## 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.