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

[Raph Frank]

2010/5/6 Peter Zbornik <pzbornik at gmail.com>:
> 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.

H is the number of members in the smallest partition.

When comparing candidate x to candidate y, there are n-1 candidates
who have already been elected.

candidate 1: called A(1)
candidate 2: called A(2)
...
candidate n-1: called A(n-1)

You must then split the voters up into n groups.

The voters in group 1 must prefer candidate 1 to candidate y
The voters in group 2 must prefer candidate 2 to candidate y
(and so on)
The voters in group n-1 must prefer candidate (n-1) to candidate y

Finally, the voters in group n must prefer candidate x to candidate y

Voters who prefer candidate y to all others cannot be placed in any group.

You then arrange the voters so that the smallest group has as many
members as possible.  There is no point in putting all the voters in
one group, as then the other groups will be smaller and it is only the
smallest group size that matters.

When you have done that, you look at size of the smallest group.  The
number of members in that group is taken as the votes for x is
preferred to y.  Assume that is 600 votes.

You then repeat the process but with x and y swapped.  The number of
members in the smallest group is taken as the votes for y is preferred
to x.  Assume that is 700 votes.

Under those assumptions, y beats x by 700 votes to 600.

You then repeat for every possible pair and elect the Schulze winner.