[EM] XA

[Forest Simmons]

Can it be the case that Approval has a better CD strategy than XA ?  Not
really, because XA voters can always vote approval style by rating only at
the extremes of zero and 100%, and in that case XA is reduced to plain
Approval.

Any strategy available in Approval is also available in XA, which has
additional options, more control levers to work with, as it were, if you
want to use them.

Note that if candidate X is rated 100% on 70 percent of the ballots and
rated zero on the other ballots, for example, then the XA score is 70
percent, identical to the Approval score.

Also, in the context of cardinal ratings, it makes no difference if X is
rated at 100 percent on 70 percent of the ballots or is rated at 70 percent
on 100 percent of the ballots.  The Cardinal Ratings score is 70 percent.

So is the XA score!

Contrast this with Bucklin which agrees in the unanimous rating case, but
gives a score of 100 percent in the other case.

The graphical picture makes this clear:

The distribution function F(x) defined as the percentage of ballots on
which X is rated above x, stair steps down from the left side to the right
side of the square with diagonal corners at (0, 0) and (1, 1).

The Bucklin score is determined by where it intersects the vertical line
 x = 1/2.

The XA score is determined by where it crosses the line y = x.

The mean Cardinal rating is simply the area between it and the x axis.

The XA score favors candidates whose distribution graph is both tall and
wide, giving equal importance to both height and width.  The further
towards the upper right corner of the square along the diagonal given by y
= x, the better the point of intersection for the candidate.

Once we understand the close relationship of Chiastic Approval with
ordinary Approval and Cardinal Ratings, we can justify the following
explanation of XA:

The purpose of XA is to maximize the effect of voter intent as expressed by
their ratings of the candidates.

For each candidate, you indicate on your XA ballot what you consider to be
an appropriate rating of merit or support on a scale of zero to 100 percent

In the XA count, your ballot gives full approval to the candidates that you
consider under-rated by the rest of the voters, and no approval to the
candidates that you consider over-rated by the rest of the voters.

The candidate with the highest (average or total) approval in the XA count
is elected.

Any suggestions for improvement?

Forest
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