[EM] a Borda-Condorcet relation

[Kristofer Munsterhjelm]

Stephen Turner wrote:
> So to go from median to mean you just keep adding one datum above
> and one below.  Thanks for that.
> 
> Your idea about a "gradual Black's" method is interesting
> for a different reason.  Black's method actually has some
> mometum behind it at the moment, as it is advocated by
> Dasgupta and Maskin.  I couldn't find the original paper online,
> but below is a link to their article in Scientific American. 
> Their justification is what they call the "majority dominance
> theorem" which says that Black's method is a counter-example
> to Arrow's theorem, and is the only such method, all this provided
> that certain not-too-onerous profile restrictions are satisfied.
> 
> http://www.scientificamerican.com/article.cfm?id=ranking-candidates-more-accurate
> 
> (What they call "majority rule" is Condorcet, and what they propose is Black. The
> article is verging on non-technical.)

They don't seem to be saying that Black is the only counter-example. 
What they're saying, at least in the Scientific American article, is 
that "whenever a method does good, majority rule (Condorcet) does, too". 
  They then suggest Black as a particularly simple modification of 
Condorcet to handle the case where there's a cycle, as well.

I think that if complexity is a problem, I'd prefer Ranked Pairs (or 
River) because it is reasonably simple and also, unlike Black, cloneproof.

> I wonder how "gradual Black" would fit in.

The gradual Black method wouldn't be simple. I'm unsure of what criteria 
it would meet, though.

If complexity were not an issue, the Smith set version could be better 
than "gradual Black". Take the Smith set with respect to f(*, *, 0). 
Intersect it with the "Smith" set with respect to f(*, *, 1), and so on 
until either you're all the way to Borda or only one candidate remains. 
It would be very slow, but interesting.