Simmons, Forest wrote: >This looks promising. I like this kind of creativity. > >Three Questions: > >1. Exactly how do you define correlation?
My suggestion is this: The "absolute Borda difference" (ABD) between two candidates on one ballot is the absolute value of the difference of their Borda scores on that Ballot. The "total absolute Borda difference" (TABD) between two candidates is the sum of their ABDs on all ballots. "Correlation" is the inverse of the TABD. (I had intended to send this on Wednesday night, only to find out that neo.tamu.edu no longer lets me log in as "dbishop". In the meantime, Ken posted his suggestion. It looks like mine is identical, but simpler.) EXAMPLE Consider the Tennessee capital election from Wikipedia. (C=Chattanooga, K=Knoxville, M=Memphis, N=Nashville) 42: M>N>C>K 26: N>C>K>M 15: C>K>N>M 17: K>C>N>M Nashville has 194 points, Chattanooga has 173, Memphis 126, and Knoxville 107. On the M>N>C>K ballots: ABD(C, K) = 1 ABD(C, M) = 2 ABD(C, N) = 1 ABD(K, M) = 3 ABD(K, N) = 2 ABD(M, N) = 1 On the N>C>K>M ballots: ABD(C, K) = 1 ABD(C, M) = 2 ABD(C, N) = 1 ABD(K, M) = 1 ABD(K, N) = 2 ABD(M, N) = 3 On the C>K>N>M ballots: ABD(C, K) = 1 ABD(C, M) = 3 ABD(C, N) = 2 ABD(K, M) = 2 ABD(K, N) = 1 ABD(M, N) = 1 On the K>C>N>M ballots: ABD(C, K) = 1 ABD(C, M) = 2 ABD(C, N) = 1 ABD(K, M) = 3 ABD(K, N) = 2 ABD(M, N) = 1 Therefore, TABD(C, K) = 1×42 + 1×26 + 1×15 + 1×17 = 100 TABD(C, M) = 2×42 + 2×26 + 3×15 + 2×17 = 215 TABD(C, N) = 1×42 + 1×26 + 2×15 + 1×17 = 115 TABD(K, M) = 3×42 + 1×26 + 2×15 + 3×17 = 233 TABD(K, N) = 2×42 + 2×26 + 1×15 + 2×17 = 185 TABD(M, N) = 1×42 + 3×26 + 1×15 + 1×17 = 152 Chattanooga and Knoxville are the most correlated pair, which shouldn't be too surprising -- they're clones. Knoxville has fewer Borda points and is eliminated. Then the ballots become: 42: M>N>C 26: N>C>M 32: C>N>M Nashville has 126 Borda points, Chattanooga has 90, and Memphis has 84. If correlations are not re-calculated after eliminations (which has the advantage of making the method second-order summable), then Chattanooga and Nashville are the most correlated. If correlations are recalculated, then the new values are: TABD(C, M) = 174 TABD(C, N) = 100 TABD(M, N) = 126 Either way, Chattanooga is eliminated, and then Nashville beats Memphis, 58-42. In this particular example, the Condorcet Winner is elected. However, this is not always the case. EXAMPLE 1: A>C>B 2: B>A>C 2: B>C>A 4: C>B>A The Condorcet Winner is C, who beats A 5-4 and beats B 6-3. However, the Borda ranking is B (12) > C (11) > A (4). The most-correlated pair is B and C, so C is eliminated, and then B wins. ---- Election-methods mailing list - see http://electorama.com/em for list info
