[EM] two bit ratings (was seven +/- two)

[Forest Simmons]

I won't beat Demorep or Richard on this, but here are my two entries: 

I. In the first method, the left bit is the approval bit, and is the only
instrumental bit, i.e. the only bit with any possible effect on the
outcome.

The right bit is purely expressive, for psychological, educational and
mandating purposes. The candidate with the greatest number of left column
ovals marked is the winner.

II. The second method (my favorite two bit method) is similar to Ranked
Pairs, only instead of winning votes or margin size determining the
strength of each pair, it's the number of ballots on which the pair
straddles the big gap, i.e. the number of ballots on which the two
candidates in the pair have ratings that differ in the most significant
bit.

If there is no pairwise unbeaten candidate, then weak pairs are zeroed out
(starting at the weakest) until there is an unbeaten candidate. 

In this method it is useful for the voter to think of the two bit ratings
as base ten expressions:

00=zero < 01=one << 10=ten < 11=eleven

The gap sizes count for nothing if there is a pairwise unbeaten candidate.

Otherwise, the relative gap sizes (big vs small or zero) help determine
the strength of the pairs.

It is necessary to compute two matrices: the pairwise margin matrix, as
well as the pair strength matrix.

Both of these matrices are n by n matrices, where n is the number of
candidates. 

The margin matrix is antisymmetric, and the strength matrix is symmetric
with zeros down the main diagonal. So there are only n(n-1) numbers to be
carried in the precinct summaries.

This method satisfies both the Condorcet Criterion and the FBC.

Because of the symmetry properties of the margin and strength matrices
it should also satisfy the Reverse Symmetry Criterion.

The reason I prefer this method above Demorep's two bit method is that for
more than three candidates, this method is likely to resolve all cycles
before reducing all of the way down to Approval.

On the other hand, Demorep's method is easier to explain.

In a future posting I'll try to come up with some good examples that
illustrate the advantages of the second method.

Meanwhile, feel free to try to discredit it with some favorite example of
your own.


Forest