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

maximize(s, {
x+y >= s+z,
y+z >= s+x,
x+z >= s+y,
y+s <= z,
x >= s+y,
x+y+z=13,
x>=0,
y>=0,
z>=0,
x/3 + y >= s+z
});

Ballots:
6: A>B>C
3: C>A>B
4: B>C>A

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...

-wds