[EM] Multiwinner Condorcet generalization on 1D politics

[Dan Bishop]

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.
>
> So let's consider a case one may use to justify Condorcet, or to 
> classify Condorcetian methods: if politics is one dimensional, and
> people prefer candidates closer to them on the line, then there will 
> be a Condorcet winner, and the CW is the candidate closest to the 
> median voter, and (if we think electing the CW is a good idea), we 
> should elect the candidate closest to the median voter.
>
> This case, or general heuristic, seems to be simple to generalize, and 
> one may do so in this manner: call a position the nth k-ile position 
> if n/k of the voters are closer to 0. Then, a multiwinner method that 
> elects k winners should, if politics is one-dimensional, pick the 
> candidate closest to the first (k+1)-ile position, then closest to the 
> second (k+1)-ile position (first candidate notwithstanding as he's 
> already elected), etc, up to k.
>
> To be more concrete, in a 2-candidate election, the first candidate 
> should be the one closest to the point where 33% of the voters are 
> below (closer to zero than) this candidate, and the second candidate 
> should be the one closest to the point where 67% of the voters are 
> below this candidate, the first candidate notwithstanding. 
This is exactly what STV-CLE does.

However, I would choose a goal of minimizing the "mean minimum political 
distance" (expected average distance between a voter and the nearest 
winning candidate). On a uniform linear spectrum, the minimum is 1/(4k), 
which occurs by electing the set {(n-1/2)/k for n=1 to k}. This reduces 
to {1/2} when k=1, so is a generalization of the Condorcet Criterion. 
However, it can also be applied to a multidimensional political spectrum.