[EM] UD Approval/Disapproval/Other

[Forest Simmons]

Here's a reasonable way to convert RCV ballots into three slot approval
ballots:

On each ballot B find the cutoff candidates C1 and C2 defined as ...
C1 is the candidate closest to Bottom on B that defeats every candidate
ranked above it. C2 is the candidate closest to Too on B that is defeated
by every candidate that it out ranks.

Every candidate that outranks C1 gets full Approval. Every candidate
outranked by C2 gets full Disapproval.

How about the  cutoff candidates themselves?

C1 gets full approval only if it outranks some candidate X1 that defeats
every candidate ranked above C1.

Similarly, C2 gets full Disapproval only if some candidate X2 ranked over
C2 defeats every candidate outranked by C2.

All candidates which by these rules are endowed with neither full Approval
nor full Disapproval status ... are in the "Other" category between
Approval and Disapproval.

These rules imply (among other things) that all of B's Too candidates are
Approved, and all of its Bottom candidates are Disapproved.

When a Condorcet Winner exists for a ballot set, it will be the cutoff
candidate C1 on every ballot B in the set ... and will not receive full
disapproval unless it has Bottom status ... (which means it outranks no
candidate).

Similarly, when a ballot Condorcet Loser exists it will serve as C2 on
every ballot, and will not receive full Approval unless it has Top status
... (which means that no candidate outranks it).

When there is no ballot CW and not all Smith members are ranked the same,
then at least one if them will receive full Approval.

When there is no ballot CLand not all anti-Smith candidates are ranked
equal, then at least one of them will receive full Disapproval.

A Smith member is a candidate having beatpaths to each of the other
candidates.

An anti-Smith member can be reached by a beatpath from each of the other
candidates.

If X is approved on ballot B, and then is raised on any other ballot, it
will still be approved on ballot B after the change.

So the rules seem to give reasonable results. Time will tell.

Have Fun!

-Forest
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