# [EM] Question about complete clone independence

Markus Schulze schulze at sol.physik.tu-berlin.de
Wed May 3 18:47:33 PDT 2000

```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