# Circular Tie solution

donald at mich.com donald at mich.com
Thu Nov 7 03:48:34 PST 1996

```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
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,

```