[EM] Preferential Party-List Proportional Representation (PPLPR)

Vidar Wahlberg canidae at exent.net
Sat Nov 1 05:58:34 PDT 2014

On Fri, Oct 31, 2014 at 11:13:33PM +0100, Kristofer Munsterhjelm wrote:
> From a very brief look at it (particularly since you gave the LCR
> example), it seems like your method passes house monotonicity. You
> end up with percentages which I assume you then run through
> something like Sainte-Laguë to get actual seat allocations.
> But in that case, the fork of LCR will mean that
> - either you don't elect C in the one-seat case
> o4
> - you elect C plus someone else in the two-seat case,
> neither of which is preferable.
> One-seat and two-seat elections are kinda extremes for multiwinner
> methods, but driving them to edge cases can reveal otherwise subtle
> problems more easily.

In elections with few winners this would indeed be a problem. However,
this is not a significant problem in elections with many winners, which
usually is common for party-list elections.
I can outline a complete election system. Let's use Norwegian parliament
election as that is familiar to us both:

169 seats to be allocated.
Have an election threshold, for example 4% vote support or require that
a party wins at least 5 seats.
Count votes using PPLPR, if a party doesn't meet the election threshold
then remove the party with least support and redo the vote counting.
Once all parties remaining meet the election threshold then you allocate
the 169 seats to the parties, using Sainte-Laguë, D'Hondt or another
method that allocates seats as closely to the percentage vote result as
For each county redo the election with the parties that won seats, this
is used for the final step which is:
Use the bi-proportional method (pukelsheim[1]) with the national result
and the county results to allocate seats in counties.

The election threshold is to avoid fragmentation (likely a larger
problem with a preferential system than plurality) and to a certain
degree cloning (as in parties splitting up, I mentioned this in my
previous post). Cloning is however a threat to the system, I'll go in
on details about this later.
Using bi-proportional seat allocation honors local wishes for
representation, while counting the end result nationally will diminish
the problem with too few seats. It will also prevent gerrymandering,
although luckily, this is not a problem in Norway.

> I'd also say that IMHO, if every voter ranks {A, B, C} next to each
> other (i.e. never rank some other party in between), and you replace
> this with some party X (e.g. if someone votes D > A > B > C > E,
> then the replaced ballots become D > X > E), then X should have the
> same support as {A, B, C} had in total. Both small-party and
> large-party bias is bias, after all. But making cloneproof methods
> can be pretty tricky, so if you'd consider trying to do that now to
> be premature optimization, I'd understand that :-)

I agree, it would be preferable if the system was cloneproof. I'm not
entirely sure if PPLPR in its current form can be modified to become
cloneproof while retaining its properties, but I'd love to be proven
wrong on that point.
However, a question here is: Does it have to be cloneproof?
I've mentioned two likely ways for parties to "game" the election.

"Extremist" parties (i.e. far left, far right, single-issue parties):
These parties tend to have strong support among their voters, while more
moderate voters who support other parties are less likely to have the
more "extreme" parties as a later preference.
Such parties could create new smaller parties whose sole purpose is to
weaken the later preference of voters, to prevent support from being
drained from the party. Using the Tennessee election as example:
42 M>N>C>K
26 N>C>K>M
15 C>K>N>M
17 K>C>N>M

Here [M]emphis is the "extremist" party, 42% of the voters prefer
Memphis while 58% would much rather have something else. With PPLPR the
result of this would be:
Party |  Result
C     |  23.31% 
K     |  17.03% 
M     |  29.63% 
N     |  30.03% 

If Memphis wants to win this election, all they have to do is create a
new "party" to enter the election. Let's add the neighbouring city
Jackson, and have one Memphis voter prefer that:
41 M>N>C>K
26 N>C>K>M
15 C>K>N>M
17 K>C>N>M
 1 J>M>N>C>K

Now, Memphis wins:
Party |  Result
C     |  20.90% 
J     |   0.62% 
K     |  16.91% 
M     |  33.08% 
N     |  28.49% 

For smaller "extremist" parties, this can to a larger degree be
prevented with an election threshold.
Not the best example as Memphis also was the biggest "party". The
important bit in this example is that Memphis ended up with a higher
percentage with the introduction of a new party, even though they lost a
first preference vote.

Large parties splitting up:
This is the greatest threat to the method as I currently see it.
Continuing with the Tennessee election example, let's say the large
party Memphis creates a new party Jackson, and instruct all their voters
to rank Jackson 2nd. Even though the other voters see through this and
rank Jackson last, the Memphis voters will still benefit greatly:
42 M>J>N>C>K
26 N>C>K>M>J
15 C>K>N>M>J
17 K>C>N>M>J

Party |  Result
C     |  19.83% 
J     |   5.50% 
K     |  16.77% 
M     |  34.01% 
N     |  23.89% 

Memphis voters move a significant portion of the votes away from the
other parties to Jackson, and at the same time the introduction of a new
party weakens later preferences and cause Memphis to keep more of its
votes. Instead of the 30.03% Memphis originally would get, they now have
34.01%, and when you include Jackson (a clone of Memphis) they end up
with 39.51%!

This cannot be prevented with an election threshold, as it would need to
be so high that it likely will remove sincere parties from the election.
A party that does a trick like this is likely going to be shunned upon,
not only by opposing parties and voters, but also by media and even
among their own voters. Those who previously ranked such a party shortly
after their first preference are likely to move this party further down
on the list. They also run the risk that the new party they formed will
deviate from their own goals and become a new, unique party.
Is this enough to discourage such strategy?
I would prefer a better counter against this strategy.

[1] http://www.math.uni-augsburg.de/stochastik/pukelsheim/2008f.pdf

Vidar Wahlberg

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