[EM] Avoiding Clone Problems in Condorcet Flavored PR methods

[Forest Simmons]

Suppose there are to be two winners in a PR election among several
candidates C1, C2, C3, C4, C5, ..., and in comparing subset {C1,C2} with
subset {C2, C3} the ballots show

170 C1>C2>C3
170 C1>C3>C2
330 C2>C3>C1
330 C3>C2>C1 .

Which of these two subsets provides better proportional representation?

Since C1 has the highest number of first place votes (relative to C2 and
C3) it would seem that C1 should be in the winning circle.

Furthermore it could be argued that C2 and C3 appear to be clones, and
that C2 would represent the clone supporters while C1 would represent the
other voters.

Therefore, it would seem that {C1,C2} provides better proportional
representation than {C2,C3}.

On the other hand, let's compare these two combinations of candidates on
how well they appear to "cover" the electorate:

It is true that the combination {C1,C2} covers 67 percent of the
electorate with a first choice.

But it covers only 17 percent of the electorate with both first and second
choice, while the other combination covers 66 percent of the electorate
with both first and second choice.

It is not clear to me from the ordinal information alone which of these
points of view is to be preferred, but to be on the safe side and avoid
clone problems, I believe that we should favor methods that give the win
to {C1,C2} in cases like this.

If we had (in addition to the ordinal information) approval information
like

170 C1>>C2>C3
170 C1>>C3>C3
330 C2>C3>>C1
330 C3>C2>>C1 ,

then it would be obvious that the clone interpretation was the correct
one, and that {C1,C2} would provide better PR than {C2,C3}.

On the other hand,

170 C1>C2>>C3
170 C1>C3>>C2
330 C2>>C3>C1
330 C3>>C2>C1 ,

for example, would tend to support {C2,C3} as providing better PR.

Since "Condorcet Flavored PR Methods" are based on pairwise comparisons of
combinations of candidates, a good method of this class should have a way
of automatically collapsing clones of the type in this example.

Any ideas on how to do that?

Forest

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