[EM] Feature extraction and criteria for multiwinner elections

[Kristofer Munsterhjelm]

Juho Laatu wrote:
> --- On Fri, 2/1/09, Kristofer Munsterhjelm <km-elmet at broadpark.no> wrote:
> 
>> Reverse Condorcet: If the election is (n-1, n) and
>> there's a Condorcet loser, all but the Condorcet loser
>> should be elected.
> 
> Example:
> - 10 Republican candidates, one Democrat candidate
> - 55% support to Republicans
> - 45% support to Democrats
> - 10 candidates will be elected
> - The Democrat candidate is a Condorcet loser
> => Should D not be elected?

To simplify,

55: R1 > R2 > D
45: D > R1 > R2

and the election is (2, 3). The integer Droop quota is 34. STV does 
this: R1 is elected, then D is elected. That seems fair, but we can't 
get around the fact that D is the Condorcet loser (R* beats D 55-to-45).

So yes, D should be elected, and thus Reverse Condorcet isn't desirable. 
Actually, since R1 and D are both supported by a Droop quota, that means 
that Reverse Condorcet is incompatible with the DPC. So much for that.

Does this also mean that it's possible to find a (2,3) "overlap" 
Yee-like diagram where, even for the ideal method (whatever it is), the 
shape between two candidates is not Voronoiesque filled with the 
composite color of the two (e.g purple for the space between red and 
blue candidates)? I'm not quite sure.

A (seemingly) reasonable generalization of the Euclidean distance 
Voronoi would be this: For each point, find the two candidate points so 
that the sum of the distances to those two points are minimized. Color p 
according to the composite color of the k closest candidates (for a 
(k,n) election). But doesn't that correspond to the election method 
where you elect the CW, then remove him and elect the next CW and 
continue like that until done? That method is not PR.

Maybe it would be easier to visualize for the 1D Left-Center-Right case. 
With a sum-of-points metric, we would get something like

L = Left (red)
C = center (green)
R = right (blue)

-1.5             0             +1.5
L                C                R
|----------------|----------------|
     L+C (yellow)     C+R (cyan)

L covers from -1.5 to -0.5 (length 1)
C covers from -0.5 to +0.5 (length 1)
R covers from +0.5 to +1.5 (length 1)

If we're dealing with a Gaussian distribution, this may be the optimum. 
Say the Gaussian is centered at 0.3. If the method elects L, it can be 
improved (support more people) by electing C or R instead. That is, if 
the diagram means that
all who are placed left of -0.5 vote L > C > R
all who are placed right of 0.5 vote R > C > L
the rest vote C > L > R or C > R > L depending on who they're closer to,

then the outcome might be right, because most voters vote C > something.

So how does that give with our more radical Left-Center-Right example?

46: Left > Center > Right
46: Right > Center > Left
  8: Center > Left > Right

which should elect Center in a single-winner election, but Left and 
Right in a multiwinner one?

That is not a single Gaussian, so it's outside of the scope of the 
extended Yee diagram. It's also outside the scope of the single-winner 
Yee diagram, but one could reason in favor of Center for the 
single-winner election as such: Consider the case where L is elected 
(WLOG). That excludes the R Gaussian and also the center part of the L 
Gaussian. However, if the method elects C, it covers the tails of both L 
and R Gaussians, and since 8 voted C > L > R, at least one of those 
tails is pretty large. Hence, one should elect C.

-

Ultimately, my error in proposing Reverse Condorcet might have been that 
I generalized the multiwinner Yee diagrams too far. At least I know how, 
now. If my analysis is correct, one shouldn't be able to find a 
non-Voronoiesque (2,3) extended Yee diagram, but that's simply because 
they don't cover all situations (because of the voter distribution 
assumptions they make).