[Election-Methods] method design challenge + new method AMP

[Jobst Heitzig]

Dear Juho,

you wrote:
> One observation on clone independence and electing a centrist
> candidate using rankings only and when one of the "extremists" has
> majority.
...
> It is thus impossible for the algorithm in this case and
> with this information (rankings only) to satisfy both requirements
> and to be fully clone independent.

D'accord. This is a good reason to consider rankings insufficient, since 
from rankings only one cannot determine whether to apparent clones are 
truly clones in the sense that they are (nearly) equivalent in all 
relevant aspects.

From ratings information, however, one can see this. Therefore I would 
not at all consider A1,A2 clones in your ratings example:
> A=100 C=55 B=0 => A1=100 A2=56 C=54 B=0
> B=100 C=55 A=0 => B=100 C=56 A1=54 A2=0

For A1,A2 to be considered clones, the ratings would have to be 
something like
51: A1 100 > A2 99 > C 55 > B 0
49: B 100 > C 55 > A1 1 > A2 0

You also seem to think so, since you wrote:
> One approach to try to avoid this problem would be to use a more
> limited clone concept: candidates that are ranked/rated equal with
> each others.

But that would never really occur in practice. I think one should define 
the notion "clone" like this: A1,A2 are clones if and only if on each 
ballot, the difference in ratings between any pair of options is 
smallest for the pair A1,A2. 

(Analogously, a set S of options should be called a clone set if and 
only if on each ballot, all rating differences between two options in S 
are smaller than all rating differences between other pairs of options. 
Even more generally, a system Y of disjoint sets S1,...,Sk of options 
could be called a clone partition if and only if on each ballot, all 
rating differences between two options which are contained in the same 
member of Y are smaller than all rating differences between other pairs 
of options.)

With this definition, the problem you described cannot really occur: 
Assume the rankings are
> 51: X1>X2>X3>X4
> 49: X4>X3>X2>X1
If X1,X2 are clones, X2 cannot be considered a good compromise since 49 
voters don't like her. Similarly, if X3,X4 are clones, X3 cannot be 
considered a good compromise since 51 voters don't like her.

Yours, Jobst
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