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

Kristofer Munsterhjelm km_elmet at t-online.de
Sat Oct 4 23:58:34 PDT 2014

```On 10/05/2014 08:02 AM, Toby Pereira wrote:
> From: Kathy Dopp <kathy.dopp at gmail.com <mailto:kathy.dopp at gmail.com>>
>
>  >>  What I'm saying is
>  >> that in your system, adding C changes which out of A or B is more
> deserving
>  >> of the final seat, which seems wrong to me.
>
>  >First, adding more voters (group C) means that the denominator of the
>  >ratio (proportion) for each (every) voting group changes.
>  >How could I change the denominator of the quantity v_i/v (the
>  >proportion of each voting group)  without changing the proportion of
>  >seats that each group should have?
>
> Well, it changes the overall proportion they should have, but it does
> not change the 5:3 correct ratio of A to B seats. I would argue that if
>
> Faction 1: w seats
> Faction 2: x seats
> Faction 3: y seats
>
> is more proportional than
>
> Faction 1: w+1 seats
> Faction 2: x-1 seats
> Faction 3: y seats
>
> then with the same voting patterns
>
> Faction 1: w seats
> Faction 2: x seats
> Faction 3: z seats
>
> is more proportional than
>
> Faction 1: w+1 seats
> Faction 2: x-1 seats
> Faction 3: z seats
>
> for any x, y, w, z. Sainte-Laguë, D'Hondt and my system fit this criterion.

That sounds like house monotonicity. I don't know if my example is
translatable to Approval voting, but this left-right-center example
seems to show that it's not always desirable for ranked voting. E.g.
something like:

43: L>C>R
34: R>C>L
12: C>R>L

If you have one seat, then C is a compromise candidate, doesn't give too
much bias to either the left or right wing, and thus provides
proportionality. But in the two-seat case, balance comes from electing L
and R; LC would bias to the left and RC would bias to the right.

For this case, let L "really" be L1, L2, L3, ..., Ln; and the same with
C and R. That means the ballots are really L1 > L2 > L3 > ... > Ln > C1
> C2 > ..., etc.

So in the one-seat case:

L: 0 seats
C: 1 seat
R: 0 seats

(w = 0, x = 1) is preferable to

L: 1 seat       w + 1
C: 0 seats      x - 1
R: 0 seats

but in the 2-seat case:

L: 0 seats      w = 0
C: 1 seat       x = 1
R: 1 seat

is not preferable to

L: 1 seat       w + 1
C: 0 seats      x - 1
R: 1 seat.

You could argue that these are "entangled" factions (each group of
voters ranks all the candidates, not just those he supports). But then
you might have to clarify it to say that it doesn't hold in the case of
entangled factions.
```