Third-order correlation
From Electowiki
Third-order correlation is a measure of Candidate correlation proposed by Dan Bishop. The name comes from the fact that the correlations can be computed with a third-order summation array.
[edit] Definitions
On a ballot, a candidate C is voted between A and B if either C is voted both strictly lower than A and strictly higher than B, or vice-versa.
The correlation of A and B with respect to C, denoted "corr(A, B) wrt C", is the proportion of the ballots on which C is not voted between A and B.
The correlation of A and B is the minimum of corr(A, B) wrt C over all candidates C in the complement of {A, B}.
[edit] Example
Imagine an election for the capital of Tennessee, a state in the United States that is over 500 miles east-to-west, and only 110 miles north-to-south. In this vote, the candidates for the capital are Memphis, Nashville, Chattanooga, and Knoxville. The population breakdown by metro area is as follows:
- Memphis: 826,330
- Nashville: 510,784
- Chattanooga: 285,536
- Knoxville: 335,749
If the voters cast their ballot based strictly on geographic proximity, the voters' sincere preferences might be as follows:
42% of voters (close to Memphis)
|
26% of voters (close to Nashville)
|
15% of voters (close to Chattanooga)
| 17% of voters (close to Knoxville)
|
Consider, for example, the correlation between Chattanooga and Memphis with respect to Knoxville. For brevity, the cities will be denoted by their initial letters.
- On the M>N>C>K ballots, K is not voted between M and C. Therefore, the 42% of the ballots with this ranking are counted in corr(C, M) wrt K.
- However, on the N>C>K>M ballots, K is voted between C and M, so these ballots do not count towards the correlation.
- The same is true for the C>K>N>M ballots.
- But on the K>C>N>M ballots, K is not voted between C and M, so these 17% of the ballots count towards the correlation.
Therefore, corr(C, M) wrt K = 42%+17% = 59%. Similarly,
- corr(C, K) wrt M = 100%
- corr(C, K) wrt N = 100%
- corr(C, M) wrt K = 59%
- corr(C, M) wrt N = 26%
- corr(C, N) wrt K = 85%
- corr(C, N) wrt M = 100%
- corr(K, M) wrt C = 41%
- corr(K, M) wrt N = 26%
- corr(K, N) wrt C = 15%
- corr(K, N) wrt M = 100%
- corr(M, N) wrt C = 74%
- corr(M, N) wrt K = 74%
The correlations between each possible pair of candidates are:
- corr(C, K) = min(100%, 100%) = 100%
- corr(C, M) = min(59%, 26%) = 26%
- corr(C, N) = min(85%, 100%) = 85%
- corr(K, M) = min(41%, 26%) = 26%
- corr(K, N) = min(15%, 100%) = 15%
- corr(M, N) = min(74%, 74%) = 74%
The most-correlated pair is Chattanooga and Knoxville.
