[EM] CR pairwise

Forest Simmons fsimmons at pcc.edu
Tue Aug 28 13:29:26 PDT 2001

Suppose that the voters are willing and able to fill out Cardinal Ratings
ballots in which each each candidate is rated on a scale of zero to 100.

A pairwise matrix M can be filled out for each completed ballot as

The entry in the i_th row and j_th column M(i,j) is a one or a zero
depending on whether or not this ballot has a higher rating for candidate
i than for candidate j .

The sum of all of these pairwise matrices can be used to find out if there
is a "beats all winner" similar to the Condorcet Winner based on ranked

Suppose there is no beats all winner (BAW).

Then on each ballot divide all of the ratings by three and round to the
nearest whole number to get a set of cruder ratings.

Calculate the pairwise matrices based on these cruder ratings.

If there is no BAW on the basis of these cruder ratings divide them by
three and round again, etc.

A BAW will be found before this process is repeated five times, because
division by three and rounding four times will convert any number between
zero and 100 into either a zero or a one.  At that stage there will be a
BAW, the candidate with the most ones.

If it takes all four steps to reach a BAW, then the BAW winner is the same
as the Approval winner where the approved candidates are the ones whose
original scores were above 40 :

100 -> 33 -> 11 -> 4 -> 1
 41 -> 14 ->  5 -> 2 -> 1
 40 -> 13 ->  4 -> 1 -> 0

You can safely rate your lesser evil at 41 under this method.

Note that this method is summable if you fill out four pairwise matrices
for each ballot along with an approval count.

The method comes very close to satisfying the Condorcet Criterion since
any CW would almost surely turn out to be a BAW in the first step.

It is also pretty obvious that the method satisfies the Favorite Betrayal

The method can be adapted to the zero to fifteen range ballot below by
dividing the ratings by two and dropping the remainder at each stage.

Equivalently, successively strike out the ratings columns from right to

No more than three steps are necessary since the method reduces to
Approval when only the first column remains.


Range Ballot for a Scale of Zero to Fifteen:

Jose Blaze    (8)  (4)  (2)  (1)
Sheila M.     (8)  (4)  (2)  (1)
Jana P. Q.    (8)  (4)  (2)  (1)
A. Ron Bla    (8)  (4)  (2)  (1) 
J. Q. Sten    (8)  (4)  (2)  (1)
Sajh Amlkj    (8)  (4)  (2)  (1) 

Each candidate's score on this ballot is the sum of the digits shaded
(with a number two pencil) to the right of the candidate's name (or

Any combination of shaded and unshaded digits produces a validly marked

If none of the digits are shaded, then the score is zero for that

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