<div>Dear Markus Schulze,</div>
<div> </div>
<div>I think got the idea of the Schulze proportional method after your definition and Raph Frank's explanation and example.</div>
<div> </div>
<div>I am however not sure that the Schulze proportional method "satisfies the proportionality criterion for the top-down approach to create party lists".</div>
<div> </div>
<div>You wrote (6.5.2010):</div>
<blockquote dir="ltr" style="MARGIN-RIGHT: 0px">
<p>a. Suppose x and y are the only hopeful candidates.</p>
<div> Suppose N is the number of voters.</div>
<div> </div>
<div> Suppose Droop proportionality for n seats requires</div>
<div> that x must be elected and that y must not be</div>
<div> elected, then we get H[A(1),...,A(n-1),x,y] > N/(n+1)</div>
<div> and H[A(1),...,A(n-1),y,x] < N/(n+1), and, therefore,</div>
<div> H[A(1),...,A(n-1),x,y] > H[A(1),...,A(n-1),y,x].</div>
<div> </div>
<div> This guarantees that the Schulze proportional</div>
<div> ranking method satisfies the proportionality</div>
<div> criterion for the top-down approach to create</div>
<div> party lists. </div></blockquote>
<div> </div>
<div>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.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>Best regards</div>
<div>Peter Zborník </div>
<div> </div>
<div> </div>
<div class="gmail_quote">On Thu, May 6, 2010 at 1:51 PM, Markus Schulze <span dir="ltr"><<a href="mailto:markus.schulze@alumni.tu-berlin.de">markus.schulze@alumni.tu-berlin.de</a>></span> wrote:
<div> </div>
<blockquote class="gmail_quote" style="PADDING-LEFT: 1ex; MARGIN: 0px 0px 0px 0.8ex; BORDER-LEFT: #ccc 1px solid">Dear Peter Zbornik,
<div> </div>
<div>in the scientific literature, candidates, who</div>
<div>have not yet been elected, are sometimes called</div>
<div>"hopeful".</div>
<div> </div>
<div>***************************</div>
<div> </div>
<div>The Schulze proportional ranking method can be</div>
<div>described as follows:</div>
<div> </div>
<div> Suppose place 1 to (n-1) have already been</div>
<div> filled. Suppose A(i) (with i = 1,...,(n-1))</div>
<div> is the candidate of place i.</div>
<div> </div>
<div> Suppose we want to fill the n-th place.</div>
<div> </div>
<div> Suppose x,y are two hopeful candidates. Then</div>
<div> H[A(1),...,A(n-1),x,y] is the largest possible</div>
<div> value such that the electorate can be divided</div>
<div> into n+1 disjoint parts T(1),...,T(n+1) such that</div>
<div> </div>
<div> 1. For all i := 1,...,n: |T(i)| >= H[A(1),...,A(n-1),x,y].</div>
<div> 2. For all i := 1,...,(n-1): Every voter in T(i)</div>
<div> prefers candidate A(i) to candidate y.</div>
<div> 3. Every voter in T(n) prefers candidate x</div>
<div> to candidate y.</div>
<div> </div>
<div> Apply the Schulze single-winner election method</div>
<div> to the matrix d[x,y] := H[A(1),...,A(n-1),x,y].</div>
<div> The winner gets the n-th place.</div>
<div> </div>
<div>***************************</div>
<div> </div>
<div>The best way to understand the Schulze proportional</div>
<div>ranking method is to investigate the properties of</div>
<div>H[A(1),...,A(n-1),x,y]. For example:</div>
<div> </div>
<div>a. Suppose x and y are the only hopeful candidates.</div>
<div> Suppose N is the number of voters.</div>
<div> </div>
<div> Suppose Droop proportionality for n seats requires</div>
<div> that x must be elected and that y must not be</div>
<div> elected, then we get H[A(1),...,A(n-1),x,y] > N/(n+1)</div>
<div> and H[A(1),...,A(n-1),y,x] < N/(n+1), and, therefore,</div>
<div> H[A(1),...,A(n-1),x,y] > H[A(1),...,A(n-1),y,x].</div>
<div> </div>
<div> This guarantees that the Schulze proportional</div>
<div> ranking method satisfies the proportionality</div>
<div> criterion for the top-down approach to create</div>
<div> party lists.</div>
<div> </div>
<div>b. Adding or removing another hopeful candidate z</div>
<div> does not change H[A(1),...,A(n-1),x,y].</div>
<div> </div>
<div>c. H[A(1),...,A(n-1),x,y] is monotonic. That means:</div>
<div> </div>
<div> Ranking candidate x higher cannot decrease</div>
<div> H[A(1),...,A(n-1),x,y]. Ranking candidate x</div>
<div> lower cannot increase H[A(1),...,A(n-1),x,y].</div>
<div> </div>
<div> Ranking candidate y higher cannot increase</div>
<div> H[A(1),...,A(n-1),x,y]. Ranking candidate y</div>
<div> lower cannot decrease H[A(1),...,A(n-1),x,y].</div>
<div> </div>
<div>d. H[A(1),...,A(n-1),x,y] depends only on which</div>
<div> candidates of {A(1),...,A(n-1),x} the individual</div>
<div> voter prefers to candidate y, but it does not</div>
<div> depend on the order in which this voter prefers</div>
<div> these candidates to candidate y.</div>
<div> </div>
<div> This guarantees that my method is not needlessly</div>
<div> vulnerable to Hylland free riding. In my paper</div>
<div> (<a href="http://m-schulze.webhop.net/schulze2.pdf" target="_blank">http://m-schulze.webhop.net/schulze2.pdf</a>), I argue</div>
<div> that other STV methods are needlessly vulnerable to</div>
<div> Hylland free riding, because the result depends on</div>
<div> the order in which the individual voter prefers</div>
<div> strong winners. In my paper, I argue that voters,</div>
<div> who understand STV well, know that it is a useful</div>
<div> strategy to give candidates, who are certain of</div>
<div> election, an insincerely low ranking. I argue</div>
<div> that, therefore, the order in which the individual</div>
<div> voter prefers strong winners doesn't contain any</div>
<div> information about the opinion of this voter, but</div>
<div> only information about how clever this voter is in</div>
<div> identifying strong winners. Therefore, the result</div>
<div> should not depend on the order in which the</div>
<div> individual voter prefers strong winners.</div>
<div> </div>
<div>
<div></div>
<div class="h5">
<div>Markus Schulze</div>
<div> </div>
<div> </div>
<div>----</div>
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<div> </div></div></div></blockquote></div>
<div> </div>