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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