[EM] Multiwinner Condorcet generalization on 1D politics

[Kristofer Munsterhjelm]

Dave Ketchum wrote:
> On Thu, 12 Feb 2009 20:31:56 +0100 Kristofer Munsterhjelm wrote:
>> I think that one problem with devising a multiwinner method is that we 
>> don't quite know what it should do. PAV type optimization methods try 
>> to fix this, but my simulations don't give them very favorable scores.
>>
>> If we are to construct a multiwinner method that degrades gracefully, 
>> we probably need to have an idea of what, exactly, it should do, 
>> beyond just satisfying Droop proportionality (for instance). The 
>> problem with building a method primarily to satisfy a certain 
>> criterion is that if the criterion is broken slightly, then the 
>> criterion does not tell us how the method should work; and therefore, 
>> we might get "discontinuous" methods where the method elects a certain 
>> set if a Droop quota supports it, but a completely different group if 
>> the Droop quota less one supports that set.
> 
> Let me try a different picture:
>      Candidates scattered across a 2-dimension space, unevenly.  There 
> could be bunches near Party 1 position, near Party 2 position, etc.
>      Voters scattered likewise.
> 
> Each voter would give top rank to the candidate they see as nearest, 
> etc., not bothering to rank those too distant.
> 
> Looking for CW in the N*N matrix:  Pick any candidate as A and see if A 
> was ranked higher than x for all A vs x:
>      More A>x than x>A for each x - if so, A is CW.
>      Has the x in x>A been an A in this search - if so, have a cycle.
>      Try again with this x becoming A.
> 
> Note that this does not require looking at every entry in the matrix - 
> move on as soon as disqualifying an A.
> 
> Seems to me 2-dimensional works better for such as three parties pulling 
> in different directions.
>      With a CW, extra winners should have looked good when looked at via 
> the matrix.

Finding a CW is relatively easy if you have the Condorcet matrix. What's 
the issue space equivalent? A beats B if more voters are close to A (and 
thus vote A before they vote B) than the other way around. Thus 
Condorcet gets around the "cloaking" problem by removing the candidates 
that aren't relevant to the pairwise comparison.

> Note that for multiwinner a 3-member cycle would almost certainly  fit 
> for 3 winners.

What do you mean by that - that if A > B > C > A, and there's a (3,n) 
election, then A, B, and C should win? There may be multiple cycles.

This brings to mind an idea for a criterion: that if we're dealing with 
a (k, n) election, and k candidates have a Droop quota strength beatpath 
to all others, those k should be elected. It seems sensible at first, 
but you can have something like:

Two centrists, nobody's first preference, then two candidates, each of 
which is supported by a Droop quota each, then two candidates one short 
of being supported by a Droop quota. The PR principle for an election of 
four would probably each wing's two candidates. This also is a good 
example of a discontinuity: a method that is of the form "elect as 
Condorcetian a candidate as possible, subject to Droop quota limits" 
would elect the centrists and one from each wing.

-

What PR really is, I think, is a synthesis problem. Each candidate 
supports a given position to some extent which may range from full 
support to no support at all. So does each voter. The ideal condition is 
a direct democracy, where everybody participates, so that the support 
for a given position is equal to the sum of the support by each voter. 
The synthesis or vector quantization problem is to find the set of 
candidates whose sum on opinion space is as similar to that of the 
people as possible, by some similarity metric like the Sainte-Lague index.

This is made more difficult because we don't have that data. The only 
thing we know about candidates' support of issues is the voters' ranked 
ballot comparisons where, discounting manipulation like mass media 
campaigns, a voter presumably rank those who support issues similar to 
what he supports, above those where that's not the case.

Thus a consistent PR method must somehow infer the nature of support so 
that we get close to the ideal. The k-ile (percentile) consideration is 
one attempt at doing so, or at finding something that is not just a PR 
criterion for a specific condition, but a metric for many conditions.

Note also that the definition of PR-as-VQ above suggests that if you 
have a true centrist, who would himself act like the people would in a 
direct democracy, then that centrist could be superior to a great number 
of right- and left-wingers (for instance). However, council sizes are 
usually fixed, so the centrist may cause an imbalance unless the other 
candidates were similarly perfect. As an example, in a two out of three, 
(2,3) election with a centrist, left-winger and right-winger, even if 
the centrist alone was perfect, the election of him in addition to 
either the left- or right-winger would bias the council unfairly to the 
left or right, respectively.

I've also not considered the competition component, which is, basically, 
that a council of some left and some right may have the left- and 
right-wingers keep each other from doing something untoward, whereas a 
council populated entirely by centrists may deal more leniently with a 
corrupt member since "he's one of us". Of course, there can also be too 
much competition, which can lead to civil wars in the very extreme case.