[EM] Simmons cloneproof method is not cloneproof

[Warren Smith]

see http://groups.yahoo.com/group/RangeVoting/message/2934
for counterexample  (plus linear program explainign how I found the 
counterexample)
wds

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Re: [EM] Simmons cloneproof method is not cloneproof

[Chris Benham]

Warren Smith wrote:

>see http://groups.yahoo.com/group/RangeVoting/message/2934
>for counterexample  (plus linear program explainign how I found the 
>counterexample)
>wds
>
> 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!!

> If we agree only to clone non-winners,
> or if, when cloning a winner, all voters agree to rank the clones
> EQUALLY, THEN Simmons is cloneproof.
>
> Proof:
> After cloning,
> the A-beats-B relations are unaffected under these constraints,
> and the number of top-rank-votes-against X are either unaltered -
> or increased (increase is only possible for nonwinner X since
> winning-X clones never pairwise-beat each other).
> QED
>
> However... this weakened kind of cloneproofness is a good
> deal less impressive than genuine cloneproofness.


So  Simmons meets Clone-Loser, but can fail Clone-Winner when there are 
three or more factions
in a top cycle and the candidates in one of those factions are in a 
sub-cycle. That is a very very mild
failure of  Clone-Independence and arguably not a practical worry.

If that is the full extent of the bad news (and maybe even if it isn't) 
then I think this method remains a
great contender (for "best practical Condorcet method") because of its  
tremendous Burial resistance
and simplicity.

Chris Benham




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Re: [EM] Simmons cloneproof method is not cloneproof

[Chris Benham]

Forest/Warren/all,
One serious disadvantage that the new Simmons method (in pure form) has 
is that it has a nasty Local IIA
problem, and fails what I might call  "Independence of/from Irrelevant 
Candidates" which says that if there is
some losing candidate X with fewer top preference votes than any other 
candidate and which is pairwise beaten
by every other candidate, then dropping X from the ballots can't change 
the winner. This is a weak criterion that
is easily met by IRV and arguably by all good methods.

02: X>A>B
24: A>B
25: C>A
49: B>C (maybe sincere is B>A or B)


A>B>C>A.  "Simmons" scores: A25,  B24, C49

B has the lowest score and so wins, but if  X is dropped from the 
ballots then B's score rises to 26 and  A wins.
Those two X supporters have a "semi-clone"  split-vote problem.

This problem can be easily patched up by first dropping from the ballots 
all non-members of the Schwartz set before
applying this simple Simmons method  (to give "Schwartz//Simmons").


Warren Smith wrote:

> actually, Simmons is NOT a Condorcet method at all,
> in the sense that it is entirely possible for a unique
> Condorcet winner W to exist, but Simmons does not select W as the unique
> winner, instead claiming that several candidates are tied for winner.
>
> This usually happens when W has zero top-rank votes.
>
> I'd pointed that out before but this makes it clearer.
>
> Similarly, Simmons does not really obey the Smith set property.
>
My suggested patch would also of course fix this problem.


Chris Benham




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