[EM] Completion methods for Smith Sets

[Forest Simmons]

On Sun, 17 Jun 2001, Michael Rouse wrote in part:

> 
> Actually, the method by itself can elect a fixed number of candidates, 
> which is why it would make a good completion rule for Condorcet winning 
> groups. Say you might have 5 positions to fill. You find the Smith set with 
> three candidates and the next "winning" Condorcet set (where no candidate 
> from outside the set can win against a candidate inside the set) of six 
> candidates. The Smith set candidates should fill the first three spots, and 
> you can use the method above on those remaining in the winning set to find 
> out who should fill the final two spots. As a side note, I think the 
> combined standard/inverse method for "completing" Condorcet groups (you 
> might check my other note for a description of both) would yield a method 
> that is generally fair and well-behaved most of the time, without too many 
> illogical or strange results.
> 

Three winner election example:

51% A > B > C > D > E > F
49% F > E > D > C > B > A

If I understand your multiwinner method, it selects A, B, and C as the
three winners.

That's great for those who take "majority rule" to the extreme.

But most of us are more interested in Proportional Representation via
multiwinner elections.

Currently STV seems to be the chief contender for PR based on ranked
preference ballots, but Demorep has posted cryptic messages that look
promising for a method based on pairwise counts, except for the
computational complexity.

Currently I'm working on a method that also suffers from computational
complexity, but for which a sequential version is tractable.

After I finish grading my finals and make up to my family for neglecting
them in the process, I'll post my progress.

Meanwhile, I hope others keep coming up with new promising ideas on all
fronts, like Craig's list PR and this crosscut idea for Condorcet
completion.

Forest