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Estimated ranking using Choice-Specific Pairwise-Score (CSPS) ranking

Choice-specific pairwise-score (CSPS) ranking is a simple and mostly fair way to estimate the popularity ranking of choices in an election or survey.

Characteristics

Easy to count and calculate: The calculation of results can be done using pen and paper and a calculator.

Handles hundreds of choices:  It is useful for quickly calculating a full, reasonably fair ranking of many choices, such as when many voters indicate their top 10 and bottom 10 choices within a list of hundreds of songs or movies.

Reasonably fair:  This method is not intended for use alone.  Yet if it were used alone, with a simple refinement, the results are fairer than top-choice methods because this method uses pairwise counts.  The results do not meet the Condorcet criteria.  However, the Condorcet winner — if there is one — can be identified easily after this ranking method has been done. 

Example

(A concise description follows this example.  The description is easier to understand after you have read this example.)

Consider an election (or survey) in which there are five choices: A, B, C, D, E, and the final ranking order is this alphabetical order.

In this example the notation A>B refers to how many voters pairwise prefer choice A over choice B, and the notation B>A refers to how many voters pairwise prefer choice B over choice A.  This notation always uses the "greater-than" symbol ">", and never uses the "less-than" symbol "<" — because, for example, B>A is used instead of A<B.

At the beginning of this ranking example, suppose that the choices are arranged in the order C, E, A, D, B.  The pairwise counts for this arrangement are shown in this pairwise matrix.

      |       |       |       |       |       |
      |   C   |   E   |   A   |   D   |   B   |
      |       |       |       |       |       |
 -----+-------+-------+-------+-------+-------+
      | \     |       |       |       |       |
  C   |   \   |  C>E  |  C>A  |  C>D  |  C>B  |
      |     \ |       |       |       |       |
 -----+-------+-------+-------+-------+-------+
      |       | \     |       |       |       |
  E   |  E>C  |   \   |  E>A  |  E>D  |  E>B  |
      |       |     \ |       |       |       |
 -----+-------+-------+-------+-------+-------+
      |       |       | \     |       |       |
  A   |  A>C  |  A>E  |   \   |  A>D  |  A>B  |
      |       |       |     \ |       |       |
 -----+-------+-------+-------+-------+-------+
      |       |       |       | \     |       |
  D   |  D>C  |  D>E  |  D>A  |   \   |  D>B  |
      |       |       |       |     \ |       |
 -----+-------+-------+-------+-------+-------+
      |       |       |       |       | \     |
  B   |  B>C  |  B>E  |  B>A  |  B>D  |   \   |
      |       |       |       |       |     \ |
 -----+-------+-------+-------+-------+-------+

The diagonal line passes through empty cells.  These cells are empty because they would represent a choice's comparison with itself, such as A>A.

The goal of these calculations is to change the sequence so that the largest pairwise counts move into the upper-right triangular area, leaving the smallest pairwise counts in the lower-left triangular area.  (This is similar to the goal of VoteFair popularity ranking.)

The first step is to calculate choice-specific scores, with each choice having a row score and a column score.  For choice A, its row score equals the sum of the pairwise counts in the row labeled A, which equals A>B + A>C + A>D + A>E.  The column score for choice A is the sum of the pairwise counts in the column labeled A, which equals B>A + C>A + D>A + E>A.  The row scores and column scores for choices B, C, D, and E are calculated similarly.

Next, all the row scores are compared to determine which choice has the largest row score.  In this example that score would be the row score for choice A (because it is first in alphabetical order).  Therefore choice A is moved into first place.  The other choices remain in the same order.  The resulting sequence is A, C, E, D, B.  Here is the pairwise matrix for the new sequence.  The pairwise counts for the ranked choice (A) are surrounded by asterisks:

      |       |       |       |       |       |
      |   A   |   C   |   E   |   D   |   B   |
      |       |       |       |       |       |
 -----*****************************************
      * \     |       |       |       |       *
  A   *   \   |  A>C  |  A>E  |  A>D  |  A>B  *
      *     \ |       |       |       |       *
 -----*-------*********************************
      *       * \     |       |       |       |
  C   *  C>A  *   \   |  C>E  |  C>D  |  C>B  |
      *       *     \ |       |       |       |
 -----*-------*-------+-------+-------+-------+
      *       *       | \     |       |       |
  E   *  E>A  *  E>C  |   \   |  E>D  |  E>B  |
      *       *       |     \ |       |       |
 -----*-------*-------+-------+-------+-------+
      *       *       |       | \     |       |
  D   *  D>A  *  D>C  |  D>E  |   \   |  D>B  |
      *       *       |       |     \ |       |
 -----*-------*-------+-------+-------+-------+
      *       *       |       |       | \     |
  B   *  B>A  *  B>C  |  B>E  |  B>D  |   \   |
      *       *       |       |       |     \ |
 -----*********-------+-------+-------+-------+

The row scores and column scores for the remaining (unranked) choices are adjusted to remove the pairwise counts that involve the just-ranked choice (A).  The removed pairwise counts are the ones surrounded by asterisks.  Specifically, after subtracting B>A, the row score for choice B becomes B>C + B>D + B>E, and after subtracting A>B, the column score for choice B becomes C>B + D>B + E>B.

From among the remaining row scores the highest score is found.  At this point let's assume that both choice B and choice C have the same highest row score.

In the case of a row-score tie, the choice with the smallest column score — from among the choices that have the same largest row score — is ranked next.  This would be choice B.  Therefore, choice B is moved to the sequence position just after choice A.  The resulting sequence is A, B, C, E, D.

Below is the pairwise matrix for the new sequence.  The pairwise counts for the ranked choices are surrounded by asterisks.

      |       |       |       |       |       |
      |   A   |   B   |   C   |   E   |   D   |
      |       |       |       |       |       |
 -----*****************************************
      * \     |       |       |       |       *
  A   *   \   |  A>B  |  A>C  |  A>E  |  A>D  *
      *     \ |       |       |       |       *
 -----*-------+-------+-------+-------+-------*
      *       | \     |       |       |       *
  B   *  B>A  |   \   |  B>C  |  B>E  |  B>D  *
      *       |     \ |       |       |       *
 -----*-------+--------************************
      *       |       * \     |       |       |
  C   *  C>A  |  C>B  *   \   |  C>E  |  C>D  |
      *       |       *     \ |       |       |
 -----*-------+-------*-------+-------+-------+
      *       |       *       | \     |       |
  E   *  E>A  |  E>B  *  E>C  |   \   |  E>D  |
      *       |       *       |     \ |       |
 -----*-------+-------*-------+-------+-------+
      *       |       *       |       | \     |
  D   *  D>A  |  D>B  *  D>C  |  D>E  |   \   |
      *       |       *       |       |     \ |
 -----*****************-------+-------+-------+

The same ranking process is repeated.  The next choice to be ranked would be choice C.  It would have the highest row score -- and the smallest column score if there is a row-score tie.  So choice C would be identified as the next choice in the ranked sequence.  After that, choice D would have the highest row score, and would be ranked next.  Finally the only remaining choice, choice E, would be ranked at the last (lowest) position.

Here is the final pairwise matrix.

      |       |       |       |       |       |
      |   A   |   B   |   C   |   D   |   E   |
      |       |       |       |       |       |
 -----*****************************************
      * \     |       |       |       |       *
  A   *   \   |  A>B  |  A>C  |  A>D  |  A>E  *
      *     \ |       |       |       |       *
 -----*-------+-------+-------+-------+-------*
      *       | \     |       |       |       *
  B   *  B>A  |   \   |  B>C  |  B>D  |  B>E  *
      *       |     \ |       |       |       *
 -----*-------+-------+-------+-------+-------*
      *       |       | \     |       |       *
  C   *  C>A  |  C>B  |   \   |  C>D  |  C>E  *
      *       |       |     \ |       |       *
 -----*-------+-------+-------+-------+-------*
      *       |       |       | \     |       *
  D   *  D>A  |  D>B  |  D>C  |   \   |  D>E  *
      *       |       |       |     \ |       *
 -----*-------+-------+-------+-------+-------*
      *       |       |       |       | \     *
  E   *  E>A  |  E>B  |  E>C  |  E>D  |   \   *
      *       |       |       |       |     \ *
 -----*****************************************

The choices are now fully ranked according to the Choice-Specific Pairwise-Count method.

If only a single winner is needed, the first-ranked choice should not necessarily be selected as the winner.  Instead, the pairwise counts should be checked for a possible Condorcet winner, which may be second or third in the CSPS ranking result.

Concise description of the calculation method

A row score and a column score is calculated for each choice.  The row score is the sum of the pairwise counts in which the specified choice is preferred over each of the other choices.  The column score is the sum of the pairwise counts in which each other choice is preferred over the specified choice.

For the choices that have not yet been ranked, all the row scores are compared to find the highest row score.  The choice that has the highest row score is moved to the most-popular or next-most popular position in the ranking results.

If more than one choice is tied with the highest row score, the choice with the smallest column score is chosen.  If more than one choice has the same row score and the same column score, the choices are regarded as tied.

After each choice has been ranked, the scores for the remaining (unranked) choices are adjusted by subtracting from all the remaining scores the pairwise counts that involve the just-ranked choice.

The process of ranking each choice and adjusting the remaining scores is repeated until only one choice remains, and it is ranked in the bottom (least-popular) position.

 

© Copyright 2011 by Richard Fobes at VoteFair.org.  This copyright applies to this description.  The choice-specific pairwise-score (CSPS) calculation method is in the public domain.

 

 

 


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