[EM] Weighted voting systems for proportional representation

[Kristofer Munsterhjelm]

Kathy Dopp wrote:
> The system you describe *is* still precinct summable in the sense of
> reporting the sums for each possible slate of candidates for each
> precinct or polling location - this is at least a whole lot fewer sums
> than the number of possible ballot choice permutations including
> partially filled out ballots that IRV/STV would require to be reported
> and sampled to be precinct summable (reporting all individual ballots'
> choices would be less to report in most cases).
> 
> To be summable, this system would require reporting (N choose S) sums
> where N is the number of total candidates in the contest and S is the
> number of seats being elected.  This is a lot of sums - but could, I
> imagine, be mathematically sampled and audited to limit the risk of
> certifying the wrong slate much more easily than IRV methods could be
> - but I'm not certain about that until I have the time to think about
> it more (not any time soon).

Ah, yes. This leads me back to an older thought that perhaps the 
criterion of summability should be refined for multiwinner methods by 
turning it into two criteria. These criteria would be:

- Weak summability: If the number of seats is fixed, one can find the 
winner of the method according to precinct sums, where the amount of 
data required for these sums grows as a polynomial with respect to the 
number of candidates, and as a polylogarithmic function with respect to 
the number of voters.

- Strong summability: Same as weak, but without the number of seats 
being fixed or known in advance.

To my knowledge, Schulze STV is weakly summable, as is this method, 
because if you fix S, N choose S is bounded by a polynomial.

When people here talk about summability for multiwinner methods, they 
usually mean strong summability, though. This is like SNTV or party 
list. If you have the Plurality counts for SNTV, it doesn't matter how 
many seats you want, you can just read off the n first Plurality 
winners. Similarly, for party list, you can just run the Sainte-Laguë 
method n times for n seats with the same input data.

Do you think weak summability is sufficient to audit multiwinner methods?