Defn of Condorcet's method

[Bruce Anderson]

In my opinion, Mike and I have resolved some of our differences concerning 
definitions of Condorcet's voting method.  My version of this partial resolution 
is as follows:

As before, let p(i,j) be the sum of the number of voters who explicitly rank i 
over j plus the number of voters who rank i and leave j unranked.  Let q(i,j) be 
the sum of the number of voters who explicitly rank i as tied with j plus the 
number of voters who leave both i and j unranked.  Let p and q be the 
corresponding arrays of values of p(i,j) and q(i,j).

Unlike before, define r = r(i,j;x) for 0 <= x <= 1 and i <> j by:
r(i,j;x) = p(i,j) + xq(i,j) if p(i,j) < p(j,i), and
r(i,j;x) = v otherwise--v is the number of voters.

Further, define s = s(i,j;x) for 0 <= x <= 1 and i <> j by:
s(i,j;x) = p(i,j) + xq(i,j) if p(i,j) > p(j,i), and
s(i,j;x) = 0 otherwise.

Define i to be a row-Condorcet(x) winner if
min/j of r(i,j;x) >= min/j of r(k,j;x) for all k <> i.

Define i to be a col-Condorcet(x) winner if
max/j of s(j,i;x) <= max/j of s(j,k;x) for all k <> i.

Lemma:
i is a row-Condorcet(x) winner if and only if
i is a col-Condorcet(1-x) winner.
The proof follows from the identity:
v = p(i,j) + q(i,j) + p(j,i).

Define M-Condorcet to be the Condorcet voting method as defined and used on EM.
Define Beats-all as before; i.e., if there is a candidate that beats all of the 
other candidates in their pairwise matchups, then Beats-all selects that 
candidate as its unique winner, otherwise Beats-all selects every candidate on 
the ballot as being a winner.

Then:  M-Condorcet,
Beats-all//col-Condorcet(0), and
Beats-all//row-Condorcet(1)
are identical voting methods.

Now define  r' = r'(i,j) for i <> j by:
r'(i,j) = p(i,j) + (1/2)q(i,j).

Define i to be a sim-Condorcet winner if
min/j of r'(i,j) >= min/j of r'(k,j) for all k <> i.

Then:  sim-Condorcet,
col-Condorcet(1/2), and
row-Condorcet(1/2)
are identical voting methods.

Finally, note that if q(i,j) = 0 for all i and j, then:
sim-Condorcet,
col-Condorcet(x), and
row-Condorcet(y)
are identical voting methods for all x and y.

Bruce