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

[Peter Zbornik]

Dear Markus Schulze, dear readers,

The example below is intriguing. But I am afraid I fail to understand
this formulation of Schulze's proportional ranking.
I would be grateful if M. Schulze or someone else, could give an
example, which could help me get it.
Specifically, I didn't understand what H[A(1),...,A(n-1),x,y] is. Is
it a function, H[A(1),...,A(n-1),x,y]=min(cardinality of T(i),
0<=i<=n+1 plus other criteria)?, I didn't get the properties of
T(n+1). Why are there n+1 partitions of the electorate and not only n?
Are hopefuls x. y two members of the set of all hopefuls? I guess yes.
Some reference to the definitions in the paper could be useful.
Thank you for you kind help.

Best regards
Peter Zbornik

2010/5/6, Markus Schulze <markus.schulze at alumni.tu-berlin.de>:
> 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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