[EM] Biproportional representation (was Re: Preferential voting system where a candidate may win multiple seats)

[Kristofer Munsterhjelm]

On 07/21/2013 01:42 PM, Vidar Wahlberg wrote:

> The biproportional apportionment system Kristoffer linked to is very
> interesting. It is slightly more complex and I fear it may be too
> complex for common people to understand (which will make it difficult to
> gain support for it), and I wonder if it may end up with exceptionally
> long calculation time when there are many districts and many parties.

I get the impression convergence time is fairly rapid, but I haven't 
actually tried it.

> Especially in Norway where amount of seats in a district may be
> radically different from the amount of votes cast in that district.
> Finnmark is a such example, which got a low population and few voters,
> but a large area, giving them relatively many seats (calculated by
> population and area) compared to amount of votes. As I've understood the
> algorithm so far it'll calculate how many seats each party wins in a
> district purely based on percentage of the votes cast there, then later
> adjusted up or down to match the real amount of seats that should be won
> in the district. Due to the low amount of votes in this district it's
> likely that only about 1-2 seats will initially be won there, meaning
> you'll have to weight up the votes in Finnmark and weight the votes down
> in another district.

To make the biproportional voting method work, the numbers should be set 
so that the rows give the correct allocation per party and the columns 
give the correct allocation per district (or vice versa, depending on 
what you assign to the rows and columns). So I think the simplest way to 
do so would be to apply the weighting to the vote numbers themselves to 
get "effective votes".

Suppose that in the 2013 election, 123000 voters voted in Østfold and 
54000 voters voted in Finnmark. Then each of the Østfold votes would 
correspond to 287727.942/123000 normalized votes, and each of the 
Finnmark votes would correspond to 162042.638/54000 normalized votes. 
Here 287727.942 is equal to the 2013 (population + area * 1.8) value for 
Østfold, and 162042.638 is the value for Finnmark.

You can see this easily enough by considering the extreme case. Consider 
what would happen if only a single voter bothered to vote in Finnmark. 
Prior to any biproportional adjustment, he would dictate what party gets 
every one of Finnmark's seats. In other words, he'd control all of 
Finnmark's weight of 162042.638, and so his vote would be worth 
162042.638/1 normalized votes. If there were two voters, each would 
control half, or 162042.638/2 normalized votes, and so on.

You can also consider this an initial reweighting to make the numbers 
work out correctly. The unweighted votes would not give the right number 
of seats per region, so one would in any case adjust the votes until 
they do give the right number of seats per region. Using the county 
"population + area * 1.8" weights directly is just a quick way of doing so.

The Norwegian case would be a little more tricky than that, though. The 
district proportions are calculated using ordinary Sainte-Laguë, but the 
party proportions are calculated using modified Sainte-Laguë. I'd 
probably just use ordinary Sainte-Laguë for both, but if that's not 
acceptable, it means the alternating process would have to use 
Sainte-Laguë along one axis and modified Sainte-Laguë along the axis, 
and so may in certain situations have trouble converging.

-

And if you want to implement the weighting, the full weighting table for 
2013 and 2017 is:

Østfold			287727.94
Akershus		575251.49
Oslo			624783.25
Hedmark			243038.30
Oppland			232595.30
Buskerud		295842.37
Vestfold		242751.85
Telemark		198440.60
Aust-Agder		129256.69
Vest-Agder		189450.48
Rogaland		469035.64
Hordaland		525927.05
Sogn og Fjordane	142220.91
Møre og Romsdal		286608.61
Sør-Trøndelag		336695.35
Nord-Trøndelag		174789.51
Nordland		308839.56
Troms			206983.44
Finnmark		162042.64

I have a spreadsheet showing the actual calculation of these numbers. 
Let me know by private mail if you want it.