[EM] Peak Approval Sorted Margins

[Forest Simmons]

This is an aggregate method that can be taken apart and reassembled to fit
the circumstances.

The longest path from start to finish begins with ranked choice ballots.

Step 1 converts the ballots to 3-slot ballots. This step can be
accomplished in two main ways (detailed presently) or bypassed entirely by
getting direct 3-slot approval input as in Ted Stern's Approval Sorted
Margins, for example.

The first main way is to distinguish Top, Bottom, and Middle positions on
the ranked ballots.

The second main way is to give Top slot status to every candidate X on
ballot B for which there is some candidate Y outranked by X, that defeats
every candidate that outranks X.

Bottom slot status goes to X if it is outranked by some Y that defeats
every candidate that X outranks.

Middle slot status goes to ranked candidates not assigned Top or Bottom
status by the above rules.

Step 2 is converting 3-slot Top, Middle, and Bottom tallies into Robert
Bristow-Johnson's  "peak approval" scores.

Let t, m, & b be the respective slot values. Then the peak approval score
is ...
(t-b)/(2-t-b) or (t-b)/(1+m)

with the David Gale value t-b as the tie breaker.

Step 3. Do peak approval sorted margins, as in any other Sorted margins
method.
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