[EM] a Borda-Condorcet relation

[Stephen Turner]

Dear EM fans!
      I was wondering if anyone can think of a source 
for the following simple observation, as it might make 
a nice little paper.  It amounts to seeing how the Borda 
count can be made Condorcet-compliant by 
replacing the mean by the median as detailed below.

All ballots are total rankings of n candidates, and 
X, Y are two distinct candidates.  For any ballot,
write s(X,Y) for (rank of Y) - (rank of X) which of 
course is the difference of their Borda scores.

The s(X,Y) are all non-zero integers, at most
n-1 in absolute value.

We will suppose that there are no pairwise (i.e.
Condorcet) ties.

Given an election profile, write b(X,Y) for the mean
of all the s(X,Y), and write c(X,Y) for the
median of the same data set.

Now b(X,Y)>0 iff X beats Y in the Borda count,
and b(X,Y)>0 for all Y iff X is the Borda winner.

Also c(X,Y)>0 iff X beats Y pairwise, and
c(X,Y)>0 for all Y iff X is the Condorcet winner.

What makes all this true is the fact that the mean of 
the differences (of the Borda scores for X and Y) equals
the difference of their means; c(X,Y) is the median of the
differences, which is not the same as the difference
of the medians.

If there is no Condorcet winner, then you
could resort to one of the established methods
to break the tie (RP, Schulze, minimax, ...)

If there can be pairwise ties then c(X,Y)>0
only implies that Y does not beat X pairwise.  There is a
range 1<=c(X,Y)<=(n/2) - 1 in which either X beats
Y or they tie - both are possible.  

As calculating the median is relatively expensive, the 
above probably is not useful as an algorithm.

Any thoughts?


      
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