[EM] Proportional election method needed for the Czech Green party - Council elections

[Peter Zbornik]

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 <
markus.schulze at alumni.tu-berlin.de> 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
>
>
> ----
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>
>
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