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

[Markus Schulze]

Dear Peter Zbornik,

in the scientific literature, candidates, who
have not yet been elected, are sometimes called
"hopeful".

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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.

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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