# 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

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of Tennessee
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters
(close to Memphis)
26% of voters
(close to Nashville)
15% of voters
(close to Chattanooga)
17% of voters
(close to Knoxville)
1. Memphis
2. Nashville
3. Chattanooga
4. Knoxville
1. Nashville
2. Chattanooga
3. Knoxville
4. Memphis
1. Chattanooga
2. Knoxville
3. Nashville
4. Memphis
1. Knoxville
2. Chattanooga
3. Nashville
4. Memphis

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,

candidate pair 42% 26% 15% 17% avg.
Memphis & Nashville 1 3 1 1 1.52
Memphis & Chattanooga 2 2 3 2 2.15
Memphis & Knoxville 3 1 2 3 2.33
Nashville & Chattanooga 1 1 2 1 1.15
Nashville & Knoxville 2 2 1 2 1.85
Chattanooga & Knoxville 1 1 1 1 1.00

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.