[EM] clone immunity definitional problems
Warren Smith
wds at math.temple.edu
Sat Jan 27 10:13:09 PST 2007
Let me explain some of the difficulties with defining clone-immunity.
I see no great difficulty defining the concept for pure-rank-order deterministic
methods. But let's throw in rank-equalities and random tie breaking and/or
other randomization. Then it gets trickier.
Problem1: Say in a vote A=B and then we clone B. Can we get the new vote B1>A>B2?
Well that would introduce the problem that it could convert a tied election (50-50 win probability)
to 100% win probability for B1, thus altering things a lot from the view of A, who feels
that cloning has changed things plenty.
Problem2: Say in a vote A=B and we clone A=B1=B2. That can convert an election with 50-50
win probability into 33-33-33. But in a probability-conserving clone definition,
Prob(A wins) = 50 should be unaltered by cloning B.
Perhaps the answer is to say that A and B were ALREADY cloned... but if so you have
to make the definition reflect that.
Markus Schulze's pdf paper he cited gives a hairy definition of
"independence of clones" in section 4.5 page 53.
Trying to decode it, he defines "cloning B" to mean (in the below, A,B, and C are distinct):
* iff A>B in in an old vote in the old election, then A>(all B clones) in the new one.
* iff B>A in in an old vote in the old election, then (all B clones)>A in the new one.
* iff C>A in in an old vote in the old election, then C>A in the new one.
* Schulze does not say anything explicitly about equalities but some facts can be deduced
because he used "iff" rather than "if." (By the way, it might be better merely to use "if"
because "iff" may lead to insurmountable problems...). Here are deduced facts:
* iff A=B in in an old vote in the old election, then A=(all B clones) in the new one.
That is a very strong demand by Schulze, and one I feel should be avoided if we can avoid
it. In other words, I think Schulze's definition is a bad definition because this deman is
way too strong. However, maybe Schulze was forced to do that because trying to weaken
it leads to insurmountable problems. If so I retract my complaint.
* If A=C in in an old vote in the old election, then A=C in the new one.
(I have no objection to that.)
Schulze then makes these demands about win probability before and after the cloning of B:
* Prob(A wins) does not increase.
* Prob(B wins)=0 before, implies Prob(any B clone wins)=0 after and that
all win-probabilities for non-B candidates are unaltered.
* I can't decode Schulze's third and last demand; too many subscripts and superscripts -
but perhaps he meant that cloning B leaves all win prbabilities unaltered for the uncloned
candidates (and consequently, the probability of B winning remains the same albeit
split among B's clones). If that is what Schulze meant, then he definitely made a bad mistake
because of problems 1 and 2 up top. These problems essentially would cause clone-immunity
under Schulze's definition to be a property that simply could never be satisfied by
any reasonable voting method - kind of a self-contradicting property.
So I'm not happy. I ran into this mess when I was trying to extend my and Forest Simmons'
proofs to cover the case of voting methods allowing rank-equalities.
Before I can prove or disprove it, we need ot have a definition of the ICC property.
Warren D Smith
http://rangevoting.org
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