[EM] Three Slot Approval Idea

[Forest Simmons]

Each ballot rates each candidate Top, Bottom, or Other (the abstention
default).

For each candidate X let TopX) be the number of ballots on which X is rated
Top, and Bot(X) the number on which X is rated Bottom.

Let RBTP(X) and RBBP(X) be the respective Random Ballot Top and Bottom
Probabilities for X, where each ballot B contributes to the respective
probabilities an amount 1/(nk) or 1/(nj), respectively, where k and j are
the respective numbers of candidates rated Top and Bottom on B with X ...
and without X the contributions are zero.

Naturally Top ratings are counted as approvals. Additionally ballot B
approves candidate X rated neither Top nor Bottom on B  if
TopLightness(B)/BottomHeaviness(B)
is less than Top(X)/Bot(X), where
TopLightness(B) is the sum of the values RBTP(Y) for Y rated Top on B, and
BottomHeaviness(B) is the sum of the values RBBP(Z) for Z rated Bottom on B.

So now Approval(X) is the number of ballots on which X is rated Top plus
the number of ballots on which X has been promoted to approval status from
the other category.

Let W=argmax Approval(X). If W is uncovered, elect it ... otherwise elect
from among those X's that cover W, the one with the greatest approval ....
unless this X is also covered ... etc.

There it is!

-Forest
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