[EM] Multi-seat Condorcet method

[Daniel Bishop]

Here is a method I developed to elect the Condorcet winner in 
single-seat elections, provide proportional representation in multi-seat 
elections, and to be fairly computationally simple.

I would like to know if there are any serious flaws, and if so, how the 
method can be improved.

AN EXAMPLE

Using some random ballots, wherein Party X={A,B,C} and Party Y={D,E,F} 
are clone sets.

A>B>C>E>F>D
A>B>C>E>F>D
A>C>B>E>F>D
A>C>B>F>E>D
B>A>C>D>E>F
B>A>C>E>F>D
C>A>B>F>D>E
C>B>A>F>E>D
D>E>F>A>B>C
D>F>E>A>C>B
F>D>E>B>A>C
F>D>E>B>A>C

First, count the first-choice votes, and elect candidates that meet 
quota (in this case, both the Hare and Droop quotas are 4 votes) as in STV.

A: 4 -> elected with a surplus of 0
B: 2
C: 2
D: 2
E: 0
F: 2

Next, construct a pairwise matrix for all non-elected candidates, 
weighting the ballots as in the Newland-Britton system.  In this 
example, A exactly met the quota, so A's ballots are not transferred at 
all, so there are effectively only 8 ballots left:

B>C>D>E>F
B>C>E>F>D
C>B>F>D>E
C>B>F>E>D
D>E>F>B>C
D>F>E>C>B
F>D>E>B>C
F>D>E>B>C

The corresponding pairwise matrix is:

  B C D E F
B 0 5 4 4 4
C 3 0 4 4 4
D 4 4 0 6 3
E 4 4 2 0 3
F 4 4 5 5 0

Next, elect the Condorcet winner (or the winner of a Condorcet 
completion method) from this matrix.  In this case, there is no 
Condorcet winner; there are 2 weak Condorcet winners, B and F.  Suppose 
that F is elected.

Weight F's ballots so that the number of total votes is reduced by one 
quota.  Note that this gives the ballots a negative weight.

B>C>D>E
B>C>E>D
C>B>D>E
C>B>E>D
D>E>B>C
D>E>C>B
D>E>B>C weight = -1
D>E>B>C weight = -1

(Note: The last 2 ballots are still considered F ballots, not D ballots, 
so at this point D is considered to have 2 votes, not zero, which would 
complicate things a great deal if D were elected.)

The new pairwise matrix is:

  B C D E
B 0 1 4 4
C 3 0 4 4
D 0 0 0 2
E 0 0 2 0

C is the new Condorcet winner, and gets elected.  The winning committee 
is ACF, which contains 2 members of Party X and 1 member of Party Y, as 
expected.