Dear Steve,
you wrote (3 May 2000):
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.
This problem has been discussed in August 1998. The definition of
clones was changed as follows (28 Aug 1998):
Definition ("clones"):
A[1],...,A[m] are a set of m clones if only if the following
two statements are valid:
(1) For every pair (A[i],A[j]) of two candidates of this set,
for every voter V, and
for every candidate C outside this set
the following two statements are valid:
(a) V strictly prefers A[i] to C,
if only if V strictly prefers A[j] to C.
(b) V strictly prefers C to A[i],
if only if V strictly prefers C to A[j].
(2) For every candidate A[k] of this set and
for every candidate D outside this set
there is at least one voter W, who either
strictly prefers A[k] to D or strictly prefers D to A[k].
The aim of statement (2) is to exclude explicitely those situations
where every voter is indifferent.
**
You wrote (3 May 2000):
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.
I agree with you. That's also the reason why I am talking so
vaguely about the "result of the elections" and not about the
"winner." Those who promote random election methods usually
consider the random distribution and not the winner to be the
"result of the elections."
Markus Schulze
[EMAIL PROTECTED]
[EMAIL PROTECTED]
[EMAIL PROTECTED]
