[EM] Approval splitting pathology

[Ted Stern]

As a long time fan of Margin Sorted Approval (AKA Approval Sorted Margins
<https://electowiki.org/wiki/Approval_Sorted_Margins>), I've been thinking
about various ways to improve it.

I started thinking about how one might implement an automatic approval
cutoff, designed to give each ballot its own strategically optimal cutoff.

One reasonable strategy in Approval Voting generally is to determine, for
each ballot, the highest ranked candidate(s) who are in the Smith Set, and
then put the Approval Cutoff below that rank.

But what happens if there is a Condorcet cycle, and the candidate who would
ordinarily win in Approval voting ("A") using that strategy is cloned ("A1,
A2, A3"). and there is a cycle in that clone set? With the strategy.above,
the approval cutoff will be placed just below the top clone on the ballot.
This would cause a split of approval for the A# candidates, affecting not
only Margin Sorted Approval but also Approval Voting more generally.

This special case seems highly unlikely, but it points to a cloning
resistance issue with any Approval augmented Condorcet method.

Thoughts?
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