[EM] Summability and proportional methods

[Richard Lung]

Every stage of an STV count sums the whereabouts of all the votes in 
their process of transfer. So it is not "scary" on that account. Gregory 
method is a standard statistical technique called weighting in 
arithmetic proportion. Statisticians are not thereby a scarified 
profession. Of course, traditional STV counting does have anomalies. And 
that is why I developed its generalisation in Binomial STV.
No need for the retrograde steps you mention.
from
Richard Lung


On 08/06/2017 17:33, Jameson Quinn wrote:
> Most proportional voting methods are not summable. Transfers, 
> reweightings, and otherwise; all of these tend to rely on following 
> each ballot through the process. This makes these methods scary for 
> election administrators.
>
> I know of 3 ways to get summability: partisan 
> categorization, delegation, and second moments. List-based methods 
> (including partially list-based ones like MMP) use partisan 
> categorization. GOLD voting 
> <http://wiki.electorama.com/wiki/Geographic_Open_List/Delegated_%28GOLD%29_voting> does 
> it by giving voters a choice between that and delegation. Asset voting 
> and variants use delegation.
>
> The other way to do it is with second moments. For instance, if voters 
> give an approval ballot of all candidates, you can record those 
> ballots in a matrix, where cell i,j records how often candidates i and 
> j are both approved on the same ballot. This matrix keeps all the 
> information about the two-way correlations between candidates, but it 
> loses most of the information about three-way correlations. For 
> instance, you can know that candidates A, B, and C each got 10 votes, 
> and that each pair of them was combined on 5 ballots, but you don't 
> know if that's 5 votes for each pair, or 5 votes for the group and 5 
> for each. Note that those two possibilities actually involve different 
> numbers of total votes --- 15 in the former, 20 in the latter. In 
> order to fix this, you can instead make separate matrices depending on 
> how many total approvals there are on each ballot --- a "matrix" for 
> all the ballots approving 1, one for all those approving 2, etc. Thus, 
> in essence, you get a 3D matrix instead of a 2D one.
>
> Once you have a matrix, you can essentially turn it back into a bunch 
> of ballots, and run whatever election method you prefer. The result 
> will be proportional insofar as the fake ballots correspond to the 
> real ballots. How much is that? Well, I can make some hand-wavy 
> arguments The basic insight of the Central Limit Theorem (CLT)  --- 
> that second moments tend to dominate third moments as the number of 
> items increases  --- would seem to be in our favor.
>
> I think this could be an interesting avenue of inquiry. But on the 
> other hand, the math involved will immediately make 99% of people's 
> eyes glaze over.
>
> If this is not possible, then the only 2 ways towards summability are 
> partisan categorization and delegation. GOLD uses both. For a 
> nonpartisan method, I don't think there's any way to be summable 
> without forcing people to delegate; and I think that forced delegation 
> is going to be a deal-breaker for some people.
>
> So I'm frustrated in trying to design a nonpartisan proportional 
> method that's as practical as GOLD and 3-2-1 are for their respective 
> use cases.
>
>
> ----
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-- 
Richard Lung.
http://www.voting.ukscientists.com
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