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

[Toby Pereira]

I feel compelled to point out that although I independently encountered this method to convert score ballots to approval ballots, I wasn't the first to do so. After finding it myself, I found this post on a forum describing the same thing at an earlier time. http://www.revleft.com/vb/showpost.php?p=2030744&postcount=12
 

 
      From: Forest Simmons <fsimmons at pcc.edu>
 To: EM <election-methods at lists.electorama.com>; Jobst Heitzig <jobst.heitzig at posteo.de>; Brian Olson <bql at bolson.org>; Andrew Jennings <elections at jenningsstory.com> 
 Sent: Thursday, 17 December 2015, 22:02
 Subject: [EM] A graphical description of Toby Pereira's Transformation of Range to Approval style ballots
   
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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