[EM] A Clone Free Metric On Ballots

Forest Simmons fsimmons at pcc.edu
Thu Mar 4 11:30:34 PST 2021

It was late last night when I wrote the first post so it was short on
explanation ... so here is a brief explanation ...

Every time one ballot ranks x before y and the other one doesn't, there is
a discrepancy that separates them in ballot space. A complete reversal to y
ahead of x is twice as serious as a mere change to equality in rank.

To assure freedom from clone bias the discrepancy is weighted by the
product P(x)P(y) where P is a probability.distribution that distributes
P(x) among x1,x2,x3, etc. when x is is replaced by a set of closely
clustered clones.

The matrix manipulation keeps track of the count of weighted
discrepancies... in particular, the entry in row x, column y of M1 is
(supposed to be) a one if x is ranked ahead or equal to y, otherwise zero.

So M1(x, y) minus M1(y, x) is one, zero, or negative one, depending on
whether x>y, x=y, or x<y. So if M, defined as

(M1 - transposeM1) - (M2 - transposeM2)

is zero, the two ballot rankings are identical.

Otherwise, the absolute values of the entries in M reflect the severity of
descrepency., which brings us to the matrix A of absolute values of the
entries in M.

Finally pre and post multiplying A by b and b', the row of benchmark
probabilites (random ballot probabilities) and its transpose puts in the
weights,  and sums the weighted discrepancies. Divide by two (or four) as
desired to normalize the distance.

I hope that helps!

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