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.