Definition of "Pairwise Method"
Bruce Anderson
landerso at ida.org
Mon Apr 15 22:36:12 PDT 1996
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
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