[EM] Higher order circular ties

[Michael Rouse]

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