[EM] Multiwinner methods and summability

[Dave Ketchum]

On Fri, 06 Feb 2009 23:14:27 +0100 Kristofer Munsterhjelm wrote:
> This is a simple question: say that you have a multiwinner method that 
> reduces to Condorcet in the single-winner case, and that this single 
> winner case uses a Condorcet matrix, so that it's properly summable in 
> this case. Then say that the method can be counted in the districts 
> using an (n+1) dimension matrix for the multiwinner election where the 
> council is of size n.

Each candidate voted for requires a row and column in the N*N matrix.

A possible optimization is to note that all candidates not yet voted for 
have identical content for their entries, and thus let them share a common 
row and column - expanding total rows and columns used as needed as votes 
get counted.

However, it is number of candidates voted for that controls needed size of 
matrix - it is only after counting all the votes that deciding which 
candidates got enough votes to be interesting is possible.

Write-ins are a special case.  If few are expected, they might be treated 
as a single candidate.  Then count can decide:
      Few, as expected - ignore.
      Too many to ignore - recount the votes, as needed.

Votes originally counted in multiple precincts or districts?  The various 
N*N matrices need not have identical sets of candidates - what I write 
above should be enough clues as to needed work in merging counts.
> 
> Is this (hypothetical) method summable? The size of the matrix depends 
> (superpolynomially) upon the number of winners to elect, but not on the 
> number of voters or candidates.

See above.
> 
> (Whether or not it's summable, it probably wouldn't be practical in the 
> real world; with a parliament or council of 100 candidates, the matrix 
> would be 101-dimensional and thus far too unwieldy.)
> 
While such a matrix may be a bit unwieldy, expecting voters to usefully 
analyze 100 candidates should result in punishing those who claim such is 
tolerable for presenting to voters.

Asking voters to choose among five seems doable; choosing among ten seems 
like about the limit for being useful.
-- 
  davek at clarityconnect.com    people.clarityconnect.com/webpages3/davek
  Dave Ketchum   108 Halstead Ave, Owego, NY  13827-1708   607-687-5026
            Do to no one what you would not want done to you.
                  If you want peace, work for justice.