[EM] A set of clone independence criteria more applicable to multiwinner

[Markus Schulze]

Dear Kristofer,

in section 9.2 of my paper "The Schulze Method of Voting",
I define independence of clones for proportional ranking
methods and I explain, why the Schulze proportional ranking
method satisfies this criterion:

https://arxiv.org/pdf/1804.02973.pdf

Suppose candidate D had the Z-th place in the proportional
ranking. Suppose candidate D is replaced by a set of clones
D(1), ..., D(n). Then the highest ranked clone D(1) must
get the Z-th place of the proportional ranking and there
must be no change in the places 1, ..., (Z-1) of the
proportional ranking.

Example: Suppose the original proportional ranking was
A, B, C, D, E, F, G, ...
Then the new proportional ranking could be
A, B, C, D(1), G, F, E, D(2), ...

One might argue that the order of the candidates between D(1)
and D(2) shouldn't change either since D(2) is the first candidate
who disrupts the original proportional ranking. However, I
believe that already the sheer existence of potential winning
sets with two candidates from the set of clones can change
the order of the candidates between D(1) and D(2).

*********************************************

Now, let's apply independence of clones for proportional
ranking methods to multi-winner election methods. So let's
say that a board of M seats is filled by calculating a
proportional ranking first and then choose the first
M candidates of this proportional ranking.

When candidate D is not elected, then he wasn't one of the
M highest ranked candidates. So when D is replaced by a set
of clones, the ordering of the M highest ranked candidates
must not change. So the winners stay the same.

When candidate D is elected, then we don't know whether
candidate D was (say) the top-ranked candidate or (say)
the M-th ranked candidate. So we cannot say anything about
whether the other candidates are still elected. We only
know that at least one of the clones D(1), ..., D(n) must
be elected.

Therefore, I believe that the maximum that we could ask for
is the following:

#########################
Independence of clones for multi-winner elections:
When a non-winner is replaced by a set of clones, then the
election result must not change. When a winner is replaced
by a set of clones, then at least one of these clones must
be elected.
#########################

Markus Schulze