[EM] electing a variable number of seats

[Kristofer Munsterhjelm]

Charlie DeTar wrote:
> Howdy,
> 
> I'm on the board of a small non-profit, and have been tasked with
> revising the portion of the bylaws that defines how to elect the board
> of directors.  Having had some exposure to better election methods
> through a colleague, I'm interested in exploring how we might use a
> ranked voting system effectively.  Most of the methods I've seen,
> however, are intended for electing a single winner -- and for the board
> of directors, we have multiple seats.  Additionally, the number of seats
> is variable.
> 
> I'm looking for methods that would more or less "optimally" (by variable
> definitions of optimal) elect a variable number of people.  "Single
> Transferable Vote" seems to be the most talked-about multi-winner ranked
> system; but the vote transfer process requires a pre-defined number of
> seats to fill.  It seems like the option to have a variable number of
> seats opens up possibilities for improving representation by adding a
> winner, or eliminating polarizing candidates by removing one.
> 
> Thoughts?

As far as I can see, there are two ways you might accomplish sufficient 
representation. The first is to have a vote about how large the board 
should be, and the second is to somehow do it in an algorithmic manner.

Let's take the first way first. It's not possible for the voters to know 
the composition of the board in advance. Thus either the method has to 
be iterated, like your old majority rule system, or the voters have to 
be provided the results for all possible board sizes. Since methods like 
STV are complex, I would suggest the latter.

A system of this form might go as follows: First the voters rank all the 
candidates. Then the system calculates the board composition for all 
sizes from 5 to 9. Finally, there is a supermajority vote for which size 
board to pick. The point of a supermajority is that if the board is 
supposed to be representative, a simple majority will not be enough 
since a majority might force a board made up only of their own 
representatives to the detriment of the minority representation.

If one uses a proportional ranking method like the one Schulze talked 
about, this might be further simplified. After the first ballot, the 
list is calculated and shown. Then one starts with the top 5 entries 
included (shown in another color) and holds a "include the next member?" 
vote. If a supermajority says no, the process is over; if it says yes, 
one includes the next member and repeats until either all 9 entries have 
been included or a later "no" stops the process. If the margin is narrow 
(say 51-49) so there's no supermajority in either direction, I'm not 
sure what to do, though.

-

The second way is more difficult because inferring proportionality or 
the kind of optimality you mention is not a simple task. I don't know of 
any formal methods to do this, but you could automate the proceed/stop 
iteration above.

One possible way of doing that would be to consider the currently 
included board's topmost ranked candidate on each ballot. If one of the 
candidates on the current size board appears in the first voter's top 
rank, then that ballot is associated with the value 1; if second rank, 
2; and so on. You could then set a threshold for the mean value of this 
across all ballots and stop the process when the mean is less than the 
threshold. The threshold may have to be adjusted for the number of 
candidates running, though.

Raph has given another idea using the effective number of political 
parties. See his post for that. The idea would not use the iteration 
procedure I have mentioned; instead, it would determine the number of 
seats after examining the ballots. As such, one could use ordinary STV 
with little problem.