[EM] A graphical description of Toby Pereira's Transformation of Range to Approval style ballots

[Forest Simmons]

In a previous posting to the EM I promised to explain the Toby Pereira
transformation of a cardinal ratings (aka "score" or "range") style ballot
into a weighted sum of approval ballots.

If you think of the ratings displayed graphically like a histogram (or
vertical bar graph), then for each distinct bar height you make an
horizontal slice of the bar graph at that height just grazing the tops of
the bars at that height.

The part of the bar graph between two consecutive slices will itself be a
bar graph in its own right with all of its bars equal in height.  Divide by
that common height h and you get a bar graph whose bars are all unit
height, i.e. a bar graph of an approval vector.

The weight of that approval vector is none other than the original height
that we divided by.  If we add the approval vectors multiplied by their
weights, we get back the original range/score vector, (assuming the
original min score to be zero).
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