# [EM] Higher order circular ties

Michael Rouse mrouse at cdsnet.net
Mon Jun 25 19:46:27 PDT 2001

```When I started looking at Condorcet winners, Smith sets, circular ties and
the like, I never saw any examples higher than a three-way tie. Just in
case someone wanted a few possibilities to try their new election method
on, here are four circular ties involving five candidates (a standard
version, an "interlaced" version, and inverse methods of both). Of course,
there are many other possibilities involving five candidates, but I don't
think they are as neatly symmetric. I also found circular ties involving
seven candidates but not six -- but that's probably because of the
geometric way I figure out the relationships.

To generate the ties, just make sure the sum of the three smallest values
are larger than the sum of the two largest -- the values 25-30-35-40-45
work well in any order, since 25+30+35>40+45. Some other interesting
possibilities are 1-2-2-2-2, 2-2-3-3-3, and 5-6-7-8-9. With a little effort
you can create circular ties where the sum of the two largest values is
greater than the sum of the three smallest, but this way you don't need to
look at every possibility to see if the tie unravels.

Tie 1 (standard)
A	B	C	D	E
B	C	D	E	A
C	D	E	A	B
D	E	A	B	C
E	A	B	C	D

Tie 2 (interlaced)
A	B	C	D	E
C	D	E	A	B
E	A	B	C	D
B	C	D	E	A
D	E	A	B	C

Tie 3 (inverse of tie 2)
A	B	C	D	E
D	E	A	B	C
B	C	D	E	A
E	A	B	C	D
C	D	E	A	B

Tie 4 (inverse of tie 1)
A	B	C	D	E
E	A	B	C	D
D	E	A	B	C
C	D	E	A	B
B	C	D	E	A

Here are the ten relations for each tie (I hope they show up correctly):
Tie 1	Tie 2	Tie 3	Tie 4
A>B	A>C	A>B	A>D
A>C	A>E	A>D	A>E
B>C	B>A	B>C	B>A
B>D	B>D	B>E	B>E
C>D	C>B	C>A	C>A
C>E	C>E	C>D	C>B
D>E	D>A	D>B	D>B
D>A	D>C	D>E	D>C
E>A	E>D	E>A	E>C
E>B	E>B	E>C	E>D

I thought I might as well post them because it might save someone some
time. It's a fun way of generating some voting paradoxes. If a voting
method handles a five-way circular tie without blowing up, it'll probably
handle anything (grin).

Mike Rouse
mrouse at cdsnet.net

```