[EM] Fwd: Proportionality proof of Schulze proportional ranking

[Raph Frank]

On Thu, May 6, 2010 at 10:53 AM, Markus Schulze
<markus.schulze at alumni.tu-berlin.de> wrote:
>   Then the n-th seat goes to a candidate B
>   such that the set {A(1),...,A(n-1),B} satisfies
>   Droop proportionality for n seats.

Expanding this using the (k+1) solid coalitions definition.

If a group of voters representing a fraction, F, of the total votes,
V, ranks all the candidates in a set, G, ahead of all other
candidates, then at least

round_down( (n+1)*F/V )

of the candidates elected in the first n ranks must come from G (or
all of the candidates in G).

Note:   round_down(x) returns the largest integer less than x.

I think this rule might be to strong though.

Your rule just means that if there exists at least one candidate with
a quota, then you must elect one of them.  It doesn't require the
method to look ahead to later rounds.  Once a candidate is elected,
the candidate stays elected.

If there were votes

40: L>C>R
20: C>L>R
40: R>C>L

In round 1, the threshold is 0.5 of the vote

L,C represent a solid coalition with 0.6 of the votes, so they are
entitled to at least 1 seat.

L or C could be elected.

Schulze proportional rankings would elect C, as C is the condorcet winner.

In round 2, the threshold drops to 0.33 of the vote.

L is a coalition with 0.4 of the vote
R is a coalition with 0.4 of the vote

Electing only one of them violates Droop proportionality.

The "bottom-up approach", where you elect the full council and then
elect progressively smaller councils, would meet the criterion.
However, I agree that it is still inferior, as it isn't guaranteed to
place the condorcet winner in first place.