[EM] proportional constraints - help needed

[Peter Zbornik]

Dear all,

We recently managed, after some effort to elect some people in our
party using STV (five of seven board members of the Czech Green Party
and more recently some people to lead the Prague organisation etc.).
We used standard fractional STV, with strict quotas, valid empty
ballots, Hagenbach-Bischoff quota, no Meek.
It was the first bigger usage of STV in the Czech republic.
As a footnote, I would like to add, that one big advantage of
proportional election methods, is that it elects "the best people",
i.e. meaning the people, who have the biggest support in the
organisation.

Now we would like to go on using STV for primary elections to party
lists in our party.
I have a good idea on how to do it using proportional ranking, but am
not entirely confident in how to implement the gender quotas.
So here I would like to ask you, the experts, for help.
I have only found some old papers in election-methods, but they are
not of any great help to resolve the following problem, unfortunately.

The problem (after a slight simplification) is as follows:
We want to elect five seats with any proportional ranking method (like
Schulze proportional ranking, or Otten's top-down or similar), using
the Hagenbach-Bischoff quota
(http://en.wikipedia.org/wiki/Hagenbach-Bischoff_quota) under the
following constraints:
Constraint 1: One of the first two seats has to go to a man and the
other seat has to go to a woman.
Constraint 2: One of seat three, four and five has to go to a man and
one of those seats has to go to a woman.
Say the "default" proportional ranking method elects women to all five
seats, and thus that we need to modify it in a good way in order to
satisfy the constraints.

Now the question is: How should the quoted seats be distributed in
order to insure
i] that the seats are quoted-in fairly proportionally between the
voters (i.e. the same voters do not get both quoted-in seats) and at
the same time
ii] that the proportional ranking method remains fairly proportional?

---

Here is how I have been thinking about the problem myself.
I am not sure, however, that my line of thinking is the best or the
only one, so please read with a healthy amount of scepticism.
The problem can be re-formulated as follows.
Which method would make sure,
1) that a large number of voters do not get both of the quoted seats?
2) that the quoted seat is by default seat two and five, unless there
are compelling reasons to quote-in seat three or four (or, less
probably, seat one)?

There is a trade-off between questions 1) and 2) above, i.e.:
a) if seat two and five are quoted, then a large number of the voters
might get both the quoted seats - which would lead the quotas to be
non-proportionally distributed, making some voters dissatisfied.
b) assume we always quote in seat two (this could, but need not be
necessary). If we, by using some appropriate proportionality measure
(has to be defined), quote-in the candidate at seat three, four or
five, then a fraction of one vote might decide, that the quoted-in
seat should be seat number three instead of seat five, or the rule
could "prefer" quoting in at place three, instead of place five, as
place three would need to have higher support, than place five. Such a
quota rule would ignore the fact, that place three is more important
than place five, i.e. that the disturbance in the proportionality of
the proportional ranking would be higher, if the candidate would be
quoted in at seat three than seat five.

I.e. we search for
a) a quota proportionality measure and
b) a proportional ranking measure and
c) a rule, which "optimises" both the "quota proportionality" and the
"proportional ranking proportionality".

I am sure the above was not entirely easy to digest.
I am happy to take your questions and will do my best to clarify.
Any references to relevant papers would be more than welcome.

Best regards
Peter Zborník