[EM] Question about complete clone independence
Steve Eppley
SEppley at alumni.caltech.edu
Wed May 3 18:35:53 PDT 2000
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)
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