[EM] Thoughts on De-Cloning

[Forest Simmons]

The most common way of converting a clone dependent method into a clone
free method is to replace a sum with a max or min. For example clone
dependent Borda ... which minimizes the sum of pairwise oppositions gets
converted to MinMaxPO. Similarly the version of Copeland that minimizes the
number of defeats gets converted to minimizing the maximum defeat cost.

Let's try this idea for de-cloning the Kendall-tau metric:

The Kendall-tau cost of converting one ranking of the alternatives into
another is simply the number of swaps or transpositions of adjacent pairs
required. To convert this naive metric into a clone free metric, focus on
the cost of the most expensive swap in the conversion.

In this context the cost assessment of a single swap XY to YX should be
jointly proportional to the approval of X and the disapproval of Y, since X
gets lowered and Y gets raised.

In the Universal Domain context, in place of approval and disapproval,
respectively, the most natural cost factors would be the min pairwise
support and the max pairwise opposition, respectively.

This version of Swap Cost is probably the easiest version to sell because
of its close relationship to the familiar MinMax idea.

My other (original) Swap Cost measure is based on the favorite and
anti-favorite lottery probabilities as outlined below; using lottery
probabilities where the sum of the clone probabilities equals the
probability of the cloned alternative ... is the only other fruitful
de-cloning general strategy that I know of.

Here's the example of that strategy for my original version of "Swap Cost",
for the purpose of de-cloning Kendall-tau:

To declone KY, just replace the Kendall-tau metric with a clone independent
metric. My proposal for such a metric is the Swap Cost metric:

The cost of an elementary order reversal (transposition) XY to YX in a
permutation(i.e. a complete ranking) of the alternatives is the product
fX*f'Y, where the respective factors are the favorite and anti-favorite
lottery probabilities for X and Y, respectively.

The Swap Cost of changing one permutation (complete ranking) into another
is the sum of the costs of the elementary swaps required.

To see the clone independence, let X and Y be replaced with clone sets
{X_k} and {Y_k}, respectively.  Then the product fX*f'Y becomes
(Sum fX_j)*(Sum fY_k), which expands to
Sum (over j and k) of fX_j*fY_k,
which is precisely the sum of the elementary swap costs involved.

That's it. So from now on let's make a point of using the MinMax measure of
Swap Cost in place of the
clone dependent Kendall-tau metric.

Again, note that this new MinMax Swap Cost metric is related to Kendall-tau
in the same basic way that MMPO is related to Borda and Copeland.

-Forest
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