Borda distance correlation

Borda distance correlation is a measure of candidate correlation proposed by Ken Kuhlman and Dan Bishop.

Definitions

 * The Borda distance between two candidates on one ballot is the absolute value of the difference of their Borda scores on that ballot.
 * The total Borda distance between two candidates is the sum of their Borda distances on each ballot.
 * The average Borda distance is the total Borda distance divided by the number of ballots.
 * The most-correlated pair of candidates is the one with the lowest average Borda distance.

Example
On the Memphis>Nashville>Chattanooga>Knoxville ballots, the Borda scores are:


 * Memphis: 3
 * Nashville: 2
 * Chattanooga: 1
 * Knoxville: 0

Thus, the Borda distances on these ballots are:


 * between Memphis and Nashville: abs(3-2) = 1
 * between Memphis and Chattanooga: abs(3-1) = 2
 * between Memphis and Knoxville: abs(3-0) = 3
 * between Nashville and Chattanooga: abs(2-1) = 1
 * between Nashville and Knoxville: abs(2-0) = 2
 * between Chattanooga and Knoxville: abs(1-0) = 1

Similarly, on the Nashville>Chattanooga>Knoxville>Memphis ballots, the Borda distances are:


 * between Nashville and Chattanooga: 1
 * between Nashville and Knoxville: 2
 * between Nashville and Memphis: 3
 * between Chattanooga and Knoxville: 1
 * between Chattanooga and Memphis: 2
 * between Knoxville and Memphis: 1

On the Chattanooga>Knoxville>Nashville>Memphis ballots:


 * between Chattanooga and Knoxville: 1
 * between Chattanooga and Nashville: 2
 * between Chattanooga and Memphis: 3
 * between Knoxville and Nashville: 1
 * between Knoxville and Memphis: 2
 * between Nashville and Memphis: 1

And on the Knoxville>Chattanooga>Nashville>Memphis ballots:


 * between Knoxville and Chattanooga: 1
 * between Knoxville and Nashville: 2
 * between Knoxville and Memphis: 3
 * between Chattanooga and Nashville: 1
 * between Chattanooga and Memphis: 2
 * between Nashville and Memphis: 1

In summary,

The clone pair of Chattanooga and Knoxville has the lowest average Borda distance (1.00), so is the most-correlated pair. Memphis and Knoxville have the highest average Borda distance (2.33) and so are the least-correlated pair.