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

[Markus Schulze]

Dear Peter Zbornik,

I wrote (6 May 2010):

> 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.

You wrote (6 May 2010):

> 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?

H[A(1),...,A(n-1),x,y] is a real number. My mail above is
supposed to be a definition for H[A(1),...,A(n-1),x,y].

There are n+1 partitions because there can also be
some voters who prefer candidate y to every candidate
in {A(1),...,A(n-1),x}. The voters in T(n+1) are those
who prefer candidate y to every candidate in
{A(1),...,A(n-1),x}.

Markus Schulze