[EM] Multiwinner Condorcet generalization on 1D politics

[Dan Bishop]

Kristofer Munsterhjelm wrote:
> 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.
>
STV-CLE just happens to work the best when the political spectrum is 
one-dimensional: Candidates are eliminated at the ends of the spectrum 
until someone has a quota, and the process continues until candidates 
are neatly spaced a quota apart.

But with multiple dimensions, the CLs' votes get split among multiple 
candidates, so you have to eliminate more candidates until someone meets 
quota.  This creates a much stronger centrist bias than the 
1-dimensional case.

>> 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}.
>
This is true.  But it doesn't mean that MMPD minimization is a bad 
criterion, just that we need to consider other criteria as well.  Just 
like the Condorcet Criterion: It's possible for a method to comply while 
being non-monotonic.  That doesn't mean the Condorcet Criterion is 
defective, just that we should prefer that a method should meet both 
Condorcet AND Monotonicity.