Definition of "Pairwise Method"

[Bruce Anderson]

On Apr 15,  5:41am, Bruce Anderson wrote:
> I picture calculating winners according to (the now defined) pairwise methods 
> by using the array:
> r(i,j) = p(i,j) + xq(i,j), 
> where either x = 0, or x = 1/2, or x = 1.  When it is important to distinguish 
> among them, I would specifically refer to the Condorcet(0), Condorcet(1/2), or 
> Condorcet(1) method.  I say that i is a Condorcet winner if it has the largest 
> minimum over j of r(i,j).  I use x = 1/2; but Mike (and now, hopefully, 
> everyone else on this list who DOES NOT explicitly DISTINGUISH among the 
> possibilities) uses x = 1.  I think there should be no problem here as long as 
> I keep it clear which is which whenever it makes a difference.
>-- End of excerpt from Bruce Anderson

All this is true.  In particular, Mike uses what I call x = 1 here as part of 
calculating the winners according to his definition of the Condorcet voting 
method.  However, Mike (and I presume others) do not JUST use what I call the 
Condorcet(1) voting method.  Instead, what Mike calls the Condorcet voting 
method is what I would call the Beats-all//Condorcet(1) voting method, where the 
Beats-all voting method is defined as follows.  If there is a candidate who 
pairwise beats every other candidate, then Beats-call chooses that candidate as 
its unique winner; otherwise (i.e., if no candidate pairwise beats every other 
candidate), then Beats-call chooses every candidate on the ballot as being tied 
as its winners.

This statement is just a clarification, not (I think) a disagreement.  Still, it 
is an important clarification because, in my notation, Beats-all//Condorcet(1) 
is NOT the same as Condorcet(1) and, for that matter, Beats-all//Condorcet(0) is 
NOT the same as Condorcet(0).  However, Beats-all//Condorcet(1/2) is the same as 
Condorcet(1/2).  Further, since Smith//Beats-all is the same as Smith, 
Smith//Beats-all//Condorcet(1) is the same as Smith//Condorcet(1).

For whatever it's worth, note that Smith//Smith is the same as Smith, and that 
Beats-all//Beats-all is the same as Beats-all; but that 
Condorcet(1/2)//Condorcet(1/2) is not the same as Condorcet(1/2), and I strongly 
suspect (but have not worked out an example) that Condorcet(1)//Condorcet(1) is 
not the same as Condorcet(1).  Certainly, Copeland//Copeland is not the same as 
Copeland.

Bruce