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?
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