[EM] Proof idea that IRV can't be summable

[Kristofer Munsterhjelm]

On 03/12/2020 08.31, Richard Lung wrote:
> 
> Of course, all that learning won't alter what John Stuart Mill knew over
> 150 years ago that maiorocracy is not democracy. The fixation on single
> winner methods, (the monarchism hang-over) is candidate-centred not
> voter-centered or politician-based not people-based.

I have a hunch that anything that passes Droop proportionality must fail
strong summability: that it's impossible to find a Droop-proportional
multiwinner method where you can construct an array of numbers
polynomial in the number of candidates, and then later use that array to
find the outcome for any number of seats. The same approach could
possibly be used to prove this, although it would be a lot harder.

Proving multiwinner STV non-summable would not be too much harder than
what I did in the post you replied to. There's nothing there that has to
necessarily be single-winner. It's not clear whether the corresponding
regions should be of candidates, or councils, though. (E.g. do we
consider the region "A is on the council" as some analog of a win-or-tie
region for A, or should the region be "the two-seat council is AB"?)

If told to create something democratic without concern to current
constraints, I'd probably just skip right to sortition. This would not
just invalidate single-winner methods, but voting altogether; except,
possibly, the method the assembly itself uses to decide.

-km