Wait a minute - Simmons' "cloneproof" method is not really cloneproof!
warren_d_smith31 wds at math.temple.edu
Sun, 31 Dec 2006 18:40:47 -0000
x+y >= s+z,
y+z >= s+x,
x+z >= s+y,
y+s <= z,
x >= s+y,
x/3 + y >= s+z
A wins under Simmons voting since
A beats B pairwise ==> 6 ballots count against B
C beats A pairwise ==> 3 ballots count against A
B beats C pairwise ==> 4 ballots count against C
Now add two clones of A in a Condorcet cycle.
Then A1 is beat pairwise by A2 with 1/3 of the 6 of the
A-top ballots, i.e. 3, and ditto A2 and A3, all have 2
A-top ballots against them.
Plus, all the Ak have got C's toprank votes
against them, which is 3. So in total, each A-clone
has 5 ballots against it, while C has only 4
ballots against it.
Hence C is now the winner thanks to A's cloning.
So SIMMONS IS NOT CLONEPROOF!!
However... if the "clones" are required to be ranked EQUAL
(not in a cycle) then probably it is cloneproof in that weaker sense.
If so, perhaps Simmnos will prove that...