[EM] Correlated Instant Borda Runoff, without Borda

[Dan Bishop]

Ken Kuhlman wrote:
> PS:  I had a conjecture that the pairwise matrix & independence matrix
> combined contained enough information to re-construct the original
> ballots (assuming fully ranked ballots).    Would anyone be interested
> in evaluating it?  I could easily be embarrassingly wrong again, but
> if not, it would be pretty exciting...

It's true when there are 3 candidates.  The ballots

a: A>B>C
b: A>C>B
c: B>A>C
d: B>C>A
e: C>A>B
f: C>B>A

produce the system of equations

[ 1  1 -1 -1  1 -1]  [a]     [mAB]
[ 1  1  1 -1 -1 -1]  [b]     [mAC]
[ 1 -1  1  1 -1 -1]  [c]  =  [mBC]
[ 1  0  1  0  1  1]  [d]     [cAB]
[ 0  1  1  1  1  0]  [e]     [cAC]
[ 1  1  0  1  0  1]  [f]     [cBC]

where

mAB = pairwise margin of victory of A over B
mAC = pairwise margin of victory of A over C
mBC = pairwise margin of victory of B over C
cAB = corr(A, B) wrt C
cAC = corr(A, C) wrt B
cBC = corr(B, C) wrt A

The left-hand matrix is invertible, so therefore the original ballots 
can be reconstructed from the parwise array plus the correlation array.