[EM] Multiwinner Condorcet generalization on 1D politics
Dan Bishop
danbishop04 at gmail.com
Sat Feb 14 20:48:47 PST 2009
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.
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