[EM] Correlated Instant Borda Runoff, without Borda
Dan Bishop
daniel-j-bishop at neo.tamu.edu
Fri Dec 23 13:15:20 PST 2005
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.
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