that the Schulze proportional ranking method satisfies the proportionality criterion for the top-down approach to create party lists. If I have understood you correctly, you only show "that the Schulze proportional ranking method satisfies the proportionality criterion for the top-down approach to create party lists" for the special case where there are only two hopefuls x and y. If I am correct, then it would be helpful if you could provide a full proof, or further explanation, which shows that "the proportionality criterion for the top-down approach to create party lists" is satisfied for any number of hopefuls. Best regards Peter Zborník On Thu, May 6, 2010 at 1:51 PM, Markus Schulze < [email protected]> wrote: > Dear Peter Zbornik, > > in the scientific literature, candidates, who > have not yet been elected, are sometimes called > "hopeful". > > *************************** > > The Schulze proportional ranking method can be > described as follows: > > Suppose place 1 to (n-1) have already been > filled. Suppose A(i) (with i = 1,...,(n-1)) > is the candidate of place i. > > Suppose we want to fill the n-th place. > > Suppose x,y are two hopeful candidates. Then > H[A(1),...,A(n-1),x,y] is the largest possible > value such that the electorate can be divided > into n+1 disjoint parts T(1),...,T(n+1) such that > > 1. For all i := 1,...,n: |T(i)| >= H[A(1),...,A(n-1),x,y]. > 2. For all i := 1,...,(n-1): Every voter in T(i) > prefers candidate A(i) to candidate y. > 3. Every voter in T(n) prefers candidate x > to candidate y. > > Apply the Schulze single-winner election method > to the matrix d[x,y] := H[A(1),...,A(n-1),x,y]. > The winner gets the n-th place. > > *************************** > > The best way to understand the Schulze proportional > ranking method is to investigate the properties of > H[A(1),...,A(n-1),x,y]. For example: > > a. Suppose x and y are the only hopeful candidates. > Suppose N is the number of voters. > > Suppose Droop proportionality for n seats requires > that x must be elected and that y must not be > elected, then we get H[A(1),...,A(n-1),x,y] > N/(n+1) > and H[A(1),...,A(n-1),y,x] < N/(n+1), and, therefore, > H[A(1),...,A(n-1),x,y] > H[A(1),...,A(n-1),y,x]. > > This guarantees that the Schulze proportional > ranking method satisfies the proportionality > criterion for the top-down approach to create > party lists. > > b. Adding or removing another hopeful candidate z > does not change H[A(1),...,A(n-1),x,y]. > > c. H[A(1),...,A(n-1),x,y] is monotonic. That means: > > Ranking candidate x higher cannot decrease > H[A(1),...,A(n-1),x,y]. Ranking candidate x > lower cannot increase H[A(1),...,A(n-1),x,y]. > > Ranking candidate y higher cannot increase > H[A(1),...,A(n-1),x,y]. Ranking candidate y > lower cannot decrease H[A(1),...,A(n-1),x,y]. > > d. H[A(1),...,A(n-1),x,y] depends only on which > candidates of {A(1),...,A(n-1),x} the individual > voter prefers to candidate y, but it does not > depend on the order in which this voter prefers > these candidates to candidate y. > > This guarantees that my method is not needlessly > vulnerable to Hylland free riding. In my paper > (http://m-schulze.webhop.net/schulze2.pdf), I argue > that other STV methods are needlessly vulnerable to > Hylland free riding, because the result depends on > the order in which the individual voter prefers > strong winners. In my paper, I argue that voters, > who understand STV well, know that it is a useful > strategy to give candidates, who are certain of > election, an insincerely low ranking. I argue > that, therefore, the order in which the individual > voter prefers strong winners doesn't contain any > information about the opinion of this voter, but > only information about how clever this voter is in > identifying strong winners. Therefore, the result > should not depend on the order in which the > individual voter prefers strong winners. > > Markus Schulze > > > ---- > Election-Methods mailing list - see http://electorama.com/em for list info > >
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