[EM] Associational Proportional Representation (APR)

Kristofer Munsterhjelm km_elmet at t-online.de
Sun Oct 26 07:15:42 PDT 2014

On 10/26/2014 06:01 AM, Richard Fobes wrote:
> I'm responding (via Bcc) to Steve Bosworth's earlier reply to my
> responses, which he repeated in a direct message that is copied below. I
> no longer have a copy of the forum message, so please pardon the
> creation of a new thread about a conversation in progress.  For context,
> see below.

I've also written a few replies to Steve. Since you've crossposted to 
this list, and since what I found out might be of interest to EM in 
general, I suppose I can recap.

As far as I understood, the election aspect of APR for n seats involves 
using IRV-style elimination until n candidates remain; these n 
candidates are then elected and given weight proportional to the number 
of (remaining) first preference votes they have gathered.

I replied by saying this is very similar to plurality elimination party 
list, only that instead of allocating seats to parties with the number 
of seats being fixed, we're allocating weights to winners with the 
number of winners being fixed.

Thus, there are three sorts of potential issues with the method: those 
that it shares with plurality elimination party list, those that it 
doesn't, and those that arise from using weighting to begin with.

In my replies, I've brought most of my attention to the first category. 
Basically, weighting can't perfectly compensate for errors in which 
winners are picked. Consider, for instance, if there's only one seat, 
and only right-wing or left-wing candidates around. The position will 
either be left-biased or right-biased, and weighting can't help with it 
because there's nothing to form an equilibrium with. Similarly, if there 
are more factions than there are seats, some of them will fail to be 
represented unless the method finds proper compromises.

IRV is protected from the greatest failures here (you can't easily pack 
the council with left-wingers at lower weight to keep third factions 
out, since IRV is cloneproof), but it can still suffer from center 
squeeze. In a one-seat situation (e.g. the good old Left-Center-Right), 
IRV can fail to find the center. I suggested BTR-IRV (and STV-ME in 
general) as a simple fix.

Steve replied by saying that it seems like a problem that only directly 
appears in one-seat elections. I agreed, but in my most recent reply, I 
wrote that the problem can appear in similar guises in multiwinner methods.

Consider, for instance, a system with polarized candidates: in a 1D 
spatial model, most of them hang around either 0 to 0.4 or 0.6 to 1.0.
Then (assuming for the simplicity of the model) a uniform distribution 
of voters, in an ordinary multiwinner election, one would prefer half of 
the candidates to be left-wing and half of the candidates to be right; 
but I also think one'd prefer there to be centrism with respect to the 
wings themselves. That is, the center of the left-wing winners as a 
group should be around 1/3 and the center of the right-wing winners as a 
group should be around 2/3. But IRV can fail this if more extreme 
"loud-spoken" candidates get more of the left/right wing votes.

Similarly, for picking a number of candidates in a weighted voting 
system, the IRV-based method might pick the loud candidates on the 
respective wings; and if it retains both loud ones and moderates, it 
might give the moderates lesser weight because the weighting is based on 
first preference votes.

In a two-seat case, say that an extremist is elected from both sides 
(e.g. at 0.1 and 0.9 respectively). Then the weighting will ensure (and 
this can probably be proven by some analog of mutual majority) that the 
right-wing winner is reasonably balanced with respect to the left-wing 
winner: his weight will be the ratio of right-wingers to leftists. But 
people near 0.5 on the axis won't feel particularly represented, and the 
method would have done better if it had picked winners at 1/3 and 2/3 


This raises the question of where the optimal winners for weighted PR 
should be placed. In particular, in an 1D spatial model (left-right 
axis), it seems fair that the respective winners get a weight equal to 
the proportion of voters that are closer to them than to anybody else. 
But now we're much more free to place the winners anywhere on the axis 
because the relative weight will sort itself out by the definition above 
(unlike ordinary unweighted multiwinner elections).

The reasoning that we prefer moderates (but not too moderate ones) to 
extremists to minimize tension could be codified like this: minimize the 
sum of distances from voters to their representative. In the one-seat 
case, that reduces to finding the candidate closest to the median voter, 
like Condorcet does. But this is odd, because that's what you'd expect 
out of an ordinary (unweighted) multiwinner method as well! So if both 
the criteria I mentioned are fair, then weighting doesn't provide much 
to a multiwinner method if it already is very good; instead, it mostly 
helps salvage multiwinner methods that aren't.

But having the same criteria for candidate placement for unweighted and 
weighted multiwinner is odd in another way as well. It would mean that 
the property I mentioned in an earlier post of mine is desirable - that 
if you have a party list election with lots of seats and a fixed 
constraint n on the number of winning parties, the same parties should 
be elected as if you were to run an unweighted multiwinner n-seat 
election. These are interesting questions to think about.

Some unweighted multiwinner methods would be relatively easily 
retrofitted to handle weighted PR elections. Clustering methods like 
Monroe's would work easily, for instance: just give each winner a weight 
equal to the size of his cluster, and possibly remove the constraint on 
cluster size equality.

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