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

Kathy Dopp kathy.dopp at gmail.com
Sun Oct 5 18:26:42 PDT 2014

```On Sun, Oct 5, 2014 at 3:58 PM, Toby Pereira <tdp201b at yahoo.co.uk> wrote:

>
> The example wasn't about three or four seats. It was about four seats both
> times and who was more deserving of the fourth seat out of A and B with or
> without C's inclusion. (It doesn't matter that you might argue that C is
> more deserving than either when present because we can still order
> non-winning allocations in terms of proportionality.)

Well, then you were still arguing for a non-proportionate allocation
of seats in the instance including group C then, because group C wins
a seat if the allocation is proportionate, causing the example to be
about allocating 3 seats if C is included and 4 otherwise. I was, in
both cases, comparing the proportionate allocation of seats to the
proportionate allocation of seats in both scenarios.

>
> OK - I should have added that it is specifically with factions 1 and 2
> having the same voting patterns.

I believe you DID say that last time. An example would help.

>
> For example:
>
> 2 to elect
>
> Faction 1: 302
> Faction 2: 100
> Faction 3: 1
>
> The proportional allocations are:
>
> Faction 1: 1.499
> Faction 2: 0.496
> Faction 3: 0.005
>
> Faction 1 would win both seats. However, if we have:
>
> Faction 1: 302
> Faction 2: 100
> Faction 3: 3
>
> The proportional allocations are:
>
> Faction 1: 1.491
> Faction 2: 0.494
> Faction 3: 0.015
>
> This would make it one seat all between factions 1 and 2. This is why I
> would argue that there are better definitions of proportionality than your
> method.
>

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
Town of Colonie, NY 12304
"A little patience, and we shall see ... the people, recovering their
true sight, restore their government to its true principles." Thomas
Jefferson

Fundamentals of Verifiable Elections
http://kathydopp.com/wordpress/?p=174

View my working papers on my SSRN:
http://ssrn.com/author=1451051
```