# [EM] Associational Proportional Representation (APR) (Kristofer Munsterhjelm) 26

Forest Simmons fsimmons at pcc.edu
Sat Nov 1 14:22:30 PDT 2014

```Kristofer,

Condorcet when the issue space is not one dimensional:

Suppose that the population of a town is concentrated near the vertices of
an equilateral triangle with no single vertex having a majority of the
citizens in its vicinity, and that there are four proposed cites for a
community center X, Y, Z, and C, where  C is the center of the inscribed
circle, i.e. the intersection of the three angle bisectors, and  X, Y, and
Z are locations such that the respective sides of the triangle form
perpendicular bisectors of the segments CX, CY, and CZ.

Assuming that the voters like to be close to the community center, the
point C is the Condorcet preference.

But if we move X, Y, and Z to points X', Y', and Z' just a few yards closer
to C along the segments XC, YC, ands ZC, respectively, C suddenly becomes
the Condorcet Loser., even though it still has a significantly smaller
average distance to the three vertices of the triangle than any of X', Y',
or Z'.

In this example we can use the distances from the voters to the candidate
positions as the potential costs to the voters, and the opposites of these
costs as the utilities.

Note that the example still works if the triangle is not equilateral. But
in the case of an equilateral triangle, when a majoritarian tie between X',
Y', and Z' is broken at random, the expected cost to a voter is about 33
percent greater than if point C is chosen.

Btw, for the benefit of those who think that single winner methods can
always be replaced by multi-winner PR methods (and I know you're lurking
out there!) any good single winner lottery method would select C with
certainty!  Any multi-winner PR method for choosing representatives to
figure out a solution would just be a way of procrastinating the journey
towards a failed solution.

Forest

On Sat, Nov 1, 2014 at 1:16 AM, Kristofer Munsterhjelm <km_elmet at t-online.de
> wrote:

> On 10/31/2014 11:32 PM, Forest Simmons wrote:
>
>>
>> Kristofer wrote ...
>>
>> (huge skip)
>>
>>     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.
>>
>>
>> It seems to me that methods like APR that rely exclusively on ordinal
>> information (rankings) cannot detect "distances."  For that we need some
>> measure of intensity of preference like that provided by Approval and
>> other Score based methods.
>>
>
> I'm not sure, actually. Consider Condorcet on a 1D spatial model. By
> Black's single-peakedness theorem, the candidate closest to the median is
> the CW, and the median is the single point that minimizes the sum of
> distances to every other point along the line. Yet Condorcet has no idea of
> distance beyond what the ranked votes tell it.
>
> It might be the case that this can't be generalized to multiwinner
> methods. But it does at least show that it's not obviously impossible.
>
>
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