[EM] Multiwinner Condorcet generalization on 1D politics

[Kristofer Munsterhjelm]

Dan Bishop wrote:
> Kristofer Munsterhjelm wrote:
>> 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.

James-Green Armytage devised some examples where STV-CLE seems to work 
poorly. Are those problems with STV-CLE in particular, or with my 
"percentile voter" generalization? I can't say for sure because his 
examples don't fit into a 1D space.

> 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.

The problem with measuring closest distance is that the closest 
candidates can cloak disproportionality further out. Consider something 
like this:

  0| .x. A ... B ... C ... D ... E .y. |1

Half the electorate is at position x, closer to A than to any other 
candidate, and the other half is at position y, closer to E than to any 
other candidate. Say you're going to elect four candidates. Obviously, A 
and E should be in the outcome. To be fair, the next ones should be D 
and B, but you can pick any without changing the mean minimum political 
distance. For instance, you could be biased towards E and pick {A, D, E}.