[EM] Question about complete clone independence

[Steve Eppley]

Hi,

I've been looking at ways to strengthen "independence from 
clones." My primary intent is to show that independence from 
clones is a robust property of certain voting procedures.  (To 
be a bit more specific, a set of "similar alternatives" would be 
defined based on certain aggregate properties of the votes which 
are unaffected by small perturbations of the votes, any set of 
clones would also be a set of similar alternatives, and voting 
procedures which are independent from similar alternatives would 
thus be robustly independent from clones.)

I've encountered a problem having to do with "complete" 
independence (i.e., independence even when there are pairties 
and/or equal size majorities, which would be rare events in 
large public elections).  The problem doesn't seem to vanish 
entirely when restricting attention to clones rather than the 
broader "similar alternatives."  So I want to doublecheck my 
knowledge of clones by asking fellow subscribers of the EM list.

I haven't managed to read every message in EM, and my memory of 
those I've read isn't perfect, so I'm unsure whether the 
definitions in EM of clones and independence from clones have 
changed significantly since the following was posted by Markus 
in 1997:

   Definition:  A[1],...,A[m] are a set of m twins
   if & only if for each pair (A[i],A[j]) of two
   candidates of the set of m twins, for each voter V,
   and for each candidate C outside the set of m twins,
   the following statements are true:

   V prefers A[i] to C if & only if V prefers A[j] to C.
   V prefers C to A[i] if & only if V prefers C to A[j].

   Definition:  A voting method meets the "Generalized
   Independence from Twins Criterion" (GITC) if & only if
   additional twins cannot change the result of the elections.
   (If one twin is elected instead of another twin, then
   this is not regarded as a change of the result.)

Consider the following scenario:  There are three candidates 
x, y, and z, exactly one must be elected, and every voter is 
indifferent between every candidate.  As I understand the 
definition of clones (a.k.a. twins), there are 4 sets of clones 
which contain at least two alternatives:

  {x,y,z}, {x,y}, {y,z}, {x,z}

The first set is "trivial" since all alternatives are in the 
set.  Consider one of the non-trivial clone sets, say {x,y}:  
If only x and z compete, then z has a 1/2 chance of winning, 
given a single-winner voting procedure which satisfies anonymity 
and neutrality.  Adding y (which is a clone of x) decreases z's 
chance of winning to 1/3.  So unless the definition of clones is 
modified, the completeness of the independence is questionable.  
Perhaps there are other scenarios exhibiting this problem which 
do not depend on massive indifference.

I'd also like to suggest that the definition of independence be 
rephrased, if it has not already been, so that it requires the 
*probability* of election of a non-clone remain unchanged when 
clones are added.  Otherwise, it's hard to say whether voting 
procedures which are not completely deterministic can rigorously 
satisfy the criterion; outcomes cannot be directly compared when 
randomness is involved, but probabilities of outcomes can be 
directly compared.


---Steve     (Steve Eppley    seppley at alumni.caltech.edu)