[EM] Grade Voting revisited

[Forest Simmons]

Suppose that the ballots allowed voters to grade the candidates in terms
of the familiar system of A, B, C, D, F.

Many methods have been suggested on the basis of these grade ballots,
including Condorcet methods, Approval based on the greatest number of
above D grades, highest grade point average, etc.

Here's another possibility:

First calculate the most-grades-above-C winner C1, then the fewest below C
winner C2, and then the highest grade point winner C3.

If these are all the same, then this candidate wins, period.

Otherwise, on each ballot set the approval cutoff at the position

                       (c1+c2+c3)/3,

where c1, c2, and c3 represent the grade values assigned to the respective
candidates on that ballot.

The approval winner based on approval cutoffs calculated in this way is
the method winner.

The method is summable in a data structure of a size equal to the fourth
power of the number of candidates.

Rationale: if all three of C1, C2, and C3 are the same, then that
candidate is a strong, stable winner.  If, on the other hand, they are not
the same, the average of their positions on a ballot tells the voter of
that ballot approximately what can be expected of an approval winner, and
so gives a reasonable position for an approval cutoff.

Forest