[EM] Martin Harper's "Majority Potential" Idea Applied to Social Orderings

[Forest Simmons]

The following is Martin Harper's "Majority Potential" idea adapted to
the purpose of finding a "social ordering" by applying his idea to "order
space" instead of issue space, which was the original context.

Suppose that we have a set M of marked fully ranked preference ballots of
the type contemplated for use with IRV, Borda, or Condorcet.  For now,
let's say that neither equal ranking nor truncation is allowed.

Now we want to make an head-to-head comparison between two social
orderings  Order1 and Order2, to see which of the two orders better
represents the set M of individual orderings.

There are many ways to do this comparison.  But for now suppose that we
have some way of measuring distance between permutations (i.e. orderings)
of the candidates.

There are many ways of measuring distance between orderings.  For
definiteness let us focus on the Kemeny distance, which is the minimum
number of transpositions required to convert one order (of a pair) into
the other.

There are many ways to make head-to-head comparisons using this distance.
For example, we could follow Kemeny's lead and say that Order1 beats
Order2 if and only if the sum of the distances from members of M to Order1
is smaller than the sum of distances to Order2.

But let's follow Martin Harper's majority idea instead of Kemeny's idea,
i.e. let's take the Harper fork in the road rather than the Kemeny fork:

We'll say that Order1 beats Order2 if and only if the number of members of
M that are closer to Order1 than to Order2 is greater than the number of
members of M that are closer to Order2 than to Order1.

In other words, among those members of M that have a preference (measured
by the Kemeny distance), the majority prefer the first order over the
second order.

Now that we have a way of making head-to-head comparisons between social
orderings, let us construct the pairwise margins matrix.  The entry in the
i_th row and j_th column is the number of orderings in M that prefer
Order_i over Order_j.

Note that if there are N candidates there are N! permutations of the
candidates, so this pairwise matrix has N!*N! entries.

There are many possible ways of using this pairwise matrix, but let's
follow Martin Harper's suggestion and go with the Copeland winner.

In other words, the order with the greatest difference between the number
of head-to-head wins and head-to-head losses is the winning social
ordering.

[End of description of method.]

Remarks:

(1) If we had followed the Kemeny fork in the road, this Copeland winner
would be identical to the Kemeny social ordering.

(2) As Martin Harper pointed out long ago, the usual defects of Copeland's
method are over-come because the objects being compared head-to-head are
uniformly distributed.

(3) Martin's idea can be adapted to create another Condorcet flavored
Proportional Representation method, about which I will say more later.

(4) If there are four candidates there are 23*24/2=276 head-to-head
comparisons.  In general there are (N!-1)*N!/2 comparisons, so the method
is not intended for use in public elections.


Forest