[EM] General PR question (from Andy Jennings in 2011)

[Toby Pereira]

From: Kathy Dopp <kathy.dopp at gmail.com>

 >OK Toby.  You're right re. the *remainder method*.>Thus, the remainder method is *NOT* equivalent to my method of
>minimizing the sum because my method selects 1 candidate each for
>Factions 1 and 2 for *both* scenarios you mention.>Thank you for discovering this difference between the remainder method
>and my method of minimizing:>Sum(v_i/v *Absolute(v_i/v - s_i/s))>After looking at this example, I believe an improvement to my method
>would be to minimize the sum:>Sum(v_i *Absolute(v_i/v - s_i/s))>so that we do not have to hang on to quite so many decimal places to
>see which set of winning candidates minimizes the sum and is, thus,
>the most proportionate set of winning candidates.>My formula, thus, gives:
>100.5012 for the following allocation>302  1 seat
>100  1 seat
>1    0 seat
>100.5037  for the following allocation (higher, thus *not* the most
>proportionate)>302  2 seat
>100  0 seat
>1    0 seat
>99.52593  for the following allocation>302  1 seat
>100  1 seat
>3      0 seat
>101.5185  for the following allocation (higher, thus *not* the most
>proportionate)>302  2 seat
>100  0 seat
>3      0 seat
>Your example was helpful in showing that my method works, whereas the
>remainder method does not always work, and in prompting me to multiply
>the formula times the constant total number of voters to make it
>slightly easier to use.>My method of minimizing my formula will ALWAYS select the most
>proportionately fair set of winning candidates for any approval voting
>election, whether or not there is candidate overlapping support among
>groups or not (so for both party list systems and for general approval voting).
>Thanks for trying to shoot holes in my method and, thus, help to show
>how consistently it works and help me find ways to improve it.-- >Kathy Dopp
I haven't checked your numbers so I assume you're correct, but in the first example, you can see that it's much closer (100.5012 v 100.5037) than in the second (99.52593 v 101.5185), so it suggests it's possible to contrive an example where the result would swap. So:
2 to elect
303: 2 seats100: 0 seats1: 0 seats
This gives 100.505
303: 1 seat100: 1 seat1: 0 seats
This gives 101, so giving the larger faction both seats is more proportional.
And then
303: 2 seats100: 0 seats3: 0 seats
This gives 101.502
303: 1 seat100: 1 seat3: 0 seats
This gives 100.002, so in this case the two largest factions receive one seat each. Also, I would suggest that the larger faction should win both seats in both cases because I would consider 300 voters to 100 to be the exact point where there would be a tie between the 2/0 and 1/1 allocations. 303 to 100 would therefore shift it in favour of the larger faction.
But also, I'm still unclear how to translate it into approval voting not along party lines. The (v_i/v - s_i/s) figure for a voting group will work towards giving the same proportion of seats as the proportion that group is of the whole voting population. But when you have
Faction 1: A, B, C, D, EFaction 2: A, B, C, D, F
It doesn't make sense to use that figure. Assuming the groups are of equal size, your formula would suggest that they should only have half the candidates each, whereas in reality if there are say four to elect, they could all get 100% (A, B, C, D).
Toby
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