[EM] Weighted voting systems for proportional representation

[Juho Laatu]

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?
> 
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