[EM] Weighted voting systems for proportional representation

[Jameson Quinn]

I agree with Juho's argument that all methods are "summable" these days.
However, I'd suggest a few vaguely-defined related criteria. The general one
is "auditable" - is it possible to gain confidence that the result is
correct through some process besides a full recount? This breaks down into
"human-auditable" (can you gain confidence through some simple arithmetic
process which doesn't require a computer? Such a process can involve
summary-statistics.); and "sampling-auditable" (can you gain an arbitrary
degree of confidence in the result by recounting a random sample of the
votes? Are the sample sizes required "reasonable"?).

STV does pretty poorly on these criteria. Since a tie at any stage can have
extreme effects on later stages; and electing A,B,C is likely to lead to
different results later on than electing C,B,A; the system is "unstable".
The number of possible ties at all stages is order N³, the cube of the
number of candidates. And the voters divide into N camps at each stage.
Thus, in general, if there are fewer than about N⁴ voters, you can expect to
come arbitrarily close to a tie at some stage, which would mean that you
have to recount all the votes to be confident of the result.

Party list systems do the best on all these criteria, because they are
traditionally summable.

Things like Schulze-STV, AT-TV, and SODA-PR fall somewhere in the middle. If
summary statistics of votes are published, there are some easy "sanity
checks" which would catch the clumsier frauds in the counting process. And a
sampled recount would be enough to confirm the validity of these summary
statistics. Some forms of fraud would still take a full recount to detect;
but in general, such frauds would be logistically difficult.

I'm doing a lot of hand-waving in that last paragraph, though. The fact is,
to really be fully auditable in both senses, you need a system which is
summable in the traditional, low-order-polynomial-of-candidates, sense.

JQ

2011/7/24 Juho Laatu <juho4880 at yahoo.co.uk>

> One approach to summability and auditing is to say that the target is to
> allow the district to count the votes and later check that at the top level
> their votes (or the votes of all districts) were counted correctly, AND to
> allow the top level to check that the districts will report their results
> correctly (they may e.g. want the districts to tell their final results
> before they hear what the results of the other districts were). In addition
> to these also the general public and independent auditors will benefit of
> having the results in a format that allows them to recount and check the
> results and that allows them to audit all the districts (and subdistricts,
> and top level) independently one by one. The key point is thus to provide
> transparency and easy and local checks in all directions.
>
> If this is what we want, then an interesting new feature is the new
> information society and its technical capabilities. That makes almost all
> methods "summable" in the above mentioned sense. Let's say we use STV. With
> current technology it is not a big problem to require all poll stations to
> record all their ballots in some digital format. And it is easy to check, if
> needed, that the physical ballots actually correspond to the reported
> results. It is also easy to check at the top level that the results are
> correct since the digital space taken by few million STV ballots is not that
> big, and the time that computers take to check the results is not that big.
>
> One may thus say that current digital storage, transport and computation
> capabilities can make almost any method "summable" in the discussed way.
>
> But there are still some problems left. One could say that the summability
> criterion also includes a requirement of being able to sum up the votes so
> that the summed up votes will hide details of the individual votes. This may
> be needed to provide privacy and to avoid unwanted phenomena like coercion
> and vote selling. In this sense the above mentioned treatment of the STV
> votes may not be sufficient. The number of candidates may be high enough and
> the length of the ballots high enough to allow identification of individual
> ballots. If we want to guarantee also privacy some additional tricks are
> needed in this case.
>
> My point was anyway that it isa also possible to divide this summability /
> auditing / privacy requirement in two parts so that it will consist of the
> 1) easy verifiability and 2) privacy parts. New information technology may
> redefine the rules to some extent. Verification is easier, but on the other
> hand privacy may be more problematic since it is now easier to share all the
> information (partially intentionally to guarantee better verifiability). The
> tricky part is actually the privacy part. We may nowadays have e.g. interest
> to break the individual ballots in smaller and more numerous parts instead
> of trying to sum them up in smaller space.
>
> Juho
>
>
>
> On 24.7.2011, at 11.53, Kristofer Munsterhjelm wrote:
>
> > 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?
> >
> > ----
> > Election-Methods mailing list - see http://electorama.com/em for list
> info
>
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20110724/efd96e7c/attachment-0004.htm>