Volume 15, Number 2, Summer 2007
CANONICAL INVARIANTS FOR
THREECANDIDATE PREFERENCE
RANKINGS
MARLOS VIANA
Abstract. It is shown that the data space for the threecandidate Condorcet Rule can be decomposed as the sum of
two onedimensional and one twodimensional permutation invariant subspaces. The nontrivial onedimensional invariant
describes the variation in total number of votes between fully distinct preferences and preferences that agree on the ranking of
exactly one candidate. The twodimensional invariant describes the voting difference between the extreme (win vs. show) rankings for any two candidates. In contrast, the data space for the
original voting data has one additional twodimensional invariant subspace corresponding to win vs. place (or place vs. show)
data for any two candidates. Canonical bases for these subspaces are constructed, interpreted, and graphically displayed
as invariant plots. Permutation invariant distances among data points in the invariant subspaces are obtained. The presented
dataanalytic methodology equally applies to ranking data of an arbitrary number of choices. Related applications to racetrack betting, short DNA words, and linear geometric optics
are outlined.
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