Circular Tie solution

[donald at mich.com]

Dear members of the election-methods list,

I was studying the example of a circular tie by DEMOREP1 - thank you
DEMOREP1 - and I came up with a way to solve the tie.


                             H and G        H and S        G and S
DEMOREP1's                   Pairing        Pairing        Pairing
Example
35 HG                        35H            35H            35G
33 GS                            33G            33S        33G
32 SH                        32H                32S            32S
H beats G, 67 to 33         --------        -------        -------
G beats S, 68 to 32          67H 33G        35H 65S        68G 32S
S beats H, 65 to 35
Circular Tie---  H>G>S>H

The Solution of the Circular Tie: We are going to combine the Vote-Sums
created by the pairing - but we first remove the duplicate Vote-Sums.


                             H and G        H and S        G and S
                             Pairing        Pairing        Pairing

                             35H           [35H]           35G
                                 33G            33S       [33G]
                             32H                32S           [32S]
                             --------        -------        -------
                       The Vote-Sums in brackets are duplicate Vote-Sums

Below are the Vote-Sums with the duplicates removed:

              H and G        H and S        G and S
              Pairing        Pairing        Pairing

              35H           [   ]           35G
                  33G            33S       [   ]
              32H                32S           [   ]
             --------        -------        -------

We now collect the Vote-Sums of H, G, and S as shown below:

                         H      G      S

                        35H    35G
                               33G    33S
                        32H           32S
                       ----   ----   ----
                        67H    68G    65S

We have a WINNER! Candidate G with 68

Now - it is possible to arrive at these same results by another way. We
start with DEMOREP1's example and instead of working the candidates in
pairs we work them all together as shown below:

            35 HG       35H    35G
            33 GS              33G    33S
            32 SH       32H           32S
                       ----   ----   ----
                        67H    68G    65S

By working the candidates all together we can by-pass the pairing and the
dropping of the duplicate Vote-Sums.

This method will avoid a possible circular tie - so I think you should make
this change in your Condorcet method.

Donald,