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

[Juho]

Based on my best understanding of the requirements here is an exact  
but partial definition of the ideal method, with the assumption that  
all votes are sincere.

V1 = set of ranked ballots that indicate who would be the best  
president / vice presidents
V2 = set of ranked ballots that indicate who would be the best council  
members

1) If there is a Condorcet winner in V1, that candidate will be the  
president (P)
2) If there is no Condorcet winner in V1, then ... will be the  
president (P)

3) Elect the first vice president (VP1) so that the pair P+VP1 is as  
proportional as possible based on V1
4) Elect other possible vice presidents one by one so that at each  
round the set of P+VP1+...+VPn is as proportional as possible based on  
V1

5) Elect the remaining council members (all at one round) so that set  
of council members (that includes P+VP1+...+VPn) is as proportional as  
possible based on V2

6) The method must guarantee that the council will have at least the  
agreed minimum number of both male and female representatives. The  
resulting distortion should be minimized.

- Maybe "as proportional as possible" is clear enough so that I don't  
need to define it here :-)
- Minimal distortion caused by the male/female rule is a more vague  
concept. It could mean minimal changes in the most important seats or  
in all the seats in average. The required forced selections could be  
pushed to the last seats or spread to all of them.
- If this is a correct reflection of the requirements then this  
definition hopefully helps in discussing the properties of different  
candidate methods or method categories
- I used the assumption that the method uses ranked ballots, but that  
should not exclude other approaches (=> differences to be described)
- Note that V1 could be chosen to be the same as V2 but that means a  
minor deviation from the ideal method
- Note also that this definition means that the proportionality of the  
full council is not perfect since the P+VPs are elected using a  
proportional ranking based method
- I assumed sincere votes. Possible strategic concerns (free riding,  
burying,...) might lead to using some other method than the one that  
gives optimal results with sincere votes. This doesn't seem to be a  
strong trend however.

The only detailed proposal so far is the one that Markus Schulze  
proposed in http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/026087.html 
. I'll comment it shortly in the light of the definition above.
- In the proposed method V1=V2
- It is Condorcet compliant (1)
- In (2) it follows the logic on the (single-winner) Schulze method  
(good or bad)
- It is quite good in (3) and (4) (but see also the male/female rule  
below)
- It does not strictly follow (5) since it uses a proportional ranking  
based approach for the whole council (the results may not be radically  
different though)
- In (6) the distortion is not minimal. The method could e.g. change  
the third candidate to opposite sex needlessly (the whole council  
could contain sufficient number of both sexes also without that change).

- There are also many other proportional ranking based methods or  
variants of this proposal that would meet the criteria the same way or  
better. One could e.g. improve the male/female algorithm, or use some  
other Condorcet method than the (single-winner) Shulze method below  
the proportional ranking part.
- Another direction would be to use different approaches in the P+VPs  
election (that according to the requirements above should pretty much  
follow the "proportional ranking style") and in the "rest of the  
council" election. The proportional ranking only approach is simpler  
but is that a good enough reason to allow the minor distortion in  
proportionality?

Juho